UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and...

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COMMUN. MATH. SCI. c 2019 International Press Vol. 17, No. 6, pp. 1595–1624 UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ EQUATIONS WITHOUT THERMAL DIFFUSION * NICOLE BOARDMAN , RUIHONG JI , HUA QIU § , AND JIAHONG WU Abstract. This paper focuses on the general d-dimensional (d 2) Boussinesq equations with the fractional dissipation (-Δ) α u and without thermal diffusion. Our primary goal here is the uniqueness of weak solutions to this partially dissipated system in the weakest possible setting. The issue of the uniqueness of weak solutions is very important and can be quite difficult as in the case of the Leray- Hopf weak solutions to the 3D Navier-Stokes equations. We present two main results. The first is the global existence and uniqueness of weak solutions which assesses the global existence of L 2 -weak solutions for any α> 0 and the uniqueness of the weak solutions when α 1 2 + d 4 for d 2. Especially the 2D Boussinesq equations without thermal diffusion have unique and global L 2 weak solutions. The second result establishes the zero thermal diffusion limit with an explicit convergence rate for the aforementioned weak solutions. This convergence result appears to be the very first one on weak solutions of partially dissipated Boussinesq systems. Keywords. Besov space; Boussinesq equation; Uniqueness; Weak solution. AMS subject classifications. 35A05; 35Q35; 76D03. 1. Introduction This paper concerns itself with the d-dimensional (d-D) Boussinesq system t u + u ·∇u = -ν (-Δ) α u -∇P + θe d , x R d ,t> 0, t θ + u ·∇θ =0, x R d ,t> 0, ∇· u =0, x R d ,t> 0, (u,θ)| t=0 =(u 0 0 ), x R d , (1.1) where u, P and θ represent the velocity field, the pressure and the temperature, respec- tively, and ν> 0 denotes the kinematic viscosity and e d = (0, 0, ··· , 1) is the unit vector in the vertical direction. Here the fractional Laplacian (-Δ) α with α> 0 is defined via the Fourier transform, F ((-Δ) α f )(ξ )=(4π 2 |ξ | 2 ) α F (f )(ξ ), where F (f )(ξ ) denotes the Fourier transform of f , F (f )(ξ )= Z R d f (x) e -2πix·ξ dx. We may sometimes use Λ = (-Δ) 1 2 . When θ = 0, (1.1) reduces to the generalized Navier- Stokes equations. * Received: July 12, 2018; Accepted (in revised form): July 1, 2019. Communicated by Yaguang Wang. Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA (nicki.board [email protected]). Corresponding author. Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, P.R. China ([email protected]). § Department of Mathematics, South China Agricultural University, Guangzhou 510642, P.R. China ([email protected]). Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA (jiahong.wu @okstate.edu). 1595

Transcript of UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and...

Page 1: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

COMMUN MATH SCI ccopy 2019 International Press

Vol 17 No 6 pp 1595ndash1624

UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQEQUATIONS WITHOUT THERMAL DIFFUSIONlowast

NICOLE BOARDMANdagger RUIHONG JIDagger HUA QIUsect AND JIAHONG WUpara

Abstract This paper focuses on the general d-dimensional (dge2) Boussinesq equations with thefractional dissipation (minus∆)αu and without thermal diffusion Our primary goal here is the uniquenessof weak solutions to this partially dissipated system in the weakest possible setting The issue of theuniqueness of weak solutions is very important and can be quite difficult as in the case of the Leray-Hopf weak solutions to the 3D Navier-Stokes equations We present two main results The first isthe global existence and uniqueness of weak solutions which assesses the global existence of L2-weaksolutions for any αgt0 and the uniqueness of the weak solutions when αge 1

2+ d

4for dge2 Especially

the 2D Boussinesq equations without thermal diffusion have unique and global L2 weak solutionsThe second result establishes the zero thermal diffusion limit with an explicit convergence rate forthe aforementioned weak solutions This convergence result appears to be the very first one on weaksolutions of partially dissipated Boussinesq systems

Keywords Besov space Boussinesq equation Uniqueness Weak solution

AMS subject classifications 35A05 35Q35 76D03

1 IntroductionThis paper concerns itself with the d-dimensional (d-D) Boussinesq system

parttu+u middotnablau=minusν(minus∆)αuminusnablaP +θed xisinRd tgt0

parttθ+u middotnablaθ= 0 xisinRd tgt0

nablamiddotu= 0 xisinRd tgt0

(uθ)|t=0 = (u0θ0) xisinRd

(11)

where u P and θ represent the velocity field the pressure and the temperature respec-tively and ν gt0 denotes the kinematic viscosity and ed= (00middotmiddot middot 1) is the unit vectorin the vertical direction Here the fractional Laplacian (minus∆)α with αgt0 is defined viathe Fourier transform

F((minus∆)αf)(ξ) = (4π2|ξ|2)αF(f)(ξ)

where F(f)(ξ) denotes the Fourier transform of f

F(f)(ξ) =

intRdf(x)eminus2πixmiddotξ dx

We may sometimes use Λ = (minus∆)12 When θ= 0 (11) reduces to the generalized Navier-

Stokes equations

lowastReceived July 12 2018 Accepted (in revised form) July 1 2019 Communicated by YaguangWangdaggerDepartment of Mathematics Oklahoma State University Stillwater OK 74078 USA (nickiboard

manokstateedu)DaggerCorresponding author Geomathematics Key Laboratory of Sichuan Province Chengdu University

of Technology Chengdu 610059 PR China (jiruihong09cdutcn)sectDepartment of Mathematics South China Agricultural University Guangzhou 510642 PR China

(tsiuhuascaueducn)paraDepartment of Mathematics Oklahoma State University Stillwater OK 74078 USA (jiahongwu

okstateedu)

1595

1596 BOUSSINESQ EQUATIONS

The Boussinesq equations model large scale atmospheric and oceanic flows that areresponsible for cold fronts and the jet stream (see eg the books by Gill [21] Pedlosky[42] and Majda [39]) In addition the Boussinesq system also plays an importantrole in the study of the Rayleigh-Benard convection one of the most commonly studiedconvection phenomena (see eg [141520]) The first equation in (11) reflects Newtonrsquossecond law with the left-hand side being the acceleration and the right-hand side beingthe forces due to viscosity the pressure gradient and the buoyancy The term θedmodels the buoyancy in the direction of gravitational force The temperature differencegenerates density difference which in turn generates the buoyancy force The secondequation in (11) simply states that the temperature is transported by the velocity field

Although the diffusion process is normally modeled by the standard Laplacian op-erator there are geophysical circumstances in which the Boussinesq equations withfractional Laplacian arise Flows in the middle atmosphere travelling upwards undergochanges due to the changes in atmospheric properties although the incompressibilityand Boussinesq approximations are applicable The effect of kinematic and thermaldiffusion is attenuated by the thinning of atmosphere This anomalous attenuation canbe modeled using the space fractional Laplacian (see [1221])

The Boussinesq system retains some key features of the 3D Euler and Navier-Stokes equations such as the vortex stretching mechanism The inviscid 2D Boussi-nesq equations are identical to the Euler equations for the 3D axisymmetric swirlingflows [40] Equation (11) is a partially dissipated system with no thermal diffusionFundamental issues on the Boussinesq system with partial or fractional dissipationsuch as the global existence uniqueness and regularity problem have attracted enor-mous interests during the last fifteen years and significant progress has been made[1ndash58ndash111316ndash1923ndash3133ndash3846ndash495254]

Our goal here is twofold The first is to establish the global existence and uniqueness

of weak solutions of (11) with initial data u0isinL2(Rd)θ0isinL2(Rd)capL4dd+2 (Rd) Our

key point here is the uniqueness of solutions in a very weak setting for a partiallydissipated system Although the global regularity of this partially dissipated system insmoother functional settings has been extensively investigated the issue concerned hereis quite different The uniqueness in the weakest possible functional settings is whatwe care about here The issue of the uniqueness of weak solutions is very importantand can be quite difficult as in the case of the Leray-Hopf weak solutions of the 3DNavier-Stokes equations The Boussinesq system concerned here involves only partialdissipation and the solution space appears to be the weakest setting in which one canprove the uniqueness

Our second goal is to understand the zero thermal diffusion limit of the fully dissi-pative Boussinesq equations

parttu(η) +u(η) middotnablau(η) =minusν(minus∆)αu(η)minusnablaP (η) +θ(η)ed xisinRd tgt0

parttθ(η) +u(η) middotnablaθ(η) =η∆θ(η) xisinRd tgt0

nablamiddotu(η) = 0 xisinRd tgt0

(u(η)θ(η))|t=0 = (u(η)0 θ

(η)0 ) xisinRd

(12)

and show that the solution of (12) converges strongly to the corresponding solutionof (11) with an explicit convergence rate as ηrarr0 Due to the weak initial setup

u(η)0 isinL2(Rd)θ(η)

0 isinL2(Rd)capL4dd+2 (Rd) we resort to lower regularity quantities and the

Yudovich approach Our precise results are stated in the following theorems

N BOARDMAN J JI H QIU AND J WU 1597

Theorem 11 Consider the d-D Boussinesq equations in (11)

(1) Let αgt0 and (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Let T gt0 be arbitrarily fixed Then(11) has a global weak solution (uθ) on [0T ] satisfying

uisinCw([0T ]L2)capL2(0T Hα) θisinCw([0T ]L2)capLinfin(0T L2)

(2) Let αge 12 + d

4 Assume u0isinL2(Rd) and θ0isinL2(Rd)capL4dd+2 (Rd) with nablamiddotu0 = 0

Then (11) has a unique and global weak solution (uθ) satisfying

uisinC([0T ]L2)capL2(0T Hα) uisin L1(0T B1+ d

222 )

θisinCw([0T ]L2)capLinfin(0T L2capL4dd+2 )

where the definition of L1(0T B1+ d

222 ) can be found in Section 2 Especially u

satisfies

supqge2

1radicq

int T

0

nablau(t)Lq dtltinfin

Theorem 11 assesses the global existence of weak solutions for any αgt0 and anyL2 initial data and the uniqueness when αge 1

2 + d4 A special consequence of Theorem

11 is the global existence of Leray-Hopf weak solutions to the d-D generalized Navier-Stokes equations with any αgt0 and u0isinL2(Rd) and the uniqueness of weak solutionsof the d-D Navier-Stokes equations with αge 1

2 + d4 Even though there is no thermal

diffusion the crucial step of passing to the limit in the thermal convection term stillgoes through

Theorem 12 Let αge 12 + d

4 Assume u0 θ0 u(η)0 θ

(η)0 satisfy

u0 u(η)0 isinL2(Rd) nablamiddotu0 = 0 nablamiddotu(η)

0 = 0 θ0 θ(η)0 isinL2(Rd)capL

4dd+2 (Rd)

Let (uθ) and (u(η)θ(η)) be the corresponding weak solutions of (11) and (12) respec-

tively Then the difference (uh) with

u=u(η)minusu h=h(η)minush minus∆h(η) =θ(η) minus∆h=θ

satisfies for any tgt0

(unablah)(t)2L2 leCM (1minuseminusC0t)(u0nablah0)2L2 +ηt

)eminusC0t

(13)

where C is a pure constant M =θ02L2 +θ(η)0 2L2 and

C0 =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

We summarize closely related previous results on the Boussinesq equations withoutthermal diffusion to clarify how our theorems are different The study of the globalwell-posedness of the 2D Boussinesq equations namely (11) with d= 2 and α= 1 inthe whole space was initiated in the papers of Hou-Li [25] and of Chae [13] in whichthe global and unique solutions were obtained for the initial data (u0θ0)isinHs(R2) with

1598 BOUSSINESQ EQUATIONS

sgt2 The global existence and uniqueness of solutions to (11) with d= 3 and αge 54

was investigated by several researchers (see eg [32 43 50 51 53]) The regularityassumptions on the initial data in these papers are (u0θ0)isinHs(R3) with sgt 5

2 (sgt 54

in [32]) The 2D Boussinesq system in a bounded domain with the Dirichlet boundarycondition was first studied by Lai Pan and Zhao [33] and the global existence anduniqueness was obtained in the functional setting (u0θ0)isinH3(R2) Danchin and Paicuextended the Fujita-Kato result for the Naviver-Stokes equations to the Boussinesqsystem and as a special consequence obtained the well-posedness of the finite energysolutions for the 2D Boussinesq equations [18] The paper of Larios Lunasin andTiti [34] seriously sought the uniqueness of solutions of (11) in a weak setup Theywere able to show among many other results that u0isinH1(T2) and θ0isinL2(T2) lead toa unique and global strong solution of (11) Here T2 denotes the 2D periodic box Forthe bounded domain Ω with Dirichlet boundary conditions the work of He [22] furtherreduced the regularity assumption to (u0θ0)isinL2(Ω) and still managed to show theuniqueness There are many more interesting results on the existence and uniquenessof the solutions to (11) with intermediate regularity settings (see eg [27ndash29]) Thezero thermal diffusion limit does not appear to have been much studied especially inthe circumstance when the functional setting is weak

The proof of Theorem 11 starts with the global existence of weak solutions for anyαgt0 and (u0θ0)isinL2(Rd) This process starts with showing the global existence ofsmooth solutions (u(n)θ(n)) to a sequence of approximate systems It is then followedby establishing global uniform bounds on this sequence and obtaining the strong L2

convergence of u(n) It finishes with passing to the limit Due to the lack of thermaldiffusion there is no strong convergence in θ(n) However we can still pass to the limit inthe thermal convection term due to the strong L2 convergence of u(n) When αge 1

2 + d4

the weak solution is unique Due to the weak regularity setting of the solutions u is notLipschitz and the corresponding vorticity is not necessarily bounded The proof makesuse of the following smoothing property of the velocity

uL1

(0T B

1+ d2

22

)leC(Tu0L2 θ0L2capL

4dd+2

) (14)

and a special consequence of (14) The definition of the Besov related space

L1(0T B1+ d

222 ) is provided in Section 2 Equation (14) is proven via the Littlewood-

Paley decomposition and Besov spaces techniques The proof for the coincidence of twoweak solutions (u(1)θ(1)) and (u(2)θ(2)) is based on the bounds for the L2-norms of thedifferences

u(1)minusu(2)L2 +nablah(1)minusnablah(2)L2

where h(1) and h(2) satisfy

minus∆h(i) =θ(i) i= 12

Due to the lack of thermal diffusion and the weak regularity of θ it is not possibleto bound the difference θ(1)minusθ(2)L2 The introduction of h(1) and h(2) reduce theregularity requirements and helps facilitate the proof

To prove Theorem 12 and compare the solutions (u(η)θ(η)) of (12) and (uθ) of(11) we make use of the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

N BOARDMAN J JI H QIU AND J WU 1599

and estimate the difference

(u(η)minusu)(t)2L2 +(nablah(η)minusnablah)(t)2L2

via Yudovich techniques

The rest of this paper is divided into three sections and an appendix Section 2provides the definitions of the Littlewood-Paley decomposition as well as the functionalsettings associated with the Besov spaces and related facts Section 3 proves Theorem11 Due to the length of the proof for the global existence of weak solutions the proofof this part is given in the Appendix Section 4 proves Theorem 12

2 PreliminariesThis section provides the definitions of the Littlewood-Paley decomposition func-

tional settings associated with the Besov spaces and related facts In addition anOsgood-type inequality is also stated here for the convenience of readers More detailscan be found in several books and many papers (see eg [6 7 414445])

To introduce the Besov spaces we start with a few notations S denotes the usualSchwarz class and S prime its dual the space of tempered distributions S0 denotes a subspaceof S defined by

S0 =

φisinS

intRdφ(x)xγ dx= 0 |γ|= 012 middotmiddot middot

and S prime0 denotes its dual S prime0 can be identified as

S prime0 =S primeSperp0 =S primeP

where P denotes the space of multinomials For each jisinZ we write

Aj =ξisinRd 2jminus1le|ξ|lt2j+1

(21)

The Littlewood-Paley decomposition asserts the existence of a sequence of functionsΦjjisinZisinS such that

suppΦjsubAj Φj(ξ) =Φ0(2minusjξ) or Φj(x) = 2jdΦ0(2jx)

and

infinsumj=minusinfin

Φj(ξ) =

1 if ξisinRd 00 if ξ= 0

Therefore for a general function ψisinS we have

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for ξisinRd 0

In addition if ψisinS0 then

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for any ξisinRd

1600 BOUSSINESQ EQUATIONS

That is for ψisinS0

infinsumj=minusinfin

Φj lowastψ=ψ

and hence

infinsumj=minusinfin

Φj lowastf =f f isinS prime0 (22)

in the sense of weak-lowast topology of S prime0 For notational convenience we define

∆jf = Φj lowastf = 2jdint

Φ0(2j(xminusy))f(y)dy jisinZ (23)

The homogeneous Littlewood-Paley decomposition (22) can then be written as

f =

infinsumj=minusinfin

∆jf f isinS prime0

Definition 21 For sisinR and 1lepqleinfin the homogeneous Besov space Bspq con-sists of f isinS prime0 satisfying

fBspq equiv2js∆jfLplq ltinfin

We now choose ΨisinS such that

Ψ(ξ) = 1minusinfinsumj=0

Φj(ξ) ξisinRd

Then for any ψisinS

Ψlowastψ+

infinsumj=0

Φj lowastψ=ψ

and hence

Ψlowastf+infinsumj=0

Φj lowastf =f (24)

in S prime for any f isinS prime To define the inhomogeneous Besov space we set

∆jf =

0 if jleminus2Ψlowastf if j=minus1Φj lowastf if j= 012 middotmiddot middot

(25)

The inhomogeneous Littlewood-Paley decomposition (24) can then be written as

f =

infinsumj=minus1

∆jf f isinS prime

N BOARDMAN J JI H QIU AND J WU 1601

Definition 22 The inhomogeneous Besov space Bspq with 1lepqleinfin and sisinRconsists of functions f isinS prime satisfying

fBspq equiv2js∆jfLplq ltinfin

The Besov spaces Bspq and Bspq with sisin (01) and 1lepqleinfin can be equivalentlydefined by the norms

fBspq =

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

fBspq =fLp +

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

When q=infin the expressions are interpreted in the normal way We will also use thespace-time spaces introduced by Chemin-Lerner (see eg [6])

Definition 23 For tgt0 sisinR and 1lepqrleinfin the space-time spaces Lrt Bspq and

LrtBspq are defined through the norms

fLrt Bspq equiv2js∆jfLrtLplq

fLrtBspq equiv2js∆jfLrtLplq

Here Lrt is the abbreviation for Lr(0t) These spaces are related to the classical space-time spaces Lrt B

spq and LrtB

spq via the Minkowski inequality if rge q

Lrt BspqsubeLrt Bspq LrtB

spqsubeLrtBspq

and if rltq

Lrt BspqsupLrt Bspq LrtB

spqsupLrtBspq

Many frequently used function spaces are special cases of Besov spaces The follow-ing proposition lists some useful equivalence and embedding relations

Proposition 21 For any sisinR

Hssim Bs22 HssimBs22

For any sisinR and 1ltqltinfin

Bsqminq2 rarrW sq rarr Bsqmaxq2

In particular B0qminq2 rarrLq rarr B0

qmaxq2

For notational convenience we write ∆j for ∆j There will be no confusion if we

keep in mind that ∆j associated with the homogeneous Besov spaces is defined in (23)while those associated with the inhomogeneous Besov spaces are defined in (25) Besides

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 2: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1596 BOUSSINESQ EQUATIONS

The Boussinesq equations model large scale atmospheric and oceanic flows that areresponsible for cold fronts and the jet stream (see eg the books by Gill [21] Pedlosky[42] and Majda [39]) In addition the Boussinesq system also plays an importantrole in the study of the Rayleigh-Benard convection one of the most commonly studiedconvection phenomena (see eg [141520]) The first equation in (11) reflects Newtonrsquossecond law with the left-hand side being the acceleration and the right-hand side beingthe forces due to viscosity the pressure gradient and the buoyancy The term θedmodels the buoyancy in the direction of gravitational force The temperature differencegenerates density difference which in turn generates the buoyancy force The secondequation in (11) simply states that the temperature is transported by the velocity field

Although the diffusion process is normally modeled by the standard Laplacian op-erator there are geophysical circumstances in which the Boussinesq equations withfractional Laplacian arise Flows in the middle atmosphere travelling upwards undergochanges due to the changes in atmospheric properties although the incompressibilityand Boussinesq approximations are applicable The effect of kinematic and thermaldiffusion is attenuated by the thinning of atmosphere This anomalous attenuation canbe modeled using the space fractional Laplacian (see [1221])

The Boussinesq system retains some key features of the 3D Euler and Navier-Stokes equations such as the vortex stretching mechanism The inviscid 2D Boussi-nesq equations are identical to the Euler equations for the 3D axisymmetric swirlingflows [40] Equation (11) is a partially dissipated system with no thermal diffusionFundamental issues on the Boussinesq system with partial or fractional dissipationsuch as the global existence uniqueness and regularity problem have attracted enor-mous interests during the last fifteen years and significant progress has been made[1ndash58ndash111316ndash1923ndash3133ndash3846ndash495254]

Our goal here is twofold The first is to establish the global existence and uniqueness

of weak solutions of (11) with initial data u0isinL2(Rd)θ0isinL2(Rd)capL4dd+2 (Rd) Our

key point here is the uniqueness of solutions in a very weak setting for a partiallydissipated system Although the global regularity of this partially dissipated system insmoother functional settings has been extensively investigated the issue concerned hereis quite different The uniqueness in the weakest possible functional settings is whatwe care about here The issue of the uniqueness of weak solutions is very importantand can be quite difficult as in the case of the Leray-Hopf weak solutions of the 3DNavier-Stokes equations The Boussinesq system concerned here involves only partialdissipation and the solution space appears to be the weakest setting in which one canprove the uniqueness

Our second goal is to understand the zero thermal diffusion limit of the fully dissi-pative Boussinesq equations

parttu(η) +u(η) middotnablau(η) =minusν(minus∆)αu(η)minusnablaP (η) +θ(η)ed xisinRd tgt0

parttθ(η) +u(η) middotnablaθ(η) =η∆θ(η) xisinRd tgt0

nablamiddotu(η) = 0 xisinRd tgt0

(u(η)θ(η))|t=0 = (u(η)0 θ

(η)0 ) xisinRd

(12)

and show that the solution of (12) converges strongly to the corresponding solutionof (11) with an explicit convergence rate as ηrarr0 Due to the weak initial setup

u(η)0 isinL2(Rd)θ(η)

0 isinL2(Rd)capL4dd+2 (Rd) we resort to lower regularity quantities and the

Yudovich approach Our precise results are stated in the following theorems

N BOARDMAN J JI H QIU AND J WU 1597

Theorem 11 Consider the d-D Boussinesq equations in (11)

(1) Let αgt0 and (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Let T gt0 be arbitrarily fixed Then(11) has a global weak solution (uθ) on [0T ] satisfying

uisinCw([0T ]L2)capL2(0T Hα) θisinCw([0T ]L2)capLinfin(0T L2)

(2) Let αge 12 + d

4 Assume u0isinL2(Rd) and θ0isinL2(Rd)capL4dd+2 (Rd) with nablamiddotu0 = 0

Then (11) has a unique and global weak solution (uθ) satisfying

uisinC([0T ]L2)capL2(0T Hα) uisin L1(0T B1+ d

222 )

θisinCw([0T ]L2)capLinfin(0T L2capL4dd+2 )

where the definition of L1(0T B1+ d

222 ) can be found in Section 2 Especially u

satisfies

supqge2

1radicq

int T

0

nablau(t)Lq dtltinfin

Theorem 11 assesses the global existence of weak solutions for any αgt0 and anyL2 initial data and the uniqueness when αge 1

2 + d4 A special consequence of Theorem

11 is the global existence of Leray-Hopf weak solutions to the d-D generalized Navier-Stokes equations with any αgt0 and u0isinL2(Rd) and the uniqueness of weak solutionsof the d-D Navier-Stokes equations with αge 1

2 + d4 Even though there is no thermal

diffusion the crucial step of passing to the limit in the thermal convection term stillgoes through

Theorem 12 Let αge 12 + d

4 Assume u0 θ0 u(η)0 θ

(η)0 satisfy

u0 u(η)0 isinL2(Rd) nablamiddotu0 = 0 nablamiddotu(η)

0 = 0 θ0 θ(η)0 isinL2(Rd)capL

4dd+2 (Rd)

Let (uθ) and (u(η)θ(η)) be the corresponding weak solutions of (11) and (12) respec-

tively Then the difference (uh) with

u=u(η)minusu h=h(η)minush minus∆h(η) =θ(η) minus∆h=θ

satisfies for any tgt0

(unablah)(t)2L2 leCM (1minuseminusC0t)(u0nablah0)2L2 +ηt

)eminusC0t

(13)

where C is a pure constant M =θ02L2 +θ(η)0 2L2 and

C0 =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

We summarize closely related previous results on the Boussinesq equations withoutthermal diffusion to clarify how our theorems are different The study of the globalwell-posedness of the 2D Boussinesq equations namely (11) with d= 2 and α= 1 inthe whole space was initiated in the papers of Hou-Li [25] and of Chae [13] in whichthe global and unique solutions were obtained for the initial data (u0θ0)isinHs(R2) with

1598 BOUSSINESQ EQUATIONS

sgt2 The global existence and uniqueness of solutions to (11) with d= 3 and αge 54

was investigated by several researchers (see eg [32 43 50 51 53]) The regularityassumptions on the initial data in these papers are (u0θ0)isinHs(R3) with sgt 5

2 (sgt 54

in [32]) The 2D Boussinesq system in a bounded domain with the Dirichlet boundarycondition was first studied by Lai Pan and Zhao [33] and the global existence anduniqueness was obtained in the functional setting (u0θ0)isinH3(R2) Danchin and Paicuextended the Fujita-Kato result for the Naviver-Stokes equations to the Boussinesqsystem and as a special consequence obtained the well-posedness of the finite energysolutions for the 2D Boussinesq equations [18] The paper of Larios Lunasin andTiti [34] seriously sought the uniqueness of solutions of (11) in a weak setup Theywere able to show among many other results that u0isinH1(T2) and θ0isinL2(T2) lead toa unique and global strong solution of (11) Here T2 denotes the 2D periodic box Forthe bounded domain Ω with Dirichlet boundary conditions the work of He [22] furtherreduced the regularity assumption to (u0θ0)isinL2(Ω) and still managed to show theuniqueness There are many more interesting results on the existence and uniquenessof the solutions to (11) with intermediate regularity settings (see eg [27ndash29]) Thezero thermal diffusion limit does not appear to have been much studied especially inthe circumstance when the functional setting is weak

The proof of Theorem 11 starts with the global existence of weak solutions for anyαgt0 and (u0θ0)isinL2(Rd) This process starts with showing the global existence ofsmooth solutions (u(n)θ(n)) to a sequence of approximate systems It is then followedby establishing global uniform bounds on this sequence and obtaining the strong L2

convergence of u(n) It finishes with passing to the limit Due to the lack of thermaldiffusion there is no strong convergence in θ(n) However we can still pass to the limit inthe thermal convection term due to the strong L2 convergence of u(n) When αge 1

2 + d4

the weak solution is unique Due to the weak regularity setting of the solutions u is notLipschitz and the corresponding vorticity is not necessarily bounded The proof makesuse of the following smoothing property of the velocity

uL1

(0T B

1+ d2

22

)leC(Tu0L2 θ0L2capL

4dd+2

) (14)

and a special consequence of (14) The definition of the Besov related space

L1(0T B1+ d

222 ) is provided in Section 2 Equation (14) is proven via the Littlewood-

Paley decomposition and Besov spaces techniques The proof for the coincidence of twoweak solutions (u(1)θ(1)) and (u(2)θ(2)) is based on the bounds for the L2-norms of thedifferences

u(1)minusu(2)L2 +nablah(1)minusnablah(2)L2

where h(1) and h(2) satisfy

minus∆h(i) =θ(i) i= 12

Due to the lack of thermal diffusion and the weak regularity of θ it is not possibleto bound the difference θ(1)minusθ(2)L2 The introduction of h(1) and h(2) reduce theregularity requirements and helps facilitate the proof

To prove Theorem 12 and compare the solutions (u(η)θ(η)) of (12) and (uθ) of(11) we make use of the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

N BOARDMAN J JI H QIU AND J WU 1599

and estimate the difference

(u(η)minusu)(t)2L2 +(nablah(η)minusnablah)(t)2L2

via Yudovich techniques

The rest of this paper is divided into three sections and an appendix Section 2provides the definitions of the Littlewood-Paley decomposition as well as the functionalsettings associated with the Besov spaces and related facts Section 3 proves Theorem11 Due to the length of the proof for the global existence of weak solutions the proofof this part is given in the Appendix Section 4 proves Theorem 12

2 PreliminariesThis section provides the definitions of the Littlewood-Paley decomposition func-

tional settings associated with the Besov spaces and related facts In addition anOsgood-type inequality is also stated here for the convenience of readers More detailscan be found in several books and many papers (see eg [6 7 414445])

To introduce the Besov spaces we start with a few notations S denotes the usualSchwarz class and S prime its dual the space of tempered distributions S0 denotes a subspaceof S defined by

S0 =

φisinS

intRdφ(x)xγ dx= 0 |γ|= 012 middotmiddot middot

and S prime0 denotes its dual S prime0 can be identified as

S prime0 =S primeSperp0 =S primeP

where P denotes the space of multinomials For each jisinZ we write

Aj =ξisinRd 2jminus1le|ξ|lt2j+1

(21)

The Littlewood-Paley decomposition asserts the existence of a sequence of functionsΦjjisinZisinS such that

suppΦjsubAj Φj(ξ) =Φ0(2minusjξ) or Φj(x) = 2jdΦ0(2jx)

and

infinsumj=minusinfin

Φj(ξ) =

1 if ξisinRd 00 if ξ= 0

Therefore for a general function ψisinS we have

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for ξisinRd 0

In addition if ψisinS0 then

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for any ξisinRd

1600 BOUSSINESQ EQUATIONS

That is for ψisinS0

infinsumj=minusinfin

Φj lowastψ=ψ

and hence

infinsumj=minusinfin

Φj lowastf =f f isinS prime0 (22)

in the sense of weak-lowast topology of S prime0 For notational convenience we define

∆jf = Φj lowastf = 2jdint

Φ0(2j(xminusy))f(y)dy jisinZ (23)

The homogeneous Littlewood-Paley decomposition (22) can then be written as

f =

infinsumj=minusinfin

∆jf f isinS prime0

Definition 21 For sisinR and 1lepqleinfin the homogeneous Besov space Bspq con-sists of f isinS prime0 satisfying

fBspq equiv2js∆jfLplq ltinfin

We now choose ΨisinS such that

Ψ(ξ) = 1minusinfinsumj=0

Φj(ξ) ξisinRd

Then for any ψisinS

Ψlowastψ+

infinsumj=0

Φj lowastψ=ψ

and hence

Ψlowastf+infinsumj=0

Φj lowastf =f (24)

in S prime for any f isinS prime To define the inhomogeneous Besov space we set

∆jf =

0 if jleminus2Ψlowastf if j=minus1Φj lowastf if j= 012 middotmiddot middot

(25)

The inhomogeneous Littlewood-Paley decomposition (24) can then be written as

f =

infinsumj=minus1

∆jf f isinS prime

N BOARDMAN J JI H QIU AND J WU 1601

Definition 22 The inhomogeneous Besov space Bspq with 1lepqleinfin and sisinRconsists of functions f isinS prime satisfying

fBspq equiv2js∆jfLplq ltinfin

The Besov spaces Bspq and Bspq with sisin (01) and 1lepqleinfin can be equivalentlydefined by the norms

fBspq =

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

fBspq =fLp +

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

When q=infin the expressions are interpreted in the normal way We will also use thespace-time spaces introduced by Chemin-Lerner (see eg [6])

Definition 23 For tgt0 sisinR and 1lepqrleinfin the space-time spaces Lrt Bspq and

LrtBspq are defined through the norms

fLrt Bspq equiv2js∆jfLrtLplq

fLrtBspq equiv2js∆jfLrtLplq

Here Lrt is the abbreviation for Lr(0t) These spaces are related to the classical space-time spaces Lrt B

spq and LrtB

spq via the Minkowski inequality if rge q

Lrt BspqsubeLrt Bspq LrtB

spqsubeLrtBspq

and if rltq

Lrt BspqsupLrt Bspq LrtB

spqsupLrtBspq

Many frequently used function spaces are special cases of Besov spaces The follow-ing proposition lists some useful equivalence and embedding relations

Proposition 21 For any sisinR

Hssim Bs22 HssimBs22

For any sisinR and 1ltqltinfin

Bsqminq2 rarrW sq rarr Bsqmaxq2

In particular B0qminq2 rarrLq rarr B0

qmaxq2

For notational convenience we write ∆j for ∆j There will be no confusion if we

keep in mind that ∆j associated with the homogeneous Besov spaces is defined in (23)while those associated with the inhomogeneous Besov spaces are defined in (25) Besides

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 3: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1597

Theorem 11 Consider the d-D Boussinesq equations in (11)

(1) Let αgt0 and (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Let T gt0 be arbitrarily fixed Then(11) has a global weak solution (uθ) on [0T ] satisfying

uisinCw([0T ]L2)capL2(0T Hα) θisinCw([0T ]L2)capLinfin(0T L2)

(2) Let αge 12 + d

4 Assume u0isinL2(Rd) and θ0isinL2(Rd)capL4dd+2 (Rd) with nablamiddotu0 = 0

Then (11) has a unique and global weak solution (uθ) satisfying

uisinC([0T ]L2)capL2(0T Hα) uisin L1(0T B1+ d

222 )

θisinCw([0T ]L2)capLinfin(0T L2capL4dd+2 )

where the definition of L1(0T B1+ d

222 ) can be found in Section 2 Especially u

satisfies

supqge2

1radicq

int T

0

nablau(t)Lq dtltinfin

Theorem 11 assesses the global existence of weak solutions for any αgt0 and anyL2 initial data and the uniqueness when αge 1

2 + d4 A special consequence of Theorem

11 is the global existence of Leray-Hopf weak solutions to the d-D generalized Navier-Stokes equations with any αgt0 and u0isinL2(Rd) and the uniqueness of weak solutionsof the d-D Navier-Stokes equations with αge 1

2 + d4 Even though there is no thermal

diffusion the crucial step of passing to the limit in the thermal convection term stillgoes through

Theorem 12 Let αge 12 + d

4 Assume u0 θ0 u(η)0 θ

(η)0 satisfy

u0 u(η)0 isinL2(Rd) nablamiddotu0 = 0 nablamiddotu(η)

0 = 0 θ0 θ(η)0 isinL2(Rd)capL

4dd+2 (Rd)

Let (uθ) and (u(η)θ(η)) be the corresponding weak solutions of (11) and (12) respec-

tively Then the difference (uh) with

u=u(η)minusu h=h(η)minush minus∆h(η) =θ(η) minus∆h=θ

satisfies for any tgt0

(unablah)(t)2L2 leCM (1minuseminusC0t)(u0nablah0)2L2 +ηt

)eminusC0t

(13)

where C is a pure constant M =θ02L2 +θ(η)0 2L2 and

C0 =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

We summarize closely related previous results on the Boussinesq equations withoutthermal diffusion to clarify how our theorems are different The study of the globalwell-posedness of the 2D Boussinesq equations namely (11) with d= 2 and α= 1 inthe whole space was initiated in the papers of Hou-Li [25] and of Chae [13] in whichthe global and unique solutions were obtained for the initial data (u0θ0)isinHs(R2) with

1598 BOUSSINESQ EQUATIONS

sgt2 The global existence and uniqueness of solutions to (11) with d= 3 and αge 54

was investigated by several researchers (see eg [32 43 50 51 53]) The regularityassumptions on the initial data in these papers are (u0θ0)isinHs(R3) with sgt 5

2 (sgt 54

in [32]) The 2D Boussinesq system in a bounded domain with the Dirichlet boundarycondition was first studied by Lai Pan and Zhao [33] and the global existence anduniqueness was obtained in the functional setting (u0θ0)isinH3(R2) Danchin and Paicuextended the Fujita-Kato result for the Naviver-Stokes equations to the Boussinesqsystem and as a special consequence obtained the well-posedness of the finite energysolutions for the 2D Boussinesq equations [18] The paper of Larios Lunasin andTiti [34] seriously sought the uniqueness of solutions of (11) in a weak setup Theywere able to show among many other results that u0isinH1(T2) and θ0isinL2(T2) lead toa unique and global strong solution of (11) Here T2 denotes the 2D periodic box Forthe bounded domain Ω with Dirichlet boundary conditions the work of He [22] furtherreduced the regularity assumption to (u0θ0)isinL2(Ω) and still managed to show theuniqueness There are many more interesting results on the existence and uniquenessof the solutions to (11) with intermediate regularity settings (see eg [27ndash29]) Thezero thermal diffusion limit does not appear to have been much studied especially inthe circumstance when the functional setting is weak

The proof of Theorem 11 starts with the global existence of weak solutions for anyαgt0 and (u0θ0)isinL2(Rd) This process starts with showing the global existence ofsmooth solutions (u(n)θ(n)) to a sequence of approximate systems It is then followedby establishing global uniform bounds on this sequence and obtaining the strong L2

convergence of u(n) It finishes with passing to the limit Due to the lack of thermaldiffusion there is no strong convergence in θ(n) However we can still pass to the limit inthe thermal convection term due to the strong L2 convergence of u(n) When αge 1

2 + d4

the weak solution is unique Due to the weak regularity setting of the solutions u is notLipschitz and the corresponding vorticity is not necessarily bounded The proof makesuse of the following smoothing property of the velocity

uL1

(0T B

1+ d2

22

)leC(Tu0L2 θ0L2capL

4dd+2

) (14)

and a special consequence of (14) The definition of the Besov related space

L1(0T B1+ d

222 ) is provided in Section 2 Equation (14) is proven via the Littlewood-

Paley decomposition and Besov spaces techniques The proof for the coincidence of twoweak solutions (u(1)θ(1)) and (u(2)θ(2)) is based on the bounds for the L2-norms of thedifferences

u(1)minusu(2)L2 +nablah(1)minusnablah(2)L2

where h(1) and h(2) satisfy

minus∆h(i) =θ(i) i= 12

Due to the lack of thermal diffusion and the weak regularity of θ it is not possibleto bound the difference θ(1)minusθ(2)L2 The introduction of h(1) and h(2) reduce theregularity requirements and helps facilitate the proof

To prove Theorem 12 and compare the solutions (u(η)θ(η)) of (12) and (uθ) of(11) we make use of the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

N BOARDMAN J JI H QIU AND J WU 1599

and estimate the difference

(u(η)minusu)(t)2L2 +(nablah(η)minusnablah)(t)2L2

via Yudovich techniques

The rest of this paper is divided into three sections and an appendix Section 2provides the definitions of the Littlewood-Paley decomposition as well as the functionalsettings associated with the Besov spaces and related facts Section 3 proves Theorem11 Due to the length of the proof for the global existence of weak solutions the proofof this part is given in the Appendix Section 4 proves Theorem 12

2 PreliminariesThis section provides the definitions of the Littlewood-Paley decomposition func-

tional settings associated with the Besov spaces and related facts In addition anOsgood-type inequality is also stated here for the convenience of readers More detailscan be found in several books and many papers (see eg [6 7 414445])

To introduce the Besov spaces we start with a few notations S denotes the usualSchwarz class and S prime its dual the space of tempered distributions S0 denotes a subspaceof S defined by

S0 =

φisinS

intRdφ(x)xγ dx= 0 |γ|= 012 middotmiddot middot

and S prime0 denotes its dual S prime0 can be identified as

S prime0 =S primeSperp0 =S primeP

where P denotes the space of multinomials For each jisinZ we write

Aj =ξisinRd 2jminus1le|ξ|lt2j+1

(21)

The Littlewood-Paley decomposition asserts the existence of a sequence of functionsΦjjisinZisinS such that

suppΦjsubAj Φj(ξ) =Φ0(2minusjξ) or Φj(x) = 2jdΦ0(2jx)

and

infinsumj=minusinfin

Φj(ξ) =

1 if ξisinRd 00 if ξ= 0

Therefore for a general function ψisinS we have

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for ξisinRd 0

In addition if ψisinS0 then

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for any ξisinRd

1600 BOUSSINESQ EQUATIONS

That is for ψisinS0

infinsumj=minusinfin

Φj lowastψ=ψ

and hence

infinsumj=minusinfin

Φj lowastf =f f isinS prime0 (22)

in the sense of weak-lowast topology of S prime0 For notational convenience we define

∆jf = Φj lowastf = 2jdint

Φ0(2j(xminusy))f(y)dy jisinZ (23)

The homogeneous Littlewood-Paley decomposition (22) can then be written as

f =

infinsumj=minusinfin

∆jf f isinS prime0

Definition 21 For sisinR and 1lepqleinfin the homogeneous Besov space Bspq con-sists of f isinS prime0 satisfying

fBspq equiv2js∆jfLplq ltinfin

We now choose ΨisinS such that

Ψ(ξ) = 1minusinfinsumj=0

Φj(ξ) ξisinRd

Then for any ψisinS

Ψlowastψ+

infinsumj=0

Φj lowastψ=ψ

and hence

Ψlowastf+infinsumj=0

Φj lowastf =f (24)

in S prime for any f isinS prime To define the inhomogeneous Besov space we set

∆jf =

0 if jleminus2Ψlowastf if j=minus1Φj lowastf if j= 012 middotmiddot middot

(25)

The inhomogeneous Littlewood-Paley decomposition (24) can then be written as

f =

infinsumj=minus1

∆jf f isinS prime

N BOARDMAN J JI H QIU AND J WU 1601

Definition 22 The inhomogeneous Besov space Bspq with 1lepqleinfin and sisinRconsists of functions f isinS prime satisfying

fBspq equiv2js∆jfLplq ltinfin

The Besov spaces Bspq and Bspq with sisin (01) and 1lepqleinfin can be equivalentlydefined by the norms

fBspq =

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

fBspq =fLp +

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

When q=infin the expressions are interpreted in the normal way We will also use thespace-time spaces introduced by Chemin-Lerner (see eg [6])

Definition 23 For tgt0 sisinR and 1lepqrleinfin the space-time spaces Lrt Bspq and

LrtBspq are defined through the norms

fLrt Bspq equiv2js∆jfLrtLplq

fLrtBspq equiv2js∆jfLrtLplq

Here Lrt is the abbreviation for Lr(0t) These spaces are related to the classical space-time spaces Lrt B

spq and LrtB

spq via the Minkowski inequality if rge q

Lrt BspqsubeLrt Bspq LrtB

spqsubeLrtBspq

and if rltq

Lrt BspqsupLrt Bspq LrtB

spqsupLrtBspq

Many frequently used function spaces are special cases of Besov spaces The follow-ing proposition lists some useful equivalence and embedding relations

Proposition 21 For any sisinR

Hssim Bs22 HssimBs22

For any sisinR and 1ltqltinfin

Bsqminq2 rarrW sq rarr Bsqmaxq2

In particular B0qminq2 rarrLq rarr B0

qmaxq2

For notational convenience we write ∆j for ∆j There will be no confusion if we

keep in mind that ∆j associated with the homogeneous Besov spaces is defined in (23)while those associated with the inhomogeneous Besov spaces are defined in (25) Besides

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 4: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1598 BOUSSINESQ EQUATIONS

sgt2 The global existence and uniqueness of solutions to (11) with d= 3 and αge 54

was investigated by several researchers (see eg [32 43 50 51 53]) The regularityassumptions on the initial data in these papers are (u0θ0)isinHs(R3) with sgt 5

2 (sgt 54

in [32]) The 2D Boussinesq system in a bounded domain with the Dirichlet boundarycondition was first studied by Lai Pan and Zhao [33] and the global existence anduniqueness was obtained in the functional setting (u0θ0)isinH3(R2) Danchin and Paicuextended the Fujita-Kato result for the Naviver-Stokes equations to the Boussinesqsystem and as a special consequence obtained the well-posedness of the finite energysolutions for the 2D Boussinesq equations [18] The paper of Larios Lunasin andTiti [34] seriously sought the uniqueness of solutions of (11) in a weak setup Theywere able to show among many other results that u0isinH1(T2) and θ0isinL2(T2) lead toa unique and global strong solution of (11) Here T2 denotes the 2D periodic box Forthe bounded domain Ω with Dirichlet boundary conditions the work of He [22] furtherreduced the regularity assumption to (u0θ0)isinL2(Ω) and still managed to show theuniqueness There are many more interesting results on the existence and uniquenessof the solutions to (11) with intermediate regularity settings (see eg [27ndash29]) Thezero thermal diffusion limit does not appear to have been much studied especially inthe circumstance when the functional setting is weak

The proof of Theorem 11 starts with the global existence of weak solutions for anyαgt0 and (u0θ0)isinL2(Rd) This process starts with showing the global existence ofsmooth solutions (u(n)θ(n)) to a sequence of approximate systems It is then followedby establishing global uniform bounds on this sequence and obtaining the strong L2

convergence of u(n) It finishes with passing to the limit Due to the lack of thermaldiffusion there is no strong convergence in θ(n) However we can still pass to the limit inthe thermal convection term due to the strong L2 convergence of u(n) When αge 1

2 + d4

the weak solution is unique Due to the weak regularity setting of the solutions u is notLipschitz and the corresponding vorticity is not necessarily bounded The proof makesuse of the following smoothing property of the velocity

uL1

(0T B

1+ d2

22

)leC(Tu0L2 θ0L2capL

4dd+2

) (14)

and a special consequence of (14) The definition of the Besov related space

L1(0T B1+ d

222 ) is provided in Section 2 Equation (14) is proven via the Littlewood-

Paley decomposition and Besov spaces techniques The proof for the coincidence of twoweak solutions (u(1)θ(1)) and (u(2)θ(2)) is based on the bounds for the L2-norms of thedifferences

u(1)minusu(2)L2 +nablah(1)minusnablah(2)L2

where h(1) and h(2) satisfy

minus∆h(i) =θ(i) i= 12

Due to the lack of thermal diffusion and the weak regularity of θ it is not possibleto bound the difference θ(1)minusθ(2)L2 The introduction of h(1) and h(2) reduce theregularity requirements and helps facilitate the proof

To prove Theorem 12 and compare the solutions (u(η)θ(η)) of (12) and (uθ) of(11) we make use of the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

N BOARDMAN J JI H QIU AND J WU 1599

and estimate the difference

(u(η)minusu)(t)2L2 +(nablah(η)minusnablah)(t)2L2

via Yudovich techniques

The rest of this paper is divided into three sections and an appendix Section 2provides the definitions of the Littlewood-Paley decomposition as well as the functionalsettings associated with the Besov spaces and related facts Section 3 proves Theorem11 Due to the length of the proof for the global existence of weak solutions the proofof this part is given in the Appendix Section 4 proves Theorem 12

2 PreliminariesThis section provides the definitions of the Littlewood-Paley decomposition func-

tional settings associated with the Besov spaces and related facts In addition anOsgood-type inequality is also stated here for the convenience of readers More detailscan be found in several books and many papers (see eg [6 7 414445])

To introduce the Besov spaces we start with a few notations S denotes the usualSchwarz class and S prime its dual the space of tempered distributions S0 denotes a subspaceof S defined by

S0 =

φisinS

intRdφ(x)xγ dx= 0 |γ|= 012 middotmiddot middot

and S prime0 denotes its dual S prime0 can be identified as

S prime0 =S primeSperp0 =S primeP

where P denotes the space of multinomials For each jisinZ we write

Aj =ξisinRd 2jminus1le|ξ|lt2j+1

(21)

The Littlewood-Paley decomposition asserts the existence of a sequence of functionsΦjjisinZisinS such that

suppΦjsubAj Φj(ξ) =Φ0(2minusjξ) or Φj(x) = 2jdΦ0(2jx)

and

infinsumj=minusinfin

Φj(ξ) =

1 if ξisinRd 00 if ξ= 0

Therefore for a general function ψisinS we have

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for ξisinRd 0

In addition if ψisinS0 then

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for any ξisinRd

1600 BOUSSINESQ EQUATIONS

That is for ψisinS0

infinsumj=minusinfin

Φj lowastψ=ψ

and hence

infinsumj=minusinfin

Φj lowastf =f f isinS prime0 (22)

in the sense of weak-lowast topology of S prime0 For notational convenience we define

∆jf = Φj lowastf = 2jdint

Φ0(2j(xminusy))f(y)dy jisinZ (23)

The homogeneous Littlewood-Paley decomposition (22) can then be written as

f =

infinsumj=minusinfin

∆jf f isinS prime0

Definition 21 For sisinR and 1lepqleinfin the homogeneous Besov space Bspq con-sists of f isinS prime0 satisfying

fBspq equiv2js∆jfLplq ltinfin

We now choose ΨisinS such that

Ψ(ξ) = 1minusinfinsumj=0

Φj(ξ) ξisinRd

Then for any ψisinS

Ψlowastψ+

infinsumj=0

Φj lowastψ=ψ

and hence

Ψlowastf+infinsumj=0

Φj lowastf =f (24)

in S prime for any f isinS prime To define the inhomogeneous Besov space we set

∆jf =

0 if jleminus2Ψlowastf if j=minus1Φj lowastf if j= 012 middotmiddot middot

(25)

The inhomogeneous Littlewood-Paley decomposition (24) can then be written as

f =

infinsumj=minus1

∆jf f isinS prime

N BOARDMAN J JI H QIU AND J WU 1601

Definition 22 The inhomogeneous Besov space Bspq with 1lepqleinfin and sisinRconsists of functions f isinS prime satisfying

fBspq equiv2js∆jfLplq ltinfin

The Besov spaces Bspq and Bspq with sisin (01) and 1lepqleinfin can be equivalentlydefined by the norms

fBspq =

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

fBspq =fLp +

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

When q=infin the expressions are interpreted in the normal way We will also use thespace-time spaces introduced by Chemin-Lerner (see eg [6])

Definition 23 For tgt0 sisinR and 1lepqrleinfin the space-time spaces Lrt Bspq and

LrtBspq are defined through the norms

fLrt Bspq equiv2js∆jfLrtLplq

fLrtBspq equiv2js∆jfLrtLplq

Here Lrt is the abbreviation for Lr(0t) These spaces are related to the classical space-time spaces Lrt B

spq and LrtB

spq via the Minkowski inequality if rge q

Lrt BspqsubeLrt Bspq LrtB

spqsubeLrtBspq

and if rltq

Lrt BspqsupLrt Bspq LrtB

spqsupLrtBspq

Many frequently used function spaces are special cases of Besov spaces The follow-ing proposition lists some useful equivalence and embedding relations

Proposition 21 For any sisinR

Hssim Bs22 HssimBs22

For any sisinR and 1ltqltinfin

Bsqminq2 rarrW sq rarr Bsqmaxq2

In particular B0qminq2 rarrLq rarr B0

qmaxq2

For notational convenience we write ∆j for ∆j There will be no confusion if we

keep in mind that ∆j associated with the homogeneous Besov spaces is defined in (23)while those associated with the inhomogeneous Besov spaces are defined in (25) Besides

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 5: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1599

and estimate the difference

(u(η)minusu)(t)2L2 +(nablah(η)minusnablah)(t)2L2

via Yudovich techniques

The rest of this paper is divided into three sections and an appendix Section 2provides the definitions of the Littlewood-Paley decomposition as well as the functionalsettings associated with the Besov spaces and related facts Section 3 proves Theorem11 Due to the length of the proof for the global existence of weak solutions the proofof this part is given in the Appendix Section 4 proves Theorem 12

2 PreliminariesThis section provides the definitions of the Littlewood-Paley decomposition func-

tional settings associated with the Besov spaces and related facts In addition anOsgood-type inequality is also stated here for the convenience of readers More detailscan be found in several books and many papers (see eg [6 7 414445])

To introduce the Besov spaces we start with a few notations S denotes the usualSchwarz class and S prime its dual the space of tempered distributions S0 denotes a subspaceof S defined by

S0 =

φisinS

intRdφ(x)xγ dx= 0 |γ|= 012 middotmiddot middot

and S prime0 denotes its dual S prime0 can be identified as

S prime0 =S primeSperp0 =S primeP

where P denotes the space of multinomials For each jisinZ we write

Aj =ξisinRd 2jminus1le|ξ|lt2j+1

(21)

The Littlewood-Paley decomposition asserts the existence of a sequence of functionsΦjjisinZisinS such that

suppΦjsubAj Φj(ξ) =Φ0(2minusjξ) or Φj(x) = 2jdΦ0(2jx)

and

infinsumj=minusinfin

Φj(ξ) =

1 if ξisinRd 00 if ξ= 0

Therefore for a general function ψisinS we have

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for ξisinRd 0

In addition if ψisinS0 then

infinsumj=minusinfin

Φj(ξ)ψ(ξ) = ψ(ξ) for any ξisinRd

1600 BOUSSINESQ EQUATIONS

That is for ψisinS0

infinsumj=minusinfin

Φj lowastψ=ψ

and hence

infinsumj=minusinfin

Φj lowastf =f f isinS prime0 (22)

in the sense of weak-lowast topology of S prime0 For notational convenience we define

∆jf = Φj lowastf = 2jdint

Φ0(2j(xminusy))f(y)dy jisinZ (23)

The homogeneous Littlewood-Paley decomposition (22) can then be written as

f =

infinsumj=minusinfin

∆jf f isinS prime0

Definition 21 For sisinR and 1lepqleinfin the homogeneous Besov space Bspq con-sists of f isinS prime0 satisfying

fBspq equiv2js∆jfLplq ltinfin

We now choose ΨisinS such that

Ψ(ξ) = 1minusinfinsumj=0

Φj(ξ) ξisinRd

Then for any ψisinS

Ψlowastψ+

infinsumj=0

Φj lowastψ=ψ

and hence

Ψlowastf+infinsumj=0

Φj lowastf =f (24)

in S prime for any f isinS prime To define the inhomogeneous Besov space we set

∆jf =

0 if jleminus2Ψlowastf if j=minus1Φj lowastf if j= 012 middotmiddot middot

(25)

The inhomogeneous Littlewood-Paley decomposition (24) can then be written as

f =

infinsumj=minus1

∆jf f isinS prime

N BOARDMAN J JI H QIU AND J WU 1601

Definition 22 The inhomogeneous Besov space Bspq with 1lepqleinfin and sisinRconsists of functions f isinS prime satisfying

fBspq equiv2js∆jfLplq ltinfin

The Besov spaces Bspq and Bspq with sisin (01) and 1lepqleinfin can be equivalentlydefined by the norms

fBspq =

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

fBspq =fLp +

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

When q=infin the expressions are interpreted in the normal way We will also use thespace-time spaces introduced by Chemin-Lerner (see eg [6])

Definition 23 For tgt0 sisinR and 1lepqrleinfin the space-time spaces Lrt Bspq and

LrtBspq are defined through the norms

fLrt Bspq equiv2js∆jfLrtLplq

fLrtBspq equiv2js∆jfLrtLplq

Here Lrt is the abbreviation for Lr(0t) These spaces are related to the classical space-time spaces Lrt B

spq and LrtB

spq via the Minkowski inequality if rge q

Lrt BspqsubeLrt Bspq LrtB

spqsubeLrtBspq

and if rltq

Lrt BspqsupLrt Bspq LrtB

spqsupLrtBspq

Many frequently used function spaces are special cases of Besov spaces The follow-ing proposition lists some useful equivalence and embedding relations

Proposition 21 For any sisinR

Hssim Bs22 HssimBs22

For any sisinR and 1ltqltinfin

Bsqminq2 rarrW sq rarr Bsqmaxq2

In particular B0qminq2 rarrLq rarr B0

qmaxq2

For notational convenience we write ∆j for ∆j There will be no confusion if we

keep in mind that ∆j associated with the homogeneous Besov spaces is defined in (23)while those associated with the inhomogeneous Besov spaces are defined in (25) Besides

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 6: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1600 BOUSSINESQ EQUATIONS

That is for ψisinS0

infinsumj=minusinfin

Φj lowastψ=ψ

and hence

infinsumj=minusinfin

Φj lowastf =f f isinS prime0 (22)

in the sense of weak-lowast topology of S prime0 For notational convenience we define

∆jf = Φj lowastf = 2jdint

Φ0(2j(xminusy))f(y)dy jisinZ (23)

The homogeneous Littlewood-Paley decomposition (22) can then be written as

f =

infinsumj=minusinfin

∆jf f isinS prime0

Definition 21 For sisinR and 1lepqleinfin the homogeneous Besov space Bspq con-sists of f isinS prime0 satisfying

fBspq equiv2js∆jfLplq ltinfin

We now choose ΨisinS such that

Ψ(ξ) = 1minusinfinsumj=0

Φj(ξ) ξisinRd

Then for any ψisinS

Ψlowastψ+

infinsumj=0

Φj lowastψ=ψ

and hence

Ψlowastf+infinsumj=0

Φj lowastf =f (24)

in S prime for any f isinS prime To define the inhomogeneous Besov space we set

∆jf =

0 if jleminus2Ψlowastf if j=minus1Φj lowastf if j= 012 middotmiddot middot

(25)

The inhomogeneous Littlewood-Paley decomposition (24) can then be written as

f =

infinsumj=minus1

∆jf f isinS prime

N BOARDMAN J JI H QIU AND J WU 1601

Definition 22 The inhomogeneous Besov space Bspq with 1lepqleinfin and sisinRconsists of functions f isinS prime satisfying

fBspq equiv2js∆jfLplq ltinfin

The Besov spaces Bspq and Bspq with sisin (01) and 1lepqleinfin can be equivalentlydefined by the norms

fBspq =

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

fBspq =fLp +

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

When q=infin the expressions are interpreted in the normal way We will also use thespace-time spaces introduced by Chemin-Lerner (see eg [6])

Definition 23 For tgt0 sisinR and 1lepqrleinfin the space-time spaces Lrt Bspq and

LrtBspq are defined through the norms

fLrt Bspq equiv2js∆jfLrtLplq

fLrtBspq equiv2js∆jfLrtLplq

Here Lrt is the abbreviation for Lr(0t) These spaces are related to the classical space-time spaces Lrt B

spq and LrtB

spq via the Minkowski inequality if rge q

Lrt BspqsubeLrt Bspq LrtB

spqsubeLrtBspq

and if rltq

Lrt BspqsupLrt Bspq LrtB

spqsupLrtBspq

Many frequently used function spaces are special cases of Besov spaces The follow-ing proposition lists some useful equivalence and embedding relations

Proposition 21 For any sisinR

Hssim Bs22 HssimBs22

For any sisinR and 1ltqltinfin

Bsqminq2 rarrW sq rarr Bsqmaxq2

In particular B0qminq2 rarrLq rarr B0

qmaxq2

For notational convenience we write ∆j for ∆j There will be no confusion if we

keep in mind that ∆j associated with the homogeneous Besov spaces is defined in (23)while those associated with the inhomogeneous Besov spaces are defined in (25) Besides

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 7: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1601

Definition 22 The inhomogeneous Besov space Bspq with 1lepqleinfin and sisinRconsists of functions f isinS prime satisfying

fBspq equiv2js∆jfLplq ltinfin

The Besov spaces Bspq and Bspq with sisin (01) and 1lepqleinfin can be equivalentlydefined by the norms

fBspq =

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

fBspq =fLp +

(intRd

(f(x+ t)minusf(x)Lp)q

|t|d+sqdt

)1q

When q=infin the expressions are interpreted in the normal way We will also use thespace-time spaces introduced by Chemin-Lerner (see eg [6])

Definition 23 For tgt0 sisinR and 1lepqrleinfin the space-time spaces Lrt Bspq and

LrtBspq are defined through the norms

fLrt Bspq equiv2js∆jfLrtLplq

fLrtBspq equiv2js∆jfLrtLplq

Here Lrt is the abbreviation for Lr(0t) These spaces are related to the classical space-time spaces Lrt B

spq and LrtB

spq via the Minkowski inequality if rge q

Lrt BspqsubeLrt Bspq LrtB

spqsubeLrtBspq

and if rltq

Lrt BspqsupLrt Bspq LrtB

spqsupLrtBspq

Many frequently used function spaces are special cases of Besov spaces The follow-ing proposition lists some useful equivalence and embedding relations

Proposition 21 For any sisinR

Hssim Bs22 HssimBs22

For any sisinR and 1ltqltinfin

Bsqminq2 rarrW sq rarr Bsqmaxq2

In particular B0qminq2 rarrLq rarr B0

qmaxq2

For notational convenience we write ∆j for ∆j There will be no confusion if we

keep in mind that ∆j associated with the homogeneous Besov spaces is defined in (23)while those associated with the inhomogeneous Besov spaces are defined in (25) Besides

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

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[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 8: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1602 BOUSSINESQ EQUATIONS

the Fourier localization operators ∆j the partial sum Sj is also a useful notation Foran integer j

Sjequivjminus1sumk=minus1

∆k

where ∆k is given by (25) For any f isinS prime the Fourier transform of Sjf is supportedon the ball of radius 2j and

Sjf(x) = 2djΨ(2jx)lowastf(x) = 2djint

Ψ(2j(xminusy))f(y)dy

The operators ∆j and Sj defined above satisfy the following properties

∆j∆kf = 0 if |kminusj|ge2 and ∆j(Skminus1f∆kf) = 0 if |kminusj|ge3

Bernsteinrsquos inequalities are useful tools on Fourier localized functions and theseinequalities trade integrability for derivatives The following proposition providesBernstein-type inequalities for fractional derivatives

Proposition 22 Let αge0 Let 1leple qleinfin1) If f satisfies

supp f subξisinRd |ξ|leK2j

for some integer j and a constant Kgt0 then

(minus∆)αfLq(Rd)leC1 22αj+jd( 1pminus

1q )fLp(Rd)

2) If f satisfies

supp f subξisinRd K12jle|ξ|leK22j

for some integer j and constants 0ltK1leK2 then

C1 22αjfLq(Rd)le(minus∆)αfLq(Rd)leC2 22αj+jd( 1pminus

1q )fLp(Rd)

where C1 and C2 are constants depending on αp and q only

We shall also use Bonyrsquos notion of paraproducts to decompose a product into threeparts

f g=Tf g+Tg f+R(fg)

where

Tf g=sumj

Sjminus1f∆jg R(fg) =sumj

sumkgejminus1

∆kf∆kg

with ∆k = ∆kminus1 +∆k+∆k+1 Finally we state an Osgood-type inequality to be usedin the subsequent sections (see eg [6])

Lemma 21 Let agt0 and 0le t0ltT Let ρ be a measurable function from [t0T ] to[0a] Let γ(t)gt0 be a locally integrable function on [t0T ] Let φge0 be a continuousand non-decreasing function on [0a] Assume that ρ satisfies for some constant c

ρ(t)le c+

int t

t0

γ(s)φ(ρ(s))ds for ae tisin [t0T ]

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 9: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1603

Then if cgt0 we have for ae tisin [t0T ]

minusM(ρ(t))+M(c)leint t

t0

γ(τ)dτ

where

M(x) =

int a

x

dr

φ(r)

If c= 0 and int a

0

dr

φ(r)=infin

then ρ(t) = 0 ae tisin [t0T ]

3 Proof of Theorem 11This section proves Theorem 11 Naturally the proof is divided into two main

parts The first part is the proof of the global existence of weak solutions of (11) withany αgt0 This is accomplished in Proposition 31 which is stated in this section hereSince the proof of Proposition 31 is lengthy we leave it to the Appendix The secondpart is the proof of the uniqueness of weak solutions of (11) when αge 1

2 + d4 In order

to prove the uniqueness we first prove a major smoothing estimate for the velocity fieldin Proposition 32

We start with the definition of weak solutions of (11) with any αgt0

Definition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0 LetT gt0 be arbitrarily fixed A pair (uθ) satisfying

uisinCw([0T ]L2)capL2(0T Hα) nablamiddotu= 0

θisinCw([0T ]L2)capLinfin(0T L2)

is a weak solution of (11) on [0T ] if (a) and (b) below hold

(a) For any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0

minusint T

0

intRdu middotparttφdxdtminus

intRdu0(x) middotφ(x0)dxminus

int T

0

intRdu middotnablaφudxdt

+

int T

0

intRd

(minus∆)α2u middot(minus∆)α2φdxdt=

int T

0

intRdθed middotφdxdt (31)

(b) For any ψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdparttψθdxdtminus

intRdθ0(x)ψ(x0)dx=

int T

0

intRdu middotnablaψθdxdt (32)

For any αgt0 and (u0θ0)isinL2(Rd) (11) always has a global weak solution In thespecial case when θequiv0 this result assesses the global existence of weak solutions of thegeneralized Navier-Stokes equations with any αgt0 and u0isinL2(Rd)

Proposition 31 Consider (11) with αgt0 and (u0θ0)isinL2(Rd) and nablamiddotu0 = 0Let T gt0 be arbitrarily fixed Then (11) has a global weak solution (uθ) as given inDefinition 31 satisfying

θ(t)L2 leθ0L2

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 10: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1604 BOUSSINESQ EQUATIONS

u(t)2L2 +2ν

int t

0

Λαu(τ)2L2 dτ le (u0L2 + tθ0L2)2

The proof of Proposition 31 is long and the details will be provided in the AppendixNext we establish a smoothing estimate for the weak solution shown in Proposition 31

Proposition 32 Let dge2 Consider (11) with αge 12 + d

4 Assume (u0θ0) satisfies

u0isinL2(Rd) nablamiddotu0 = 0 θ0isinL2(Rd)capL4dd+2 (Rd)

Let (uθ) be the corresponding global weak solution of (11) Then for any 0lttleT

uL1tB

1+ d2

22

le C(tu0L2 θ0L2) (33)

As a special consequence

suppge2

int t

0

nablau(τ)Lpradicp

dτ leC(tu0L2 θ0L2) (34)

Proposition 32 is proven via the Littlewood-Paley decomposition and Besov spacetechniques The proof for the 2D case is partially different from that for the generald-D case with dge3 We need a lemma for the 2D case

Lemma 31 Assume (u0θ0)isinL2(Rd) with nablamiddotu0 = 0 Consider the 2D Boussinesqequation in (11) with α= 1 Let (uθ) be the corresponding weak solution Then usatisfies int T

0

u(t)2Linfin dtle C(Tu0L2 θ0L2) (35)

Proof Applying ∆j to the velocity equation and then dotting with ∆ju yields

1

2

d

dt∆ju2L2 +ν22j∆ju2L2 =minus

intR2

∆ju middot∆j(u middotnablau)dx+

intR2

∆ju middot∆j(θe2)dx

le∆juL2∆j(u middotnablau)L2 +∆juL2∆jθL2

Eliminating ∆juL2 from each side and integrating in time yield

∆ju(t)L2 +ν

int t

0

22j∆juL2 dτ

le∆ju0L2 +

int t

0

∆j(u middotnablau)L2 dτ+

int t

0

∆jθL2 dτ

Taking the l2-norm and identifying Hs with Bs22 for sge0 we have∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

+νuL1(0tH2)

leu0L2 +

int t

0

u middotnablauL2 dτ+

int t

0

θ(τ)L2 dτ

leu0L2 +nablauL2tL

2uL2tLinfin+θL1

tL2 (36)

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 11: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1605

According to the Littlewood-Paley decomposition and by Bernsteinrsquos inequalityint t

0

u(τ)2Linfin dτ leint t

0

infinsumj=minus1

infinsumk=minus1

2j 2k ∆juL2 ∆kuL2 dτ

=H1 +H2 (37)

where

H1 =

int t

0

sum|jminusk|leN

middotmiddot middot H2 =

int t

0

sum|jminusk|gtN

middotmiddot middot

Due to |jminusk|leN the summation in H1 includes the diagonal entries j=k and 2Nsub-diagonal entries Therefore

H1 =

int t

0

infinsumj=minus1

2j∆juL2 (2jminusN∆jminusNuL2 +2jminusN+1∆jminusN+1uL2

+middotmiddot middot+ 2j+N∆j+NuL2)dτ

le 1

2

int t

0

infinsumj=minus1

(22j∆ju2L2 +22(jminusN)∆jminusNu2L2 + middotmiddot middot

+22j∆ju2L2 +22(j+N)∆j+Nu2L2

)dτ

leCNint t

0

infinsumj=minus1

22j∆ju2L2 dτ

=CN nablau2L2tL

2 (38)

where C is a pure constant independent of N The summation in H2 contains twoidentical parts and thus

H2 = 2

int t

0

sumjminuskgtN

2j ∆juL2 2k ∆kuL2 dτ

= 2

int t

0

infinsumj=N

2j∆juL2

jminusNminus1summ=minus1

2m∆muL2 dτ

le2minusN+1infinsumj=N

int t

0

22j∆juL2 dτ

jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

le2minusN+1

∥∥∥∥int t

0

22j∆juL2 dτ

∥∥∥∥l2

∥∥∥∥∥jminusNminus1summ=minus1

2m+Nminusj sup0leτlet

∆mu(τ)L2

∥∥∥∥∥l2

le2minusN+1uL1(0tH2)

∥∥∥∥ sup0leτlet

∆ju(τ)L2

∥∥∥∥l2

(39)

where we have used Youngrsquos inequality for sequence convolution Combining (36) (37)(38) (39) and Proposition 31 yieldsint t

0

u(τ)2Linfin dτ leCN nablau2L2tL

2

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 12: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1606 BOUSSINESQ EQUATIONS

+C 2minusN+1(u0L2 +nablauL2

tL2uL2

tLinfin+θL1

tL2

)2

le 1

2u2L2

tLinfin+C(tnablau2L2

tL2 u0L2 θ0L2) (310)

where we have chosen N such that

C 2minusN+1nablauL2tL

2 le1

2

(310) then yields the desired global bound in (35) This completes the proof of Lemma31

Proof (Proof of Proposition 32) Let jisinZ and jge0 Applying ∆j to the firstequation of (11) yields

partt∆ju+ν(minus∆)α∆ju=minus∆jnablaP +∆j(θed)minus∆j(u middotnablau)

Dotting with ∆ju integrating by parts and using nablamiddotu= 0 we have

1

2

d

dt∆ju2L2 +ν22αj∆ju2L2 le∆jθL2∆juL2 +I (311)

where

I=minusintRd

∆j(u middotnablau) middot∆judx

We estimate I By the notion of paraproducts provided in Section 2

I=minussum|jminusk|le3

intRd

∆j(Skminus1u middotnabla∆ku) middot∆judx

minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

In order to shift one spatial derivative to Sku via a commutator we further write thefirst term above into three terms to obtain

I= I1 +I2 +I3 +I4 +I5 (312)

where

I1 =minussum|jminusk|le3

intRd

[∆j Skminus1u middotnabla]∆ku middot∆judx

I2 =minussum|jminusk|le3

intRd

(Skminus1uminusSjminus1u) middotnabla∆j∆ku middot∆judx

I3 =minusintRdSjminus1u middotnabla∆ju middot∆judx

I4 =minussum|jminusk|le3

intRd

∆j(∆ku middotnablaSkminus1u) middot∆judx

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 13: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1607

I5 =minussumkgejminus1

intRd

∆j(∆ku middotnabla∆ku) middot∆judx

We immediately get I3 = 0 because of nablamiddotu= 0 To bound I1 we write the commutatorinto the integral form via the kernel functions defined in Section 2

[∆j Skminus1u middotnabla]∆ku

=

intRd

Φj(xminusy)Skminus1u(y) middotnabla∆ku(y)dyminusSkminus1u(x) middotintRd

Φj(xminusy)nabla∆ku(y)dy

=

intRd

Φj(xminusy)(Skminus1u(y)minusSkminus1u(x)) middotnabla∆ku(y)dy

Taking the L2-norm of the commutator

[∆j Skminus1u middotnabla]∆kuL2

lenablaSkminus1uLinfin∥∥∥∥int

Rd|Φj(xminusy)||xminusy||nabla∆ku(y)|dy

∥∥∥∥L2

lenablaSkminus1uLinfin xΦj(x)L1nabla∆kuL2

leC 2minusj nablaSkminus1uLinfin nabla∆kuL2

where we have invoked the fact

xΦj(x)L1 =

intRd|x|2dj |Φ0(2jx)|dxleC 2minusj

Therefore by Holderrsquos inequality

|I1|lesum|jminusk|le3

[∆j Skminus1u middotnabla]∆kuL2∆juL2

leC ∆juL2

sum|jminusk|le3

2minusj nablaSkminus1uLinfinnabla∆kuL2

leC ∆juL2

sum|jminusk|le3

nablaSkminus1uLinfin∆kuL2

where we have used Bernsteinrsquos inequality

2minusjnabla4kuL2 leC2kminusj4kuL2 leC4kuL2

Since the summation above is over k for |jminusk|le3 it suffices to deal with the represen-tative term with j=k Therefore without loss of generality

|I1|leC ∆ju2L2nablaSjuLinfin (313)

The estimate for I2 is easy Since Skminus1uminusSjminus1u contains the terms of the form ∆muwith m between kminus2 and jminus2 we can use Bernsteinrsquos inequality to shift one derivativefrom nabla∆ku to Skminus1uminusSjminus1u Therefore

|I2|leC ∆juL2

sum|jminusk|le3

summinkminus2jminus2lemlemaxkminus2jminus2

∆mnablauLinfin ∆kuL2

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 14: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1608 BOUSSINESQ EQUATIONS

Since the summation above is over a finite number of terms it suffices to keep the boundfor the representative term for the sake of conciseness Therefore

|I2|leCnabla∆juLinfin∆ju2L2 (314)

The estimate of I4 is direct Again we only keep the bound for the representative term

|I4|lesum|jminusk|le3

∆juL2∆j(∆ku middotnablaSkminus1u)L2

leC∆ju2L2nablaSjuLinfin (315)

By the fact that nablamiddotu= 0

I5 =minussumkgejminus1

intRd

∆jnablamiddot(∆kuotimes∆ku) middot∆judx

By Holderrsquos inequality

|I5|le C ∆juL2 2jsumkgejminus1

∆j(∆kuotimes∆ku)L2

le C ∆juL2 2jsumkgejminus1

∆kuL2∆kuLinfin (316)

Inserting the bounds of (313) (314) (315) and (316) in (312) yields

1

2

d

dt∆ju2L2 +C0 22αj∆ju2L2

le∆jθL2∆juL2 +C∆ju2L2nablaSjuLinfin

+Cnabla∆juLinfin∆ju2L2 +C2j∆juL2

sumkgejminus1

∆kuL2∆kuLinfin (317)

We further treat the right-hand side of (317) as follows We distinguish between d= 2and dge3 In the case when d= 2

nablaSjuLinfin(R2)le2jSju(t)Linfin(R2) nabla∆juLinfin(R2)le C 2j∆ju(t)Linfin(R2)

Inserting the bounds above in (317) eliminating ∆juL2 from each side and integrat-ing in time we have

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ le∆ju0L2

+

int t

0

∆jθ(τ)L2dτ+C

int t

0

2j∆juL2 Sju(τ)Linfin dτ

+C

int t

0

2j∆ju(τ)L2 ∆ju(τ)Linfin dτ+C

int t

0

2jsumkgejminus1

∆ku(τ)L2 ∆kuLinfin dτ

Taking the l2-norm of the sequence above and identifying B022 with L2 we obtain after

recalling the bound for uL2tLinfin in Lemma 31

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 15: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1609

le2u0L2 +

int t

0

θ(τ)L2dτ+C

int t

0

u(τ)Linfinnablau(τ)L2 dτ

+C

int t

0

u(τ)Linfin

∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+C nablauL2tL

2 uL2tLinfin ltinfin

where we have used the bound by Youngrsquos inequality for sequence convolution∥∥∥∥∥∥2jsumkgejminus1

∆ku(τ)L2

∥∥∥∥∥∥l2

=

∥∥∥∥∥∥sumkgejminus1

2jminusk 2k∆ku(τ)L2

∥∥∥∥∥∥l2

leC 2k∆ku(τ)L2l2 =C nablau(τ)L2

We thus have obtained (33) for the case d= 2 When dge3 we bound some of the termson the right of (317) differently By Bernsteinrsquos inequality and Sobolevrsquos inequality

nablaSjuLinfin(Rd)leC 2( 12 + d

4 )j nablaSjuL

4dd+2 (Rd)

leC 2( 12 + d

4 )j Λ 12 + d

4 uL2

nabla∆juLinfin(Rd)le C 2( 12 + d

4 )j Λ 12 + d

4 uL2

∆kuLinfin(Rd)leC 2( d4minus12 )k ∆ku

L4ddminus2 (Rd)

leC 2( d4minus12 )k Λ 1

2 + d4 uL2

Inserting the estimates above in (317) eliminating ∆juL2 from each side and inte-grating in time lead to

∆ju(t)L2 +C0 22αj

int t

0

∆ju(τ)L2 dτ

le∆ju0L2 +

int t

0

∆jθ(τ)L2dτ+C

int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

+C

int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

Taking the l2-norm of the sequence above we have

u(t)L2 +C0

∥∥∥∥22αj

int t

0

∆ju(τ)L2dτ

∥∥∥∥l2

le2u0L2 +

int t

0

θ(τ)L2dτ+J1 +J2 (318)

where

J1 = C

∥∥∥∥int t

0

∆juL22( 12 + d

4 )j Λ 12 + d

4 uL2 dτ

∥∥∥∥l2

le Cint t

0

2( 12 + d

4 )j∆juL2l2 Λ12 + d

4 uL2 dτ

= C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 16: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1610 BOUSSINESQ EQUATIONS

and

J2 = C

∥∥∥∥∥∥int t

0

2jsumkgejminus1

∆ku(τ)L22( d4minus12 )k Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

= C

∥∥∥∥∥∥int t

0

sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2 Λ 1

2 + d4 uL2 dτ

∥∥∥∥∥∥l2

le Cint t

0

∥∥∥∥∥∥sumkgejminus1

2jminusk 2( d4 + 12 )k∆ku(τ)L2

∥∥∥∥∥∥l2

Λ 12 + d

4 uL2 dτ

le Cint t

0

Λ 12 + d

4 u(τ)2L2 dτ

Inserting the bounds for J1 and J2 in (318) leads to

u(t)L2 +C0uL1tB

1+ d2

22

leu0L2 + tθ0L2 +C

int t

0

Λ 12 + d

4 u(τ)2L2 dτ

which is the desired global bound in (33) Next we show that (33) implies (34) ByBernsteinrsquos inequality

nablauLp(Rd)leinfinsum

j=minus1

nabla∆juLp(Rd)leCinfinsum

j=minus1

2j2dj ( 12minus

1p )∆juL2(Rd)

where C is a constant independent of p By Holderrsquos inequality for sequencesint t

0

nablauLpdtleinfinsum

j=minus1

2minusdp j

int t

0

2(1+ d2 )j∆juL2dτ

le

infinsumj=minus1

2minus2dp j

12 (int t

0

2(1+ d2 )j∆juL2dτ

)l2

Since infinsumj=minus1

2minus2dp j

12

leC(int infinminus1

2minus2dp (xminus1)dx

) 12

=Cradicp

(33) then implies int t

0

nablauLpdtleCradicpu

L1tB

1+ d2

22

leCradicp

where C depends on T u0L2 and θ0L2 only We thus have shown (34) Thiscompletes the proof of Proposition 32

We now prove Theorem 11

Proof (Proof of Theorem 11) Due to Proposition 31 and Proposition 32 itsuffices to show the uniqueness of the weak solutions of (11) Suppose (11) has two

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

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[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 17: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1611

weak solutions (u(1)θ(1)) and (u(2)θ(2)) with the same initial data (u0θ0) We show

that (u(1)θ(1)) and (u(2)θ(2)) must coincide To do so we consider the difference (u θ)with

u =u(1)minusu(2) θ =θ(1)minusθ(2)

Let P (1) and P (2) be the corresponding pressure terms and P =P (1)minusP (2) In additionwe introduce the lower regularity quantities h(1) and h(2) satisfying

minus∆h(1) =θ(1) minus∆h(2) =θ(2)

and set

h=h(1)minush(2)

It follows from (11) that (u θ) satisfiesparttu+u(1) middotnablau+ u middotnablau(2) +ν(minus∆)αu+nablaP = θed

parttθ+u(1) middotnablaθ+ u middotnablaθ(2) = 0

nablamiddot u= 0

(u θ)|t=0 = 0

(319)

Dotting the first equation of (319) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau(2) middot udx+

intθ middot(ed middot u)dx

=K1 +K2 (320)

where we have invoked the fact that for αge 12 + d

4 intRdu(1) middotnablau middot udx= 0

due to nablamiddotu(1) = 0 nablamiddot u= 0 andint T

0

intRd|u(1) middotnablau middot u|dxdtle

int T

0

u(1)(t)L2Λ 12 + d

4 u(t)2L2 dtltinfin

By Holderrsquos and Sobolevrsquos inequalities for dge3

|K1|leuL2nablau(2)L

4dd+2u

L4ddminus2

leCuL2 Λ 12 + d

4 u(2)L2 Λ 12 + d

4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u(2)2L2u2L2 (321)

For d= 2

|K1|leu2L4 nablau(2)L2

leuL2 nablauL2 nablau(2)L2

le ν

16nablau2L2 +C nablau(2)2L2u2L2 (322)

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 18: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1612 BOUSSINESQ EQUATIONS

By integration by parts and an interpolation inequality

|K2|=intRd|(minus∆h)(ed middot u)|dx

lenablahL2nablauL2

leCnablahL2udminus2d+2

L2 Λ12 + d

4 u4d+2

L2

leCnablahL2(uL2 +Λ 12 + d

4 uL2)

le ν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2) (323)

Dotting the second equation in (319) with h yields

1

2

d

dtnablah2L2 =minus

intRdu(1) middotnablaθ hdxminus

intRdu middotnablaθ(2)hdx =K3 +K4 (324)

We estimate K4 first The case with d= 2 is treated differently from dge3 For dge3by integration by parts Holderrsquos inequality and Sobolevrsquos inequality

|K4|leintRd|θ(2)u middotnablah|dx

leθ(2)L

4dd+2nablahL2u

L4ddminus2

leθ0L

4dd+2nablahL2Λ 1

2 + d4 uL2

le ν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2 (325)

For d= 2 by Holderrsquos inequality and Sobolevrsquos inequality

|K4|leθ(2)L2 uL2p nablahLq

leCradicpu1pL2 nablau1minus1pL2 θ0L2 nablah1minus

1p

L2 ∆h1p

L2

leCradicp(uL2 +nablauL2)nablah1minus1p

L2

le ν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2 (326)

where 1ltp qltinfin satisfy

1

p+

2

q= 1

and we have used the fact that ∆hL2 leθ(1)L2 +θ(2)L2 le2θ0L2 Recalling θ=

minus∆h and integrating by parts we have

K3 =

intRdu(1) middotnabla∆hhdx

=minusintRdpartxku

(1)j partxjpartxk hhdxminus

intRdu

(1)j partxjpartxk hpartxk hdx

=

intRdpartxku

(1)j partxk hpartxj hdx

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 19: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1613

where the repeated indices are summed and we have used nablamiddotu(1) = 0 By Holderrsquosinequality for pgt d

2 and 1p + 2

q = 1

|K3|leCnablau(1)Lpnablah2Lq

leCnablau(1)Lpnablah2minus dpL2 θ

dp

L2

leC nablau(1)Lp θ0dp

L2 nablah2minus dpL2 (327)

Adding (320) and (324) and collecting the estimates in (321) (322) (323) (325)(326) and (327) we find that for δgt0

Gδ(t) =u(t)2L2 +nablah(t)2L2 +δ

obeys the differential inequality when d= 2

d

dtGδ(t)leC

(1+nablau(2)2L2

)Gδ(t)+C

(1+nablau(1)Lp

p

)pM

1p Gδ(t)

1minus 1p (328)

and for dge3

d

dtGδ(t)leC

(1+Λ 1

2 + d4 u(2)2L2

)Gδ(t)+C

nablau(1)Lpp

pMd2p Gδ(t)

1minus d2p (329)

where we have written M =θ02L2 Optimizing the quantities pM1p Gδ(t)

1minus 1p and

pMd2p Gδ(t)

1minus d2p with respect to p we obtain

pM1p Gδ(t)

1minus 1p geeGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= ln MGδ(t)

(in the case when ln MGδ(t)

lt2 p is taken

to be 2+ln MGδ(t)

) and

pMd2p Gδ(t)

1minus d2p ge d

2eGδ(t)(lnMminus lnGδ)

with the minimum being reached at p= d2 ln M

Gδ(t) Then both (328) and (329) are

reduced to the following form

Gδ(t)leGδ(0)+C

int t

0

γ(s)φ(Gδ(s))ds

where

γ(t) =C+CΛ 12 + d

4 u(2)2L2 +Cnablau(1)Lp

p φ(r) = r+r(lnMminus lnr)

It follows from Proposition 33 thatint T

0

γ(t)dtltinfin

Let

Ω(x) =

int 1

x

dr

φ(r)=

int 1

x

dr

r+r(lnMminus lnr)= ln(1+lnMminus lnx)minus ln(1+lnM)

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 20: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1614 BOUSSINESQ EQUATIONS

It then follows from Lemma 21 that

minusΩ(Gδ(t))+Ω(Gδ(0))leint t

0

γ(s)ds

Therefore

minusln(1+lnMminus lnGδ(t))+ln(1+lnMminus lnGδ(0))leint t

0

γ(s)ds

Therefore for C(t) =int t

0γ(s)ds

Gδ(t)le (eM)1minuseminusC(t)

Gδ(0)eminusC(t)

Letting δrarr0+ and noting that G0(0) = 0 we obtain

G0(t) =u(t)2L2 +nablah(t)2L2 = 0

This completes the proof of Theorem 11

4 Proof of Theorem 12This section provides the proof of Theorem 12

Proof Let (uθ) and (u(η)θ(η)) be the weak solutions of (11) and (12) respec-

tively Then the difference (u θ) with

u=u(η)minusu θ=θ(η)minusθ

satisfies parttu+u(η) middotnablau+ u middotnablau+ν(minus∆)αu+nablaP = θed

parttθ+u(η) middotnablaθ+ u middotnablaθ=η∆θ+η∆θ

nablamiddot u= 0

(u θ)|t=0 = (u0 θ0)

(41)

where P =P (η)minusP with P (η) and P being the corresponding pressure terms of (11)and (12) respectively We introduce the lower regularity quantities h(η) and h satisfying

minus∆h(η) =θ(η) minus∆h=θ

and set

h=h(η)minush

Dotting the first equation of (41) by u and integrating by parts we have

1

2

d

dtu2L2 +νΛαu2L2 = minus

intu middotnablau middot udx+

intθ middot(ed middot u)dx

=L1 +L2 (42)

The two terms on the right of (42) can be bounded similarly in the proof of Theorem11 and we have

|L1|leν

16Λ 1

2 + d4 u2L2 +C Λ 1

2 + d4 u2L2 u2L2

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

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[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 21: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1615

and

|L2|leν

16Λ 1

2 + d4 u2L2 +C nablahL2(uL2 +nablahL2)

le ν

16Λ 1

2 + d4 u2L2 +C (u2L2 +nablah2L2)

Dotting the second equation in (41) with h yields

1

2

d

dtnablah2L2 +η∆h2L2 =L3 +L4 +L5 (43)

where

L3 =

intRdu(η) middotnablaθ hdx

L4 =

intRdu middotnablaθhdx

L5 =minusηintRd

∆θhdx

As in the proof of Theorem 11 L3 admits the following bound

|L3|leC nablau(η)Lpθ0dp

L2nablah2minus dpL2

where pgt d2 L4 can also be similarly bounded as K4 in the proof of Theorem 11 For

d= 2

|L4|leν

16nablau2L2 +u2L2 +Cpθ0

2p

L2 nablah2(1minus 1

p )

L2

and for dge3

|L4|leν

16Λ 1

2 + d4 u2L2 +C θ02

L4dd+2nablah2L2

By integration by parts and Holderrsquos inequality

|L5|leηθL2∆hL2 le η2∆h2L2 +

η

2θ2L2

Adding (42) and (43) and incorporating the bounds for L1 through L5 we find forδgt0

Eδ(t) =u(t)2L2 +nablah(t)2L2 +δ

satisfies for d= 2

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λu2L2

)Eδ(t)+C

(1+nablau(η)Lp

p

)pM

1p Eδ(t)

1minus 1p

and for dge3

d

dtEδ(t)le

η

2θ2L2 +C

(1+Λ 1

2 + d4 u2L2

)Eδ(t)+C

nablau(η)Lpp

pMd2p Eδ(t)

1minus d2p

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 22: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1616 BOUSSINESQ EQUATIONS

By following a similar procedure as in the proof of Theorem 11 and applying Lemma21 we obtain

Eδ(t)le (eM)1minuseminusC(t) (Eδ(0)+ηtθ2L2

)eminusC(t)

(44)

where M =θ02L2 +θ(η)0 2L2 denotes the sum of the initial L2-norms squared and C(t)

is the uniform bound (independent of η)

C(t) =C

int t

0

(1+Λ 1

2 + d4 u2L2 +

nablau(η)Lpp

)dτ ltinfin

Even though u(η) is the solution of (12) the bound

supqge2

int t

0

nablau(η)Lpp

dτ ltinfin

is uniform in η since it only depends on θηL2 Letting δrarr0 in (44) yields

u(t)2L2 +nablah(t)2L2 le (eM)1minuseminusC(t)(u02L2 +nablah02L2 + ηtθ2L2

)eminusC(t)

which is the desired bound (13) This completes the proof of Theorem 12

Acknowledgements Ji is partially supported by the National Natural ScienceFoundation of China NSFC (Grant No 11571243) by the Science and TechnologyDepartment of Sichuan Province (Grant No 2017JY0206) and by China ScholarshipCouncil Qiu is partially supported by the Natural Science Foundation of Guang DongProvince (Grant No 2016A030313390) and China Scholarship Council Wu is partiallysupported by NSF grant DMS 1624146 and the ATampT Foundation at Oklahoma StateUniversity

Appendix Global existence of weak solutions This section provides theproof of Proposition 31 For readersrsquo convenience we first list several simple facts tobe used in the proof The first two lemmas are Picardrsquos existence and extension results(see eg [40])

Lemma 51 (Picard Existence and Uniqueness Theorem) Let E be a Banach spaceLet OsubeE be an open subset Let F OminusrarrE be a locally Lipschitz map More preciselyfor any yisinO there is a neighborhood of y (denoted by U(y)) and L=L(yU) such that

F (y)minusF (z)EleLyminuszE forallzisinU(y)

Then for any y0isinO the ODE dydt =F (y)

y|t=0 =y0isinO(51)

has a unique local solution namely there is T gt0 and a unique solution y=y(t) satis-fying yisinC1(0T O)

Lemma 52 (Picard Extension Theorem) Assume the conditions in Lemma 51 holdand Let y=y(t) be the local solution Then either y(t) is global in time namely T =infinor for a finite T0gt0 limtrarrT0

y(t) isinO

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 23: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1617

Lemma 53 (Hodge Decomposition in Rd) For every visinL2(Rd)capCinfin(Rd) thereexist unique w and p satisfying

v=w+nablap nablamiddotw= 0

and wisinL2(Rd)capCinfin(Rd)nablapisinL2(Rd)capCinfin(Rd) and v2L2 =w2L2 +nablap2L2

We introduce a few notations f(ξ) represents the Fourier transform of f

f(ξ) =

intRdeminus2πixmiddotξf(x)dx

For a positive integer n we denote by B(0n) the ball centered at the origin with radiusn and define

Jnf(ξ) =χB(0n)(ξ) f(ξ)

In addition we write

L2n=f isinL2(Rd) supp f subB(0n)

L2nσ =f isinL2

n(Rd) nablamiddotf = 0

A special consequence of Lemma 53 is the following fact

Corollary 51 There exists a linear bounded operator P L2nrarrL2

nσ satisfying

bull For any f isinL2n PfL2 lefL2

bull For any f isinL2nσ Pf =f

Especially for any f isinL2n P2f =Pf

In addition we will also need the following Lions-Aubin compactness lemma

Lemma 54 (Lions-Aubin Compactness Lemma) Let X1 rarrX2 rarrX3 be three Banachspaces with the first embedding being compact and the second being continuous LetT gt0 For 1lepqle+infin let

W =uisinLp(0T X1)parttuisinLq(0T X3)

Then

(i) If plt+infin then the embedding of W into Lp(0T X2) is compact

(ii) If p= +infin and qgt1 then the embedding of W into C(0T X2) is compact

Lemma 54 states that any bounded sequence in W has a convergent subsequence inLp(0T X2)

We are now ready to prove Proposition 31

Proof (Proof of Proposition 31) The proof is divided into several steps Thefirst step is to show the global existence of smooth solutions to a sequence of approximatesystems The second is to establish uniform bounds for this sequence of solutions andextract a strongly convergent subsequence The third is to verify that the limit of thesubsequence is actually the weak solution

Step I The global existence of smooth solutions to an approximate system

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 24: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1618 BOUSSINESQ EQUATIONS

Let nisinN and we seek a solution (u(n)θ(n))isinL2n satisfying

parttu(n) +PJn(u(n) middotnablau(n))+ν(minus∆)αu(n) =PJn(θ(n)ed)

parttθ(n) +Jn(u(n) middotnablaθ(n)) = 0

nablamiddotu(n) = 0

u(n)(x0) =Jnu0 θ(n)(x0) =Jnθ0

(52)

We remark that functions in L2n(Rd) are smooth

L2nsubecapinfinm=0H

m

In fact let f isinL2n

f2Hm

= Σ|β|=mDβf2L2 = Σ|β|=mDβf2L2 = Σ|β|=m(2πiξ)β f2L2 le (2πn)2βf2L2

We use the Picard theorem to show that (52) has a unique global solution in L2n To

this end we first apply Lemma 51 to show (52) has a local-in-time solution We canrewrite (52) as

dy

dt=F (y)

with

Y = (u(n)θ(n))T

F (Y ) = (F1(Y )F2(Y ))T

= (minusPJn(u(n) middotnablau(n))minusν(minus∆)αu(n) +PJn(θ(n)ed)minusJn(u(n) middotnablaθ(n)))T

We set E=L2n and O=E We verify that F EminusrarrE is locally Lipschitz Assume

Y isinL2n and show F (Y )isinL2

n Clearly F (Y )isinL2(Rd) In fact

F1(Y )L2 leu(n) middotnablau(n)L2 +ν(minus∆)αu(n)L2 +θ(n)L2

leu(n)L4nablau(n)L4 +νu(n)H2α +θ(n)L2

leu(n)Hd4u(n)

H1+ d4

+νu(n)H2α +θ(n)L2

le (2πn)2(1+ d4 )u(n)2L2 +ν(2πn)2αu(n)L2 +θ(n)L2

That is F1(Y )isinL2(Rd) Similarly F2(Y )isinL2(Rd) Obviously

supp F1(Y ) supp F2(Y )subeB(0n)

Therefore F (Y )isinL2n(Rd) Next we show F (Y ) is locally Lipschitz Let Y =

(u(n)θ(n))T isinL2n and Z= (v(n)ρ(n))T isinL2

n and consider

F2(Y )minusF2(Z)L2

=minusJn(u(n) middotnablaθ(n))+Jn(v(n) middotnablaρ(n))L2

=minusJn((u(n)minusv(n)) middotnablaθ(n))minusJn(v(n) middotnabla(θ(n)minusρ(n)))L2

le(u(n)minusv(n)) middotnablaθ(n)L2 +v(n) middotnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2nablaθ(n)Linfin+v(n)Linfinnabla(θ(n)minusρ(n))L2

leu(n)minusv(n)L2θ(n)H1+ d

2+ε +v(n)

Hd2+εθ(n)minusρ(n)H1

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 25: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1619

le (2πn)1+ d2 +εθ(n)L2u(n)minusv(n)L2 +(2πn)1+ d

2 +εv(n)L2θ(n)minusρ(n)L2

leLY minusZL2

where εgt0 is a small parameter and L= 2(2πn)1+ d2 +ε(Y L2 +r) for ZminusY L2 le r

Therefore F2(Y ) is locally Lipschitz Similarly F1(Y ) is locally Lipschitz Lemma 51implies (52) has a unique local-in-time solution in L2

n

Next we use the Picard Extension Theorem Lemma 52 to show that the solutionis global in time It suffices to show that for any tleT (u(n)θ(n))L2 lt+infin This isdone by the energy method Dotting (52) by (u(n)θ(n)) yields

1

2

d

dt(u(n)2L2 +θ(n)2L2)+νΛαu(n)2L2 =M1 +M2 +M3

where ltfggt=intRd f(x)g(x)dx and

M1 =minusintRd

PJn(u(n) middotnablau(n)) middotu(n)dx

M2 =

intRd

PJn(θ(n)ed) middotu(n)dx

M3 =minusintRdJn(u(n) middotnablaθ(n)) middotθ(n)dx

We note that

M1 =minusint

PJn(u(n) middotnablau(n)) middotu(n)dx=minusintJn(u(n) middotnablau(n)) middotPu(n)dx

=minusintJn(u(n) middotnablau(n)) middotu(n)dx=minus

int(u(n) middotnablau(n)) middotu(n)dx= 0

Similarly M3 = 0 Clearly |M2|leu(n)L2θ(n)L2 Therefore

d

dt(u(n)2L2 +θ(n)2L2)+2νΛαu(n)2L2 leu(n)L2θ(n)L2

Similarly

1

2

d

dtθ(n)L2 = 0 or θ(n)(t)L2 =Jnθ0L2

Consequently

u(n)(t)L2 leJnu0L2 + tJnθ0L2 leu0L2 + tθ0L2

and

u(n)(t)2L2 +2ν

int t

0

Λαu(n)2L2dτ le (u0L2 + tθ0L2)2

Therefore (u(n)θ(n))isinL2n for all time tleT Then Lemma 52 allows us to conclude

that (u(n)θ(n)) is global in time

Step 2 Extraction of a strongly convergent subsequence

This step extracts a subsequence of u(n) that converges strongly in L2(0T L2(Rd))using the Aubin-Lions lemma In order to use the Aubin-Lions method we show that

parttu(n)isinL2(0T Hminuss) (53)

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 26: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1620 BOUSSINESQ EQUATIONS

where s= maxα1+ d2 minusα Let φisinHs We take the L2-inner product of φ and the

velocity equation in (52) leads tointRdφ middotparttu(n) dx=Q1 +Q2 +Q3

with

Q1 =minusintφ middotPJn(u(n) middotnablau(n)) dx

Q2 =minusνintφ middot(minus∆)αu(n) dx

Q3 =

intφ middotPJn(θ(n)ed) dx

Integrating by parts and applying Holderrsquos and Sobolevrsquos inequalities yield

|Q1|leu(n)2L

2ddminusαnablaPJnφ

Ldα

leC u(n)L2Λαu(n)L2 PJnφH1+ d

2minusα

leC u(n)L2Λαu(n)L2 φH1+ d

2minusα

Using integration by parts and Holderrsquos inequality we have

|Q2|leνΛαφL2Λαu(n)L2 leνφHsΛαu(n)L2

Clearly

|Q3|leφHs θ(n)L2

Therefore∣∣∣∣int φ middotparttu(n) dx

∣∣∣∣leCφHs(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

That is

parttu(n)Hminuss leC(Λαu(n)L2(1+u(n)L2)+θ(n)L2

)

Squaring and integrating in time yieldint T

0

parttu(n)2Hminuss dt

leCint T

0

(1+u(n)L2

)2

Λαu(n)2L2 dt+C

int T

0

θ(n)2L2 dt

+C

int T

0

(1+u(n)L2

)Λαu(n)L2θ(n)2L2 dt

leC sup0letleT

(1+u(n)2L2)

int T

0

Λαu(n)2L2 dt+CT sup0letleT

θ(n)L2

+C

(T sup

0letleTθ(n)L2

)middot(

sup0letleT

(1+u(n)L2)

)int T

0

Λαu(n)2L2 dt

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 27: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1621

lt+infin

Thus we have obtained (53) Since we have

u(n)isinL2(0T Hα(Rd)) parttu(n)isinL2(0T Hminuss(Rd))

and the facts that Hα(Rd) rarrL2(Rd) is locally compact and L2(Rd) rarrHminus(1+ d2minusα) is

continuous we can apply the Aubin-Lions Lemma to conclude that u(n) has a convergentsubsequence in L2(0T L2(Rd)) Let u be the limit of u(n) and θ be the weak limit ofθ(n) Clearly

θisinLinfin(0T L2(Rd)) uisinLinfin(0T L2(Rd))capL2(0T Hα(Rd))

Step 3 Passing to the limitThis step shows that (uθ) obtained in the previous step is a weak solution of (11) Itis easy to see from (52) that for any φisinCinfin0 (Rdtimes [0T )) with nablamiddotφ= 0 and for anyψisinCinfin0 (Rdtimes [0T ))

minusint T

0

intRdu(n) middotparttφdxdtminus

intRdu

(n)0 middotφ(x0)dxminus

int T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRd

Λαu(n) middotΛαφdxdt=int T

0

intRdθ(n)ed middotJnφdxdt

minusint T

0

intRdparttψθ

(n)dxdt+

intRdθ

(n)0 ψ(x0)dx=

int T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

The task is then to verify that as nrarrinfin the terms above converge to the correspond-ing terms in the definition of the weak solution given in Definition 31 We need thestrong convergence u(n)rarru in L2(0T L2) It suffices to consider the convergence ofthe nonlinear terms Let

A =minusint T

0

intRdu middotnabla(Jnφ)udxdt

A(n) =minusint T

0

intRdu(n) middotnabla(Jnφ)u(n)dxdt

and consider the difference

A(n)minusA=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnφ)u(n)dxdt

+

int T

0

intRdu middotnabla(Jnφminusφ)u(n)dxdt

+

int T

0

intRdu middotnablaφ middot(u(n)minusu)dxdt

=R1 +R2 +R3

Using Holderrsquos inequality we have

|R1|leu(n)minusuL2(Rdtimes[0T ])nablaJnφLinfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])φH2+ d2u0L2(Rdtimes[0T ])rarr0 as nrarrinfin

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 28: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1622 BOUSSINESQ EQUATIONS

Similarly

|R2|leuL2(Rdtimes[0T ])nabla(Jnφminusφ)Linfin(Rdtimes[0T ])u(n)L2(Rdtimes[0T ])

leCu0L2JnφminusφH2+ d

2u0L2rarr0 as nrarrinfin

and as nrarrinfin

|R3|leuL2(Rdtimes[0T ])nablaφLinfin(Rdtimes[0T ])u(n)minusuL2(Rdtimes[0T ])rarr0

Therefore |A(n)minusA|rarr0 as nrarrinfin The convergence of the other nonlinear terms isslightly different We do not have strong convergence in θ(n) Define

B =minusint T

0

intRdu middotnabla(Jnψ)θdxdt

B(n) =minusint T

0

intRdu(n) middotnabla(Jnψ)θ(n)dxdt

and consider the difference

B(n)minusB=minusint T

0

intRd

(u(n)minusu) middotnabla(Jnψ)θ(n)dxdt

+

int T

0

intRdu middotnabla(Jnψminusψ)θ(n)dxdt

+

int T

0

intRdu middotnablaψ middot(θ(n)minusθ)dxdt

=W1 +W2 +W3

Using Holderrsquos inequality we have

|W1|leu(n)minusuL2(Rdtimes[0T ])nablaJnψLinfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu(n)minusuL2(Rdtimes[0T ])ψH2+ d2θ0L2(Rdtimes[0T ])rarr0 as nrarrinfin

Similarly

|W2|leuL2(Rdtimes[0T ])nabla(Jnψminusψ)Linfin(Rdtimes[0T ])θ(n)L2(Rdtimes[0T ])

leCu0L2JnψminusψH2+ d

2θ0L2rarr0 as nrarrinfin

W3 is estimated differently from R3 since we do not have strong convergence in θ(n)Since L2 functions can be approximated by smooth functions with compact supportu middotnablaψ can be treated as a test function Since θ(n) converges weakly to θ we thus have

W3rarr0 as nrarr infin

Hence |B(n)minusB|rarr0 as nrarrinfin Therefore (uθ) is indeed a weak solution This com-pletes the proof of Proposition 31

REFERENCES

[1] H Abidi and T Hmidi On the global well-posedness for Boussinesq system J Diff Eqs 233199ndash220 2007 1

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 29: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

N BOARDMAN J JI H QIU AND J WU 1623

[2] D Adhikari C Cao H Shang J Wu X Xu and Z Ye Global regularity results for the 2DBoussinesq equations with partial dissipation J Diff Eqs 2601893ndash1917 2016 1

[3] D Adhikari C Cao and J Wu The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity J Diff Eqs 2491078ndash1088 2010 1

[4] D Adhikari C Cao and J Wu Global regularity results for the 2D Boussinesq equations withvertical dissipation J Diff Eqs 2511637ndash1655 2011 1

[5] D Adhikari C Cao J Wu and X Xu Small global solutions to the damped two-dimensionalBoussinesq equations J Diff Eqs 2563594ndash3613 2014 1

[6] H Bahouri J-Y Chemin and R Danchin Fourier Analysis and Nonlinear Partial DifferentialEquations Springer 2011 2 2 2

[7] J Bergh and J Lofstrom Interpolation Spaces An Introduction Springer-Verlag Berlin-Heidelberg-New York 1976 2

[8] A Biswas C Foias and A Larios On the attractor for the semi-dissipative Boussinesq equationsAnn Inst H Poincare Anal Non Lineaire 34381ndash405 2017 1

[9] L Brandolese and M Schonbek Large time decay and growth for solutions of a viscous Boussinesqsystem Trans Amer Math Soc 3645057ndash5090 2012 1

[10] JR Cannon and E DiBenedetto The initial value problem for the Boussinesq equations withdata in Lp in R Rautmann (ed) Approximation Methods for Navier-Stokes ProblemsLecture Notes in Math Springer Berlin 771129ndash144 1980 1

[11] C Cao and J Wu Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation Arch Ration Mech Anal 208985ndash1004 2013 1

[12] M Caputo Linear models of dissipation whose Q is almost frequency independent-II Geophy JR Astr Soc 13529ndash539 1967 1

[13] D Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms AdvMath 203497ndash513 2006 1 1

[14] P Constantin and CR Doering Heat transfer in convective turbulence Nonlinearity 91049ndash1060 1996 1

[15] CR Doering and P Constantin Variational bounds on energy dissipation in incompressibleflows III Convection Phys Rev E 535957 1996 1

[16] C R Doering J Wu K Zhao and X Zheng Long time behavior of the two-dimensional Boussi-nesq equations without buoyancy diffusion Phys D 376377144ndash159 2018 1

[17] D Cordoba C Fefferman and R de la Llave On squirt singularities in hydrodynamics SIAMJ Math Anal 36204ndash213 2004 1

[18] R Danchin and M Paicu Les theremes de Leray et de Fujita-Kato pour le systeme de Boussinesqpartiellement visqueux Bull Soc Math France 136261ndash309 2008 1 1

[19] R Danchin and M Paicu Global existence results for the anisotropic Boussinesq system indimension two Math Models Meth Appl Sci 21421ndash457 2011 1

[20] AV Getling Rayleigh-Benard Convection Structures and Dynamics Advanced Series in Non-linear Dynamics World Scientific 11 1998 1

[21] AE Gill Atmosphere-Ocean Dynamics Academic Press London 1982 1[22] L He Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains

with applicantions J Funct Anal 2623430ndash3464 2012 1[23] T Hmidi S Keraani and F Rousset Global well-posedness for a Boussinesq-Navier-Stokes

system with critical dissipation J Diff Eqs 2492147ndash2174 2010 1[24] T Hmidi S Keraani and F Rousset Global well-posedness for Euler-Boussinesq system with

critical dissipation Comm Part Diff Eqs 36420ndash445 2011 1[25] T Hou and C Li Global well-posedness of the viscous Boussinesq equations Discrete Contin

Dyn Syst 121ndash12 2005 1 1[26] L Hu and H Jian Blow-up criterion for 2-D Boussinesq equations in bounded domain Front

Math China 2559ndash581 2007 1[27] W Hu I Kukavica and M Ziane Persistence of regularity for the viscous Boussinesq equations

with zero diffusivity Asymptot Anal 91111ndash124 2015 1 1[28] W Hu I Kukavica and M Ziane On the regularity for the Boussinesq equations in a bounded

domain J Math Phys 54081507 2013 1 1[29] W Hu Y Wang J Wu B Xiao and J Yuan Partially dissipated 2D Boussinesq equations with

Navier type boundary conditions Phys D 37637739ndash48 2018 1 1[30] Q Jiu C Miao J Wu and Z Zhang The 2D incompressible Boussinesq equations with general

critical dissipation SIAM J Math Anal 463426ndash3454 2014 1[31] Q Jiu J Wu and W Yang Eventual regularity of the two-dimensional Boussinesq equations

with supercritical dissipation J Nonlinear Sci 2537ndash58 2015 1[32] Q Jiu and H Yu Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Acta Math Appl Sin 321ndash16 2016 1

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1

Page 30: UNIQUENESS OF WEAK SOLUTIONS TO THE BOUSSINESQ …uniqueness of weak solutions is very important and can be quite di cult as in the case of the Leray-Hopf weak solutions to the 3D

1624 BOUSSINESQ EQUATIONS

[33] M Lai R Pan and K Zhao Initial boundary value problem for 2D viscous Boussinesq equationsArch Ration Mech Anal 199739ndash760 2011 1 1

[34] A Larios E Lunasin and ES Titi Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion J Diff Eqs 2552636ndash2654 2013 1 1

[35] D Li and X Xu Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermaldiffusivity Dyn Par Diff Eqs 10255ndash265 2013 1

[36] J Li H Shang J Wu X Xu and Z Ye Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation Comm Math Sci 141999ndash2022 2016 1

[37] J Li ES Titi Global well-posedness of the 2D Boussinesq equations with vertical dissipationArch Ration Mech Anal 220983ndash1001 2016 1

[38] SA Lorca and JL Boldrini The initial value problem for a generalized Boussinesq modelNonlinear Anal 36457ndash480 1999 1

[39] A Majda Introduction to PDEs and Waves for the Atmosphere and Ocean Courant LectureNotes in Mathematics AMSCIMS 9 2003 1

[40] A Majda and A Bertozzi Vorticity and Incompressible Flow Cambridge University Press Cam-bridge UK 2002 1 4

[41] C Miao J Wu and Z Zhang Littlewood-Paley Theory and its Applications in Partial Differ-ential Equations of Fluid Dynamics Science Press Beijing China 2012 (in Chinese) 2

[42] J Pedlosky Geophysical Fluid Dyanmics Springer-Verlag New York 1987 1[43] H Qiu Y Du and Z Yao Local existence and blow-up criterion for the generalized Boussinesq

equations in Besov spaces Math Meth Appl Sci 3686ndash98 2013 1[44] T Runst and W Sickel Sobolev Spaces of Fractional Order Nemytskij Operators and Nonlinear

Partial Differential Equations Walter de Gruyter Berlin New York 1996 2[45] H Triebel Theory of Function Spaces II Birkhauser Verlag 1992 2[46] X Wang and JP Whitehead A bound on the vertical transport of heat in the lsquoultimatersquo state of

slippery convection at large Prandtl numbers J Fluid Mech 729103ndash122 2013 1[47] JP Whitehead and CR Doering Ultimate state of two-dimensional Rayleigh-Benard convection

between free-slip fixed-temperature boundaries Phys Rev Lett 106244501 2011 1[48] J Wu and X Xu Well-posedness and inviscid limits of the Boussinesq equations with fractional

Laplacian dissipation Nonlinearity 272215ndash2232 2014 1[49] J Wu X Xu L Xue and Z Ye Regularity results for the 2D Boussinesq equations with critical

and supercritical dissipation Commun Math Sci 141963ndash1997 2016 1[50] Z Xiang and W Yan Global regularity of solutions to the Boussinesq equations with fractional

diffusion Adv Diff Eqs 181105ndash1128 2013 1[51] K Yamazaki On the global regularity of N-dimensional generalized Boussinesq system Appl

Math 60109ndash133 2015 1[52] W Yang Q Jiu and J Wu Global well-posedness for a class of 2D Boussinesq systems with

fractional dissipation J Diff Eqs 2574188ndash4213 2014 1[53] Z Ye A note on global well-posedness of solutions to Boussinesq equations with fractional dissi-

pation Acta Math Sci Ser B Engl Ed 35B112ndash120 2015 1[54] K Zhao 2D inviscid heat conductive Boussinesq system in a bounded domain Michigan Math

J 59329ndash352 2010 1