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Transcript of arXiv:1203.3871v1 [math.AP] 17 Mar 2012 · ε) and the preceding compactness method is out of use....

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INCOMPRESSIBLE LIMIT FOR THE 2D ISENTROPIC EULER SYSTEM

WITH CRITICAL INITIAL DATA

TAOUFIK HMIDI AND SAMIRA SULAIMAN

Abstract. We study in this paper the low Mach number limit for the 2d isentropic Euler systemwith ill-prepared initial data belonging to the critical Besov space B22,1. By combining Strichartzestimates with the special structure of the vorticity we prove that the lifespan goes to infinity asthe Mach number goes to zero. We also prove the strong convergence in the space of the initialdata B22,1 of the incompressible parts to the solution of the incompressible Euler system. There areat least two main difficulties: the first one concerns the Beale-Kato-Majda criterion which is notknown to work for rough regularities. However, the second one is related to the critical aspect ofthe Strichartz norm (div v,c)L4

TL which has the scale of B

22,1 in the space variable.

Contents

1. Introduction 12. Basic Tools 52.1. Littlewood-Paley operators 52.2. Besov spaces 63. Preliminaries 83.1. Energy estimates 83.2. Strichartz estimates 94. Logarithmic estimate 105. Proofs of the main results 145.1. General statement 145.2. Proof of Theorem 1 155.3. Lifespan of the solutions 155.4. Incompressible limit 19References 21

1. Introduction

This work is devoted to the study of the isentropic Euler equations describing the evolution of acompressible fluid evolving in the whole space and under a constant entropy constraint. The systemis given by the coupled equations

(1)

(tu+ u u) + = 0, > 1t+ div ( u) = 0(u, )|t=0 = (u0, 0).

Here, u stands for the velocity field which is a time-dependent vector field over Rd, > 0 isthe density which is assumed to be nonnegative in order to avoid the vacuum. The parameter > 1 is the adiabatic exponent. The local well-posedness theory can be achieved by using the

Date: November 30, 2018.

1

http://arxiv.org/abs/1203.3871v1

symmetrization of Kawashima Makino and Ukai [20]. In other words, we introduce the sound speeddefined by,

c = 2

112

and thus the system (1) reduces to

(2)

tu+ u u+ c c = 0tc+ u c+ cdivu = 0(u, c)|t=0 = (u0, c0),

with := 12 . This form fits with the theory of the symmetric quasilinear hyperbolic systemsdeveloped by Klainerman and Majda and which answers the local well-posedness in the frameworkof Sobolev spaces Hs, s > d2 +1, see [15, 16]. It is well-known that the system (2) can be consideredas a good approximation of the incompressible Euler equations, despite we make some suitableassumptions as we will see later. The first step to reach this result is to introduce a suitablerescaling and change of variables. This can be done by making a perturbation of order from theconstant background state (0, c0):

u(t, x) = c0v(c0t, x), c(t, x) = c0 + c0c(c0t, x).

Therefore we get the system

(3)

tv + v v + cc + 1c = 0tc + v c + cdiv v + 1div v = 0(v, c)|t=0 = (v0,, c0,).

The parameter is called the Mach number and measures the compressibility of the fluid. Formore details about the derivation of this last model we refer for instance to [12, 15, 21]. Themathematical study of this model is well developed and there are a lot of papers devoted to therigorous justification of the convergence towards the incompressible Euler equations when the Machnumber goes to zero. We recall that the incompressible Euler system is given by,

(4)

tv + v v +p = 0div v = 0v|t=0 = v0.

We point out that the issue to this problem depends on several factors like the geometry of thedomain where the fluid is assumed to evolve or the state of the initial data. For example, theanalysis in the case of the full space is completely different from the torus due to the resonancephenomenon. The second factor is related to the structure of the initial data and whether theyare well-prepared or not. In the well-prepared case, the initial data are assumed to be slightlycompressible, which means that (div v0,,c0,) = O() as 0. This assumption allows toget a uniform bound for (tv), and consequently we can pass to the limit by using Aubin-Lionscompactness lemma, for more details see [15, 16]. However, in the ill-prepared case, the family(v0,, c0,) is only assumed to be bounded in some Sobolev spaces H

s with s > d2 + 1 and theincompressible parts of (v0,) converge to some vector field v0. This case is more subtle because

the time derivative tv, is of size O(1) and the preceding compactness method is out of use. To

overcome this difficulty, Ukai [27] used the dispersive effects generated by the acoustic waves inorder to prove that the compressible part of the velocity and the acoustic term vanish when goesto zero. We point out that this problem has already been studied in numerous papers, see forinstance [1, 3, 8, 10, 11, 15, 16, 17, 18, 19, 21, 24, 27].Concerning the blow up theory, it is well-known that contrary to the incompressible Euler system,the equations (1) develop in space dimension two singularities in finite time and for some smoothinitial data, see [22]. This phenomenon holds true for higher dimensions, see [25]. On the other

2

hand, it was shown in [16] that for the whole space or for the torus domain when the limit system(4) exists for some time interval [0, T0] then for ever T < T0 there exists 0 > 0 such that for < 0the solution to (3) exists till the time T . This result was established for the well-prepared case andwith sufficient smooth initial data, that is, in Hs and s > d2+2. The proof relies on the perturbation

theory which ceases to work for lower regularities d2 +2 > s >d2 +1. Consequently, if we denote by

T the lifespan of the solution (v, c) then we get in space dimension two that lim0 T = +.It seems that we can get better information about T when the initial data have some specialstructures. In [2], Alinhac proved that in space dimension two and for axisymmetric initial datathe lifespan of the solution is equivalent to 1

. In dimension three and for irrotational velocity,

Sideris [25] established that the solutions are almost global in time, that is, their lifespan are

bounded below by e1 . Finally, we mention the results of [9, 23] dealing with the global existence

under some suitable conditions on the initial data: the initial density must be small and has acompact support and the spectrum of u0 must be far away the set of the negative real numbers.We intend in this paper to deal with the incompressible limit problem in space dimension two butfor the critical Besov regularity B22,1. We emphasize that by using Strichartz estimates we can provethat for initial data with sub-critical regularity, that is Hs and s > 2, the lifespan is bounded belowby C log log 1

. We can interpret this double logarithmic growth by the fact that the Sobolev norm

of the solutions to the incompressible Euler equations has at most a double exponential growth intime. Our main result reads as follows.

Theorem 1. Let {(v0,, c0,)0

Thus, letting go to zero, we infer

lim sup0

Pv0, v0B22,1 .

qN

22q(qv0L2 + sup

0 0. To overcomethis critical aspect, we will proceed in the spirit of the the paper [14]: we can find an adequatenondecreasing profile such that limq+(q) = + and:

sup

q

(q)22q(qv0,,qc0,)L2 < +.

Then by performing energy estimate we prove that the solutions have the same additional decayin the time interval [0, T]. This means that there is no energy transfer to higher frequencies.Consequently, interpolating between this estimate and Strichartz estimate for the lower degreequantity (1v, c)L1

TL , we get that the acoustic parts vanish when the Mach number goes

to zero. However the rate convergence is implicit and can be related to the distribution on Fouriermodes of the energy of the initial data.The second difficulty that one should to deal with has an incompressible nature and concerns theBeale-Kato-Majda criterion [4]. It was initially stated for the system (4) and for the sub-criticalspaces Hs, s > d2+1. It asserts that we can propagate the initial regularity in the time interval [0, T ]provided that we can control the quantity L1

TL , where denotes the vorticity of the velocity.

As a matter of fact, we get in space dimension two the global existence of the strong solutions sincethe vorticity is only transported by the flow and thus its L norm is conserved. Unfortunately,

this criterion is not known to work in the critical Besov space Bd2+1

2,1 and it is replaced by a stronger

norm L1TB0

,1. The global existence for 2d case was solved some years ago by Vishik [28] who

was able to prove the following linear growth:

(t)B0,1

C0B0,1

(1 +

t

0v()Ld

).

The Vishiks method relies on a composition law in Besov space B0,1 with a logarithmic growth. We

emphasize that in [13], we gave an other proof which has the advantage to fit with general modelslike transport-diffusion equations. Now, in order to get a lower bound for the lifespan satisfyinglim0

T = + we should establish an analogous estimate in the compressible case. This is not aneasy task due to the nonlinearities in the vorticity equation and to the lack of the incompressibilityof the velocity. We recall that the vorticity = 1v

2 2v1 satisfies the compressible transport

equation:t + v + div v = 0.

We will prove in Theorem 2 the following key estimate: for p [1,+[

(t)B0,1

C0,B0,1

(1 + e

CvL1tLdiv v2

L1tB2pp,1

)(1 +

t

0v()Ld

).

We observe that for div v = 0 we get the Vishiks estimate. The proof will be done in the spiritof the work [13] but with slight modifications. In the first step, we use the Lagrangian coordinates

4

combined with a filtration procedure to get rid of the compressible part. In the second step weestablish a suitable splitting of the vorticity and use a dynamical interpolation method.Concerning the incompressible limit we prove first the strong convergence in Lloc(R+;L

2) by usingStrichartz estimates. This leads according to some standard interpolation inequalities to the strongconvergence in Besov spaces Bs2,1, for all s < 2. However for the strong convergence in the space of

the initial data B22,1, we establish an additional frequency decay of the energy uniformly with respectto t and . In other words, from the assumption on the initial data we can find a nondecreasing

function independent of such that (v0,, c0,) belongs to B2,2,1 , see the Definition 1. This

regularity will be conserved in the full interval [0, T] and thus the energy transfer for higherfrequencies can not occur.The paper is organized as follows: we recall in section 2 some basic results about the Littlewood-Paley operators and Besov spaces. In section 3, we gather some energy estimates for the heteroge-

neous Besov spaces Bs,2,1 and we establish some useful Strichartz estimates for the acoustic partsof the fluid. In section 4, we establish a logarithmic estimate for a compressible transport model.In section 5, we generalize the result of Theorem 1 and we give the proofs.

2. Basic Tools

In this preliminary section, we are going to recall the Littlewood-Paley operators and give some oftheir elementary properties. We will also introduce some function spaces and review some impor-tant lemmas that will be used later.

2.1. Littlewood-Paley operators. We denote by C any positive constant that may change fromline to line and C0 a real positive constant depending on the size of the initial data. We will usethe following notations: for any positive A and B, the notation A . B means that there exists apositive constant C independent on A and B and such that A 6 CB.To define Besov spaces we need to the following dyadic unity partition (see [7]). There are twononnegative radial functions D(R2) and D(R2\{0}) such that

() +

q0

(2q) = 1, R2,

qZ

(2q) = 1, R2\{0},

|p q| 2 supp (2p) supp (2q) = ,q 1 supp supp (2q) = .

Let u S (R2) we define the Littlewood-Paley operators by

1u = (D)u, q 0, qu = (2qD)u and Squ =

1pq1

pu.

We can easily check that

u =

qZ

qu, u S (R2).

Moreover, the Littlewood-Paley decomposition satisfies the property of almost orthogonality: forany u, v S (R2),

pqu = 0 if |p q| > 2p(Sq1uqv) = 0 if |p q| > 5.

5

Let us note that the above operators q and Sq map continuously Lp into itself uniformly with

respect to q and p. By the same way we define the homogeneous operators:

q Z, qu = (2qD)v and Squ =

jq1

ju.

We notice that these operators are of convolution type. For example for q Z, we havequ = 2

2qh(2q ) u, with h S, h() = ().Now we recall Bernstein inequalities, see for exapmle [7].

Lemma 1. There exists a constant C > 0 such that for all q N , k N and for every tempereddistriubution u we have

sup||=k

SquLb 6 Ck2q(k+2

(1a 1

b

))SquLa for b > a > 1

Ck2qkquLa 6 sup||=k

quLa 6 Ck2qkquLa .

2.2. Besov spaces. Now we will define the Besov spaces by using Littlewood-Paley operators. Let(p, r) [1,+]2 and s R, then the non-homogeneous Besov space Bsp,r is the set of tempereddistributions u such that

uBsp,r :=(2qsquLp

)r< +.

The homogeneous Besov space Bsp,r is defined as the set of u S (Rd) up to polynomials such that

uBsp,r :=(2qsquLp

)r(Z)

< +.

We remark that the usual Sobolev space Hs coincides with Bs2,2 for s R and the Holder space Cscoincides with Bs, when s is not an integer.The following embeddings are an easy consequence of Bernstein inequalities,

Bsp1,r1 Bs+2( 1

p2 1

p1)

p2,r2 , p1 p2 and r1 r2.Let T > 0 and 1, we denote by LTBsp,r the space of tempered distributions u such that

uLTBsp,r

:=(2qsquLp

)r

LT

< +.

Now we will introduce the heterogeneous Besov spaces which are an extension of the classical Besovspaces.

Definition 1. Let : {1} N R+ be a given function.(i) We say that belongs to the class U if the following conditions are satisfied:

(1) is a nondecreasing function.(2) There exists C > 0 such that

supxN{1}

(x+ 1)

(x) C.

(ii) We define the class U by the set of function U satisfying limx+

(x) = +.(iii) Let s R, (p, r) [1,+]2 and U . We define the heterogeneous Besov space Bs,p,r asfollows:

u Bs,p,r uBs,p,r =((q)2qsquLp

)r< +.

6

Remark 2. (1) From the part (2) of the above Definition, we see that the profile has at mostan exponential growth: there exists > 0 such that

(1) (q) (1)eq, q 1.

When the profile has an exponential growth: (q) = 2q with R+, the space Bs,p,rreduces to the classical Besov space Bs+p,r .

(2) The condition (2) seems to be necessary for the definition of Bs,p,r : it allows to get a coherentdefinition which is independent of the choice of the dyadic partition.

The following lemma proved in [14] is important for the proof of Theorem 1. Roughly speaking, itsays that any element of a given Besov space is always more regular than its prescribed regularity.

Lemma 2. Let s R, p [1,+], r [1,+[ and f Bsp,r. Then there exists a function Usuch that f Bs,p,r .

This yields to the following result, see [14].

Corollary 1. Let s R, p [1,+], r [1,+[ and (g)0

Proof. We split the velocity into incompressible and compressible parts: v = Pv + Qv. Then wehave

curl v = curlP v

We have by Bernstein ineguality, the continuity of qP : Lp Lp, p [1,] uniformly in q and

qvL 2qqL , that

PvL 1PvL +

qN

qPvL

. 1PvL2 +

qN

2qqPvL

. vL2 +

qN

qL

. vL2 + B0,1

.(5)

On the other hand, using Bernstein inequality leads to

QvL = 21div vL 121div vL +

q0

q21div vL

. vL2 + div vB0,1

.

Combining this estimate with (5), gives

vL . vL2 + div vB0,1

+ B0,1

.

This achieves the proof of the lemma.

3. Preliminaries

This section is devoted to some useful estimates for the system (3) that will be crucial for the proofof the main results. We will discuss some energy estimates for the full system (3) and give someStrichartz estimates for the acoustic operator.

3.1. Energy estimates. Here we list two energy estimates for (3). The first one is very classicaland is concerned with the L2-estimate. However the second one deals with the energy estimates

in the heterogeneous Besov space Bs,2,1 . All the spaces of the initial data are constructed over the

Lebesgue space L2 in order to remove the singular terms and get uniform estimates with respectto the Mach number . For the proof, see for instance [14].

Proposition 1. Let (v, c) be a smooth solution of (3) and U , see the Definition 1.(1) L2estimate: there exists C > 0 such that t 0

(v, c)(t)L2 C(v0,, c0,)L2eCdiv vL1

tL .

(2) Besov estimates: for s > 0 and r [1,+], there exists C > 0 such that

(v, c)(t)Bs,2,r C(v0,, c0,)Bs,2,r eCV(t)

with

V(t) := vL1tL + cL1tL .8

We emphasize that for 1 the space Bs,2,r reduces to the classical Besov space Bs2,r. Thereforewe get the known energy estimate, see for instance [12],

(6) (v, c)(t)Bs2,r C(v0,, c0,)Bs2,r eCV(t).

3.2. Strichartz estimates. Our goal is to establish some classical Strichartz estimates for theacoustic parts which are governed by wave equations. Roughly speaking, we will show that theaveraging in time of the compressible part of the velocity v and the sound speed c will lead tovanishing quantities when goes to zero. We point out that Ukai [27] used this fact but underits dispersion form in order to study the incompressible limit in the framework of the ill-preparedinitial data. There are various ways to exhibit the wave structure and we prefer here the complexversion. We start with rewriting the system (3) under the form:

(7)

tv +1c = v v cc := f

tc +1div v = v c cdiv v := g

(v, c)|t=0 = (v0,, c0,).

Denote by Qv := 1div v the compressible part of the velocity v. Then the quantity := Qv i|D|1c, with |D| = ()

12

satisfies the following wave equation

(8) (t +i

|D|) = Qf i|D|1g.

Similarly, we can easily check that the quantity

:= |D|1div v + icobeys to the wave equation

(9) (t +i

|D|) = |D|1div f + ig.

Now, we are in a position to use of the following Strichartz estimates stated in space dimensiontwo and for the proof see for instance [6, 12, 26].

Lemma 5. Let be a solution of the wave equation

(t +i

|D|) = G, |t=0 = 0.

Then there exists an absolute constant C such that for every T > 0,

LrTLp C

14 1

2p(0

B34

32p

2,1

+ GL1TB

34

32p

2,1

),

for all p [2,+] and r = 4 + 8p2 .

This result enables us to get the following estimates.

Proposition 2. Let (v0,, c0,) be a bounded family in B742,1. Then any smooth solution of (3)

defined in the time interval [0, T ] satisfies,

(Qv, c)L4TL C0

14 (1 + T )eCV(T ),

where C0 depends on the quantities sup0

Proof. We apply Lemma 5 to the equation (8) with p = +, then

L4TL .

14(0

B342,1

+ Qf i|D|1gL1TB

342,1

).

Using the continuity of the Riesz operators Q and |D|1 on the homogeneous Besov spaces,combined with the embedding B

342,1 B

342,1 and Holder inequality

L4TL .

14

((v0,, c0,)

B342,1

+ (f, g)L1TB

342,1

)

. 14

((v0,, c0,)

B342,1

+ T(f, g)LTB

342,1

).(10)

It remains to estimate (f, g)LTB

342,1

. For this purpose we use the following law product which

can be proved for example by using Bonys decomposition :

f gB

342,1

. fLgB

742,1

+ gLfB

742,1

.

Thus we get

(f, g)B

342,1

. (v, c)L(v, c)B

742,1

. (v, c)2B

742,1

,

where we have used in the last line the embedding B742,1 L. Therefore (6) yields to

(f, g)LTB

342,1

. (v0,, c0,)2B

742,1

eCV(T ).

Plugging this inequality into (10), we obtain

L4TL .

14((v0,, c0,)

B342,1

+ T(v0,, c0,)2B

742,1

eCV(T ))

. C014 (1 + T )eCV(T ).

Remark that the real part of is the compressible part of the velocity v, then

QvL4TL . C0

14 (1 + T )eCV(T ).

We can prove in the similar way that,

L4TL . C0

14 (1 + T )eCV(T )

and therefore

cL4TL . C0

14 (1 + T )eCV(T ).

The proof of the Proposition 2 is now achieved.

4. Logarithmic estimate

The purpose of this paragraph is to study the linear growth of the norm B0,1 for the followingcompressible transport model,

(11)

{tf + v f + fdiv v = 0f|t=0 = f0.

10

We point out that the dynamics of the vorticity for the system (3) is governed by an equationof type (11). In the framework of the incompressible vector fields, that is div v = 0, Vishik [28]established the following linear growth for Besov regularity with zero index,

f(t)B0,1

Cf0B0,1

(1 +

t

0v()Ld

).

A simple proof for this result was given in [13] and where we extended the Vishiks result toa transport-diffusion model. Our main goal here is to valid the previous linear growth to thecompressible model and this is very crucial for the proof of the Theorem 1. Our result reads asfollows,

Theorem 2. Let v be a smooth vector field and f be a smooth solution of the transport equation (11).Then for every 1 p < +, there exists a constant C depending only on p such that

f(t)B0,1

Cf0B0,1

(1 + e

CvL1tLdiv v2

L1tB2pp,1

)(1 +

t

0v()Ld

).

Proof. First we observe that the above estimate reduces in the incompressible case to the Vishiksestimate. Here, we will try to use the same approach of [13], but as we will see the lack of theincompressibility brings more technical difficulties. Let us denote by the flow associated to thevelocity v :

(t, x) = x+

t

0v(, (, x)

)d, (t, x) R+ R2.

We will make use of the following estimate for the flow and its inverse 1

(12)

((, 1(t, ))

)L

e t

v(s)Lds

.

We set g(t, x) = f(t, (t, x)) then it is clear that g satisfies the equation

tg(t, x) + (div v)(t, (t, x))g(t, x) = 0.

Therefore we obtain

g(t, x) = f0(x)e

t0 (div v)(,(,x))d .

It follows that

f(t, x) = f0(1(t, x))e

t0(div v)(,(,1(t,x)))d .

Set h(x) = ex 1 and W (t, x) = t0 (div v)(, (,

1(t, x)))d, then we infer

f(t, x) = f0(1(t, x))

(h(W (t, x)) + 1

).

Thus the problem reduces to the establishment of a composition law inB0,1 space. For this purpose,

we will use the following law product which can be obtained by using Bonys decomposition [5]:for 1 p

The proof of this estimate can be done as follows. By definition we get easily

h(W (t, ))B

2pp,1

n1

1

n!W n(t, )

B2pp,1

.

According to the law product

u2B

2pp,1

CuLuB

2pp,1

, p < +,

and using the induction principle we infer that for n 1,W n(t, )

B2pp,1

Cn1W (t, )n1L W (t, )B

2pp,1

.

Therefore

h(W (t, ))B

2pp,1

WB

2pp,1

n1

Cn1

n!Wn1L

WB

2pp,1

n0

CnWnL(n+ 1)!

WB

2pp,1

n0

CnWnLn!

WB

2pp,1

eCWL .

This ends the proof of the desired inequality. Thus it follows that

h(W (t, ))B

2pp,1

. W (t, )B

2pp,1

eCW (t)L

. W (t, )B

2pp,1

eC t0div v()Ld .(14)

To estimate W (t) we set k (t, x) = div v(, (, 1(t, x)), then k (t, (t, x)) = div v(, (, x))

and it follows that {tk + v k = 0k (, x) = div v(, x).

It remains to estimate k (t)B

2pp,1

. For this aim, we use Lemma 3-(1), yielding for p > 2 to

k (t)B

2pp,1

Cdiv v()B

2pp,1

e| tv(s)Lds|.

Combining the definition of W with this latter estimate we get

W (t)B

2pp,1

t

0k (t)

B2pp,1

d

C t

0div v()

B2pp,1

e tv(s)Ldsd.

Plugging this estimate into (14) we find

h(W (t, ))B

2pp,1

CeCvL1t L t

0div v()

B2pp,1

d.

12

Inserting this estimate in (13) we obtain

(15) f(t)B0,1

Cf0(1(t, ))B0,1

(1 + e

CvL1tL

t

0div v()

B2pp,1

d)

It remains to estimate f0(1(t, ))B0,1

. Set w(t, x) = f0(1(t, x)) then it satisfies the transport

equation {tw + v w = 0w(0, x) = f0(x).

We will split the initial data into Fourier modes f0 =

q1

qf0 and for each frequency q 1, we

denote by wq the unique global solution of the initial value problem

(16)

{twq + v wq = 0wq(0, ) = qf0.

According to Lemma 3-(2), we have

wq(t)Bs, Cqf0Bs,eCV (t), 0 < s < min(1, 2

p),

with V (t) := vL1tL + div vL1tB

2pp,

. Combined with the definition of Besov spaces, this implies

for j, q 1(17) jwq(t)L C2s|jq|qf0LeCV (t).By linearity and the uniqueness of the solution, it is obvious that

w =

q=1

wq.

From the definition of Besov spaces we have

w(t)B0,1

|jq|N

jwq(t)L +

|jq|

Choosing N such that

N =[CV (t)s log 2

+ 1],

we get

w(t)B0,1

Cf0B0,1

(1 +

t

0

(v()L + div v()

B2pp,

)d

).

Therefore

f0(1(t, )

)B0

,1 Cf0B0

,1

(1 +

t

0

(v()L + div v()

B2pp,

)d

)

Inserting this estimate into (15) gives

f(t)B0,1

Cf0B0,1

(1 + e

CvL1tLdiv v2

L1tB2pp,1

)(1 +

t

0v()Ld

).

This is the desired result.

5. Proofs of the main results

In this section we will firstly extend the result of Theorem 1 to the framework of the heterogeneous

Besov spaces Bs,2,1 . Secondly we will see how to derive the proof of the Theorem 1 and ultimatelywe will devote the rest of the paper to the discussion of the proof of the general statement given inTheorem 3.

5.1. General statement. We will give a generalization of the Theorem 1.

Theorem 3. Let U and {(v0,, c0,)}0

Remark 3. (1) We observe that the result for the subcritical regularities is a special case of theabove theorem. Indeed, let {(v0,, c0,)}0 2. It is plainthat the following embedding holds true: Hs B2,2,1 , with (q) = 2q and 0 < < s 2.Thus we get by applying Theorem 3 that

T = C0 log log {1

}.

This result is in agreement with known results about the lifespan for the subcritical regu-larities.

(2) If has a polynomial growth: (q) = (q + 2), > 0 then it is easily seen that Uand

T = C0 log log log{1

}.

Before going further into the details of the proof of the Theorem 3, we will first show how to deducethe result of the Theorem 1.

5.2. Proof of Theorem 1. Since

q1

22q sup0

Then it follows that

V(T ) C0(1 + T )eC(div v,c)L1

TB0,1 + L1

TB0

,1.(20)

Applying Theorem 2 gives

(t)B0,1

C0B0,1

(1 + e

CvL1tLdiv v2

L1tB2pp,1

)(1 + vL1tL)

C0(1 + eCV(t)div v2

L1tB2pp,1

)(1 + V(t)

).(21)

To estimate div vB

2pp,1

, we use the following interpolation inequality: for 2 < p N

qdiv Q vL

.

qN

2qqQvL +1

(N)

q>N

(q)22qqQvL2

. 2NQ vL +1

(N)vB2,2,1 .

16

We have used the fact that the profile is a nondecreasing function. Now integrating in time andusing Proposition 2 and Proposition 1-(2)

div vL1TB0

,1. 2NT

34 Q vL4

TL +

T

(N)vL

TB

2,2,1

C0(T

34 (1 + T ) + T

)eCV(T )

(

142N +

1

(N)

)

C0(1 + T74 )eCV(T )

(

14 2N +

1

(N)

).

By similar computations we get

cL1TB0

,1 C0(1 + T

74 )eCV(T )

(

14 2N +

1

(N)

).

Therefore

(div v,c)L1TB0

,1 C0(1 + T

74 )eCV(T )

(

142N +

1

(N)

).

Now, we choose N such that,

N log( 1

18

).

Hence we get

(div v,cL1TB0

,1 C0(1 + T

74 )eCV(T )

(

18 +

1

(log( 1

18))).

According to the Remark 2, the profile has at most an exponential growth: there exists > 0such that

x 1; (x) Cex.Hence we get

(24) 18 C

1

1 (log( 1

18).

Let = min(1, 1) then

(25) (div v,cL1TB0

,1 C0(1 + T

74 )eCV(T )(),

with

() :=1

(log( 1

18))

Plugging (25) into (23) and using Gronwalls inequality we obtain

V(T ) C0(1 + T )eC0(1+T74 )()eCV(T ) +C0

T

0

(1 + (1 + t

114 )eCV(t)()

)(1 + V(t))dt

C0eC0T exp{C0(1 + T154 )()eCV(T )}.(26)

We choose T such that,

(27) eeC0T

= 12 (),

then we claim that for small and 0 t T we have(28) eCV(t) 23 ().

17

Indeed, define

J :={t [0, T]; eCV(t)

23 ()

}.

This set is nonempty since 0 J. It is also closed by the continuity of the mapping t 7 V(t). Itremains to prove that J is an open subset of [0, T] and thus we deduce J = [0, T]. Let t J,then using (26) we get for small

(29) eCV(t) C0eexp{C0T+C0T154

13 ()}

From (27), we infer

C0T = log log 1

2 () log log (log(

1

18

))

Since

lim0

13 ()

{log log

12 ()

} 154 = 0,

then for sufficiently small and for t [0, T], we get

eCV(t) 2C012 ()

< 23 ().

This proves that t belongs to the interior of J and consequently J is an open subset of [0, T].Finally we get J = [0, T]. From the previous estimate, we obtain for all T [0, T]

(30) (1 + T154 )eCV(T )() . {log log 12 ()

} 154

13 () . 1.

Inserting (30) into (26), yields for 0 T T(31) V(T ) C0eC0T .In particular we have obtained

vL1TL C0eC0T .

Plugging the estimate (30) into (25), we get for small and for T [0, T] that

(div v,c)L1TB0

,1 C0{log log

12 ()

} 74

13 ()

C014 ().(32)

Using Proposition 2, (28) and (24) we obtain for small ,

QvL4TL + cL4

TL C0

14 (1 + T )eCV(T )

C014

23 () log log

12 ()

C043 () log log

12 ()

C0().(33)Let us now move to the estimate of the vorticity. First, recall that satisfies the compressibletransport equation

t + v + div v = 0.Consequently we get by using Gronwalls inequality that

(t)L 0,Lediv vL1

tL .

It suffices to apply (32), leading to the following estimate: for all T [0, T](T )L C0.

18

Finally, to estimate (v, c)(T )B2,2,1 we use Proposition 1 combined with (31) to find,

(v, c)(T )B2,2,1 C(v0,, c0,)B2,2,1 eCV(T )

C0eexp{C0T}.(34)

We observe that in Theorem 3 the formula for the lifespan involves the quantity log log(log 1)

instead of log log (log 18 ). This is due to the fact that for any given > 0 the function defined

by (x) := (x) belongs to the same class U.This completes the proof of the first part of Theorem 3.

5.4. Incompressible limit. In this paragraph we will sketch the proof of the second part ofTheorem 3 which deals with the incompressible limit.

Proof. We will proceed in two steps. In the first one, we will prove that for any fixed T > 0, thefamily (Pv) converges strongly in L

T L

2, when 0, to the solution v of the incompressible Eulerequations with initial data v0. It is worthy pointing out that similarly to the Remark 1-(2) we can

show that the limit v0 belongs also to the same space B2,2,1 . In the second step we will show how

to get the strong convergence in the Besov spaces B2,2,1 . We mention that the limit system (4) is

globally well-posed if v0 B2,2,1 . Indeed, combining the embedding B2,2,1 B22,1 with the Vishiks

result [28] we get that the system (4) admits a unique global solution such that v C(R+;B22,1),with the following Lipschitz bound

v(t)L C0eC0t.

This latter a priori estimate together with Proposition 1, which remains valid for (4), gives the

global persistence of the initial regularity B2,2,1 . More precisely, we get

(35) v(t)B

2,2,1

C0eexpC0t.

Let > 0 and set

:= v v, w = Pv v.

By applying Lerays projector P to the first equation of (3), we obtain

tPv + P(v v) = 0.

Taking the difference between the above equation and (4) yields to

t w + P(v ) + P( v) = 0.

We take the L2 inner product of the above equation with w, integrating by parts and using theidentities

= w +Qv, Pw = w,19

we get,

1

2

d

dtw(t)2L2 =

R2(v )Pwdx

R2( v)Pwdx

=

R2

(v (w +Qv)

)wdx

R2

((w +Qv) v

)wdx

=1

2

R2|w|2div vdx

R2(w v)wdx

R2(v Qv +Qv v)wdx

(12div v(t)L + v(t)L

)w(t)2L2

+(v(t)L2Qv(t)L + Qv(t)Lv(t)L2

)w(t)L2 .

This implies that

d

dtw(t)L2

(12div v(t)L + v(t)L

)w(t)L2

+ v(t)L2Qv(t)L + Qv(t)Lv(t)L2 .Integrating in time and using Gronwalls inequality yield to

(36) w(t)L2 (w0L2 + F(t)

)e

12div vL1

tL

+vL1tL ,

where

F(t) := vLt L2Qv(t)L1tL + QvL1tLvLt L2. vLt L2QvL1tL + QvL1tLLt L2 ,

where denotes the vorticity of v.To estimate QvL we use Bernstein inequality,

QvL 1QvL +

q0

qQvL

. QvL +

q0

qdiv vL

. QvL + div vB0,1

.

Integrating in time

QvL1tL QvL1tL + div vL1tB0,1 .By virtue of (32) and (33) one has for small and t [0, T ],

QvL1tL + QvL1tL C014 ().

On the other hand we get from Proposition 1-(1) and (32)

v(t)Lt L2 C0.To estimate (t)L2 we use the fact that the vorticity is transported for the incompressible flowand thus

(t)L2 = 0L2 .Therefore we obtain for t [0, T ] and for small ,

F(t) C014 ().

20

According to Vishiks result [28] we have for all t 0v(t)L C0eC0t.

Inserting the previous estimates in (36) we find for all t [0, T ] and small values of ,

(37) w(t)L2 C0eexpC0t(Pv0, v0L2 +

14 ()

).

This achieves the proof of the strong convergence in Lloc(R+;L2).

Let us now turn to the proof of the strong convergence in the space LT B2,2,1 . We recall that is

any element of the set U , see Definition 1, and satisfying in addition the assumption

limq+

(q)

(q)= 0.

By using (34) and the continuity of Lerays projector on the spaces B2,2,1 we get for t [0, T ] andfor small

(38) Pv(t)B2,2,1 C0eexpC0t.

Now let N N be an arbitrary number. Then by (37), (35) and (38) we obtain

w(t)B

2,2,1

=

N

q=1

(q)22qqw(t)L2 +

q>N

(q)

(q)(q)22qqw(t)L2

C0eexpC0t(Pv0, v0L2 +

14 ()

)(N)22N + N(Pv, v)(t)B2,2,1

C0eexpC0t((

Pv0, v0L2 +14 ()

)(N)22N + N

),

with

N := maxqN

(q)

(q).

We remark that the (N )N is a decreasing sequence to zero. Consequently,

lim sup0+

wLTB

2,2,1

C0eexpC0TN .

Combining this estimate with limN+

N = 0, we get

lim0+

wLTB

2,2,1

= 0.

This completes the proof of the strong convergence.

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IRMAR, Universite de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France

E-mail address: [email protected]

IRMAR, Universite de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France

E-mail address: [email protected]

22

http://arxiv.org/abs/1109.5339

1. Introduction2. Basic Tools2.1. Littlewood-Paley operators2.2. Besov spaces

3. Preliminaries3.1. Energy estimates3.2. Strichartz estimates

4. Logarithmic estimate5. Proofs of the main results5.1. General statement5.2. Proof of Theorem 15.3. Lifespan of the solutions5.4. Incompressible limit

References