Stochastic isentropic Euler equations

F. Berthelin∗and J. Vovelle†

April 27, 2015

Abstract

We study the stochastically forced system of isentropic Euler equa-tions of gas dynamics with a γ-law for the pressure. We show theexistence of martingale solutions; we also discuss the existence of in-variant measure in the concluding section.

Keywords: Stochastic partial differential equations, isentropic Euler equa-tions, kinetic formulation, entropy solutions.

MSC: 60H15, 35R60, 35L65, 76N15

Contents

1 Introduction 2

2 Notations and main result 42.1 Stochastic forcing . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Entropy Solution . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Organization of the paper and main problematic . . . . . . . 9

3 Parabolic Approximation 93.1 Solution to the parabolic problem . . . . . . . . . . . . . . . . 12

3.1.1 Time splitting . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Entropy bounds . . . . . . . . . . . . . . . . . . . . . 163.1.3 Lp estimates . . . . . . . . . . . . . . . . . . . . . . . 193.1.4 Gradient estimates . . . . . . . . . . . . . . . . . . . . 253.1.5 Positivity of the density . . . . . . . . . . . . . . . . . 28

∗Laboratoire Dieudonne, UMR 7351 CNRS, Universite de Nice Sophia Antipolis, ParcValrose, 06108 Nice Cedex 02, France. Email: Florent.Berthelin@unice.fr ; and COFFEE(INRIA Sophia Antipolis)†Universite de Lyon ; CNRS ; Universite Lyon 1, Institut Camille Jordan, 43 boulevard

du 11 novembre 1918, F-69622 Villeurbanne Cedex, France. Email: vovelle@math.univ-lyon1.fr, partially supported by ANR STOSYMAP and ERC NuSiKiMo

1

3.1.6 Bounds on higher derivatives of U τ . . . . . . . . . . . 283.1.7 Compactness argument . . . . . . . . . . . . . . . . . 353.1.8 Identification of the limit . . . . . . . . . . . . . . . . 36

4 Probabilistic Young measures 394.1 Young measures embedded in a space of Probability measures 394.2 A compactness criterion for probabilistic Young measures . . 414.3 Convergence to a random Young measure . . . . . . . . . . . 43

5 Reduction of the Young measure 435.1 Compensated compactness . . . . . . . . . . . . . . . . . . . . 43

5.1.1 Murat’s Lemma . . . . . . . . . . . . . . . . . . . . . . 435.1.2 Functional equation . . . . . . . . . . . . . . . . . . . 47

5.2 Reduction of the Young measure . . . . . . . . . . . . . . . . 485.3 Martingale solution . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Conclusion 53

A A parabolic uniformization effect 55

B Regularizing effects of the one-dimensional heat equation 59

1 Introduction

In this paper, we study the stochastically forced system of isentropic Eulerequations of gas dynamics with a γ-law for the pressure.Let (Ω,F ,P, (Ft), (βk(t))) be a stochastic basis, let T be the one-dimensionaltorus, let T > 0 and set QT := T× (0, T ). We study the system

dρ+ (ρu)xdt = 0, in QT , (1.1a)

d(ρu) + (ρu2 + p(ρ))xdt = Φ(ρ, u)dW (t), in QT , (1.1b)

ρ = ρ0, ρu = ρ0u0, in T× 0, (1.1c)

where p follows the γ-law

p(ρ) = κργ , κ =θ2

γ, θ =

γ − 1

2, (1.2)

for γ > 1, W is a cylindrical Wiener process and Φ(0, u) = 0. Therefore thenoise affects the momentum equation only and vanishes in vacuum regions.Our aim is to prove the existence of solutions to (1.1) for general initial data(including vacuum). When γ ≤ 2, we will manage to show the existence ofmartingale solutions, cf. Theorem 2.7 below.

2

There are to our knowledge no existing results on stochastically forced sys-tems of first-order conservation laws1. The question of existence of solutionsfor (1.1) is one of the first problem to be solved, and what we will do here for1 < γ ≤ 2, but there are some other ones to be considered (see the conclu-sion part, Section 6 on this subject). In the deterministic case, the existenceof (entropy) solutions has been proved by Lions, Perthame, Souganidis in[LPS96] (let us mention also the anterior papers by Di Perna [DiP83a], Ding,Chen, Luo [DCL85], Chen [Che86], Lions, Perthame, Tadmor [LPT94a]).The uniqueness of entropy solutions is still an open question nowadays.

For scalar non-linear hyperbolic equations with a stochastic forcing term,the theory has recently known a lot of developments. Well-posedness hasbeen proved in different contexts and under different hypotheses and alsowith different techniques: by Lax-Oleinik formula (E, Khanin, Mazel, Sinai[EKMS00]), Kruzhkov doubling of variables for entropy solutions (Kim [Kim03],Feng, Nualart [FN08], Vallet, Wittbold [VW09], Chen, Ding, Karlsen [CDK12],Bauzet, Vallet, Wittbold [BVW12], kinetic formulation (Debussche, Vovelle[DV10, DV], the more general results being the resolution in L1 given inDebussche, Vovelle [DV13]. Let us also mention the works of Hofmanovain this fields (extension to second-order scalar degenerate equations, conver-gence of the BGK approximation [Hof13b, DHV13, Hof13a]) and the recentworks by Lions, Perthame, Souganidis [LPS12, LPS13] on scalar conserva-tion laws with quasilinear stochastic terms.

We will show existence of martingale solutions to (1.1), see Theorem 2.7 be-low, under the restriction γ ≤ 2. The procedure is standard: we prove theconvergence of (subsequence of) solutions to the parabolic approximationto (1.1). For this purpose we have to adapt the concentration compactnesstechnique (cf. [DiP83a, LPS96]) of the deterministic case to the stochasticcase. Such an extension has already been done for scalar conservation lawsby Feng and Nualart [FN08] and what we do is quite similar. The modeof convergence for which there is compactness, if we restrict ourselves tothe alea, is the convergence in law. That is why we obtain martingale solu-tions. There is a usual trick (the Gyongy-Krylov characterization of conver-gence in probability) that allow to recover pathwise solutions once pathwiseuniqueness of solutions is known (cf. [Hof13b, Section4.5]). However for thestochastic problem (1.1) (as it is already the case for the deterministic one),no such results of uniqueness are know and we will say nothing else aboutpathwise solutions.

Besides the proof of convergence of the parabolic approximation to Prob-lem (1.1) (cf. Problem (3.1)) which needs serious adaptation due to the lossof L∞ bounds with respect to the deterministic case, a large part of our

1see however the works by Tornare and Fujita [TFY97] and by Feireisl, Maslowski,Novotny [FMN13] on stochastically forced flows with viscosity

3

analysis is devoted to the proof of existence of martingale solutions to theparabolic approximation (3.1) which is not a classical point here. What ischallenging, and differ from the deterministic case, is the issue of positivityof the density. We solve this problem by using the uniformization effectsof parabolic equations with drifts and a bound given by the entropy, in thespirit of Mellet, Vasseur[MV09], cf. Theorem A.1. We will give more detailsabout the main problematic of the paper in Section 2.4, after our frameworkhas been introduced more precisely.

2 Notations and main result

2.1 Stochastic forcing

Our hypotheses on the stochastic forcing term Φ(ρ, u)W (t) are the followingones. We assume that W =

∑k≥1 βkek where the βk are independent brow-

nian processes and (ek)k≥1 is a complete orthonormal system in a Hilbertspace U. For each ρ ≥ 0, u ∈ R, Φ(ρ, u) : U→ L2(T) is defined by

Φ(ρ, u)ek = σk(·, ρ, u) = ρσ∗k(·, ρ, u), (2.1)

where σ∗k(·, ρ, u) is a 1-periodic continuous function on R. More precisely,we assume σ∗k ∈ C(Tx × R+ × R), with the bound

G∗(x, ρ, u) :=

(∑k≥1|σ∗k(x, ρ, u)|2

)1/2

≤ D0

[1 + u2 + ργ−1

]1/2, (2.2)

and A revoir

G∗j (x, ρ, u) :=

(∑k≥1|∇jρ,uσ∗k(x, ρ, u)|2

)1/2

≤ ζ0(1 + ρ

γ−12−j), (2.3)

where D0, ζ0 are some non-negative constants, x ∈ T, ρ ≥ 0, u ∈ R andj ∈ 1, 2.

We define the auxiliary space U0 ⊃ U via

U0 =

v =

∑k≥1

αkek;∑k≥1

α2k

k2<∞

,

endowed with the norm

‖v‖2U0=∑k≥1

α2k

k2, v =

∑k≥1

αkek.

Note that the embedding U → U0 is Hilbert-Schmidt. Moreover, trajectoriesof W are P-a.s. in C([0, T ];U0) (see Da Prato, Zabczyk [DPZ92]). specifywhere it is used.

4

2.2 Notations

We denote by

U =

(ρq

), A(U) =

(q

q2

ρ + p(ρ)

), q = ρu, (2.4)

the 2-dimensional unknown and flux of the conservative part of the problem.We also set

Φ∗(U) = ρ−1Φ(U), ψk(U) =

(0

σk(U)

), Ψ(U) =

(0

Φ(U)

).

Note that Φ∗ is well defined by (2.1) and that, with the notation above,(1.1) can be more concisely rewritten as the following stochastic first-ordersystem

dU + ∂xA(U)dt = Ψ(U)dW (t). (2.5)

We will also set, for ρ ≥ 0,

G(U) = ρG∗(U). (2.6)

If E is a space of real-valued functions on T, we will denote U(t) ∈ E insteadof U(t) ∈ E × E when this occur (with E = W 1,∞(T) or E = L∞(T) forexample in Definition ??, Theorem ??, etc.)

We denote by P the predictable σ-algebra on Ω × [0, T ] generated by (Ft)and we denote by P2 the completion of P ⊗ B(T), where B(T) is the Borelσ-algebra on T.

We will also use the following notation in various estimates below:

A = O(1)B,

where A,B ∈ R+, with the meaning A ≤ CB for a constant C ≥ 0. Ingeneral, the dependence of C over the data and parameters at stake will bedetailed, see for instance Proposition 3.7 below.

2.3 Entropy Solution

In relation with the kinetic formulation for (1.1) in [LPT94b], there is afamily of entropies

η(U) =

∫Rg(ξ)χ(ρ, ξ − u)dξ, with q = ρu, (2.7)

for (1.1), where

χ(U) = cλ(ρ2θ − u2)λ+, λ =3− γ

2(γ − 1), cλ =

(∫ 1

−1(1− z2)λ+ dz

)−1,

5

sλ+ := sλ1s>0. Indeed, if g ∈ C2(R) is a convex function, then η is of classC2 on the set

U :=

U =

(ρq

)∈ R2; ρ > 0

and η is a convex function of the argument U. Formally, by the Ito Formula,solutions to (1.1) satisfy

dEη(U) + EH(U)xdt =1

2E∂2qqη(U)G2(U)dt, (2.8)

where the entropy flux H is given by

H(U) =

∫Rg(ξ)[θξ + (1− θ)u]χ(ρ, ξ − u)dξ, with q = ρu. (2.9)

Note that, by a change of variable, we also have

η(U) = ρcλ

∫ 1

−1g

(q

ρ+ zρ(γ−1)/2

)(1− z2)λ+dz (2.10)

and

H(U) = ρcλ

∫ 1

−1g

(q

ρ+ zρ(γ−1)/2

)(q

ρ+γ − 1

2zρ(γ−1)/2

)(1− z2)λ+dz.

(2.11)In particular, for g(ξ) = 1 we obtain the density η0(U) = ρ. To g(ξ) = ξcorresponds the impulsion η(U) = q and to g(ξ) = 1

2ξ2 corresponds the

energy

ηE(U) =1

2ρu2 +

κ

γ − 1ργ . (2.12)

Note the form of the energy, in particular the fact that the hypothesis (2.2)on the noise gives a bound

G2(U) =∑k≥1|Φ(ρ, u)ek|2 ≤ ρD]

0(η0(U) + ηE(U)), (2.13)

for a constant D]0 depending on D0 and γ (recall that η0(U) := ρ). If (2.8) is

satisfied with an inequality ≤, then formally (2.2) and the Gronwall Lemmagive a bound on E

∫T(η0 + ηE)(U)(t)dx in terms of E

∫T (η0 + ηE)(U)(0)dx.

Indeed, we have ∂2qqηE(U) = 1ρ , which gives

E∂2qqηE(U)G2(U) ≤ D]0E(η0(U) + ηE(U)),

where C is a constant.

We will prove rigorously uniform bounds for approximate parabolic solutionsin Section 3.1.2. The above formal computations are however sufficient forthe moment to introduce the following definition.

6

Definition 2.1 (Entropy solution). Let ρ0, u0 ∈ L∞(T) with ρ0 ≥ 0 a.e.. AP2-measurable function

U : Ω× (0, T )× T→ R+ × R

is said to be an entropy solution to (1.1) with initial datum U0 if

1. we have

Φ(U) ∈ L2(Ω× [0, T ],P, dP× dt;L2(U;L2(T))

), (2.14)

where L2(U;L2(T)) is the space of Hilbert-Schmidt operators from Uinto L2(T),

2. the bound

E ess sup0≤t≤T

∫Tη(U(t, x))dx < +∞, (2.15)

is satisfied for η = ηE, the energy defined in (2.12),

3. for any (η,H) given by (2.7)-(2.9), where g ∈ C2(R) is convex andsubquadratic2, for all t ∈ (0, T ], for all nonnegative ϕ ∈ C1(T), andnonnegative α ∈ C1

c ([0, t)), we have∫ t

0

⟨η(U)(s), ϕ

⟩α′(s) +

⟨H(U)(s), ∂xϕ

⟩α(s) ds

+

∫ t

0

⟨G2(U)∂2qη(U), ϕ

⟩α(s) ds+

⟨η(U0), ϕ

⟩α(0)

+∑k≥1

∫ t

0

⟨σk(U)∂qη(U), ϕ

⟩α(s) dβk(s) ≥ 0, (2.16)

P-almost surely.

Remark 2.2. An entropy solution U is not defined as a process. It is indeeddifficult to specify a Banach space in which the process U(t) would evolve.Indeed the natural bounds for q are in Lp(ν) spaces where the measure νdepends on ρ. Actually, even if we had found such a Banach space X, wewould still have to consider entropy solutions U as equivalence classes in,say, L1(Ω × [0, T ],P, dP × dt;X), for the simple reason that the entropysolutions that we obtain are issued from a limiting process that does not giveconvergence for every time t. Certainly, if we knew that a

U ∈ L1(Ω× [0, T ],P, dP× dt;X)

had a representative with some continuity property in time, this one maybe considered as a stochastic process in its own. However, we do not knowhow to obtain such continuity properties for entropy solutions to (1.1); thisis related to the lack of techniques for proving uniqueness.

2in the sense that g satisifes (5.1)

7

Remark 2.3. By (2.14), the stochastic integral t 7→∫ t0 Φ(U)(s)dW (s) is a

well defined process taking values in L2(T) (see [DPZ92] for detailed con-struction). There is a little redundancy here in the definition of entropy so-lutions since, apart from the predictability, the integrability property (2.14)will follow from (2.2) and the bounds (2.15), cf. (2.13).

Remark 2.4. We will construct (martingale) entropy solutions which sat-isfy the entropy bound (2.15) for all entropy η of the form (2.7) with g convexand C2 satisfying a polynomial bound

|g(ξ)| ≤ C(1 + |ξ|m), m ∈ N,

for some constant C ≥ 0. In particular the solutions will satisfy the followingbounds (with weight ρ):

E ess sup0≤t≤T

∫T1

(|u|α + ρβ)ρdx < +∞,

for all α, β ≥ 0. We will also obtain Lp bounds of the form

ess sup0≤t≤T

E‖u(t)‖Lp(T) + ess sup0≤t≤T

E∥∥∥|ρ(t)|

γ−12

∥∥∥Lp(T)

< +∞. (2.17)

PRECISIONS about this bound, refer to where it is proved

Remark 2.5. If we take (for g(ξ) = 1 and ξ respectively)

(η(U), H(U)) = (ρ, q), (η(U), H(U)) = (q,q

ρ+ p(ρ)),

we can deduce from (2.16) the weak formulation of (1.1).

Definition 2.6 (Martingale solution). Let ρ0, u0 ∈ L∞(T) with ρ0 ≥ 0 a.e.A martingale solution to (1.1) with initial datum U0 is a multiplet

(Ω,F ,P, (Ft),W,U),

where (Ω,F ,P) is a probability space, with filtration (Ft) satisfying the usualconditions, and W a (Ft)-cylindrical Wiener process, and (U(t)) defines,according to Definition 2.1, an entropy solution to (1.1) with initial datumU0.

Theorem 2.7 (Main result). Assume that the structure and growth hypothe-ses (2.2), (2.3) on the noise are satisfied. Let ρ0, u0 ∈ L∞(T) with ρ0 ≥ 0a.e. Then there exists a martingale solution to (1.1) with initial datum U0.

8

2.4 Organization of the paper and main problematic

The paper is organized as follows. In Section 3, we prove the existence ofmartingale solutions to the parabolic approximation of Problem (1.1). Thisis done by a splitting method. One difficulty here is to obtain gradient es-timates on high moments of the solution (cf. Section 3.1.4) since no L∞

bounds are satisfied due to the stochastic perturbation. An other difficultyis to obtain some positivity results on the density: we need quantitative es-timates, cf. Section 3.1.5. To that purpose we use De Giorgi type estimatesin a way developed by Mellet and Vasseur in [MV09]: this is the subjectof Appendix A. Once the existence of martingale solution to the parabolicapproximation of Problem (1.1) has been proved, we want to take the limiton the regularizing parameter to obtain a martingale solution to (1.1). Asin the deterministic case [DiP83a, DiP83b, LPS96], we use the concept ofmeasure-valued solution (Young measure) to achieve this. In Section 4 wedevelop the tools on Young measure (in our stochastic framework) whichare required. This is taken in part (but quite different) from Section 4.3in [FN08]. We also use the probabilistic version of Murat’s Lemma from[FN08, Appendix A], to identify the limiting Young measure. This is thecontent of Section 5, which requires two other fundamental tools: the analy-sis of the consequences of the div-curl lemma in [LPS96, Section I.5] and anidentification result for densely defined martingales from [Hof13b, AppendixA]. We obtain then the existence of a martingale solution to (1.1). In Sec-tion 6 we conclude on some open questions about Problem (1.1) (uniquenessinlaw, existence of an invariant measure). As explained above, we need atsome point some bounds from below on solutions to (1-dimensional here)parabolic equations, which are developed in Appendix A. We also needsome regularity results, with few variations, on the (1-dimensional) heatsemi-group, and those are given in Appendix B.

Acknowledgements

We thank warmly Martina Hofmanova, for her help with Section 3, andFranco Flandoli, who suggested us the use of the splitting method in Sec-tion 3 (cf. Theorem ??). We also thank an anonymous referee, whoseearnest work helped us to improve our paper.

3 Parabolic Approximation

Let ε > 0. In this section, we prove the existence of a solution to thefollowing problem.

9

dUε + ∂xA(Uε)dt = ε∂2xUεdt+ Ψ(Uε)dW (t), (3.1a)

Uε|t=0 = Uε

0. (3.1b)

Recall that U and A(U) are defined by

U =

(ρq

), A(U) =

(q

q2

ρ + p(ρ)

).

Definition 3.1 (Pathwise solution to the parabolic approximation). LetUε

0 ∈ (W 1,∞(T))2 and T > 0. A process (Uε(t))t∈[0,T ] with values in(L2(T))2 is said to be a pathwise solution to (3.1) if it is a predictable processsuch that

1. there exists an adapted random process (cε(t)) with values in (0,+∞)such that, almost surely,

ρε, qε ∈ C([0, T ];W 1,2(T)), ρε ∈ Lγ(QT ), ρε ≥ cε a.e. in QT ,(3.2)

2. for all test function ϕ ∈ C∞(T;R2), almost surely, the following equa-tion is satisfied⟨

Uε(t), ϕ⟩

=⟨U0, ϕ

⟩−∫ t

0

⟨∂xA(Uε)− ∂2xUε, ϕ

⟩ds

+

∫ t

0

⟨Ψ(Uε) dW ε(s), ϕ

⟩, t ∈ [0, T ].

We will prove the existence of pathwise solutions to the parabolic stochasticproblem (3.1) satisfying uniform (or weighted) estimates with respect to ε.If η is an entropy function given by (2.7) with a convex function g of classC2, we denote by

Γη(U) =

∫Tη(U(x))dx,

the total entropy of a function U : T→ R2.

Theorem 3.2 (Existence of pathwise solution to (3.1)). Let Uε0 ∈ (W 1,∞(T))2

satisfy ρε0 ≥ c0 a.e. in T, for a positive constant c0. Then the problem (3.1)admits a pathwise solution Uε, which has the following property:

1. it satisfies the following Lp estimates: for all p ≥ 1,

sup0≤t≤T

E‖uε(t)‖Lp(T) + sup0≤t≤T

E∥∥∥|ρε(t)| γ−1

2

∥∥∥Lp(T)

= O(1), (3.3)

where O(1) depends on γ, on the constant D0 in (2.2), on p, ‖u0‖L∞,‖ρ0‖L∞ and T ,

10

2. it satisfies the following gradient estimates: for all α, β ≥ 0,

εE∫∫

QT

(|uε|α|ρε|γ−2 + |ρε|γ−2+β

)|∂xρε|2dxdt = O(1), (3.4)

and

εE∫∫

QT

(|uε|αρε + |ρε|1+β

)|∂xuε|2dxdt = O(1), (3.5)

where O(1) depends on T , γ, on the constant D0 in (2.2) and on afinite number of quantities EΓηj (U

ε0), for entropies ηj associated to

various power-like functions |ξ|rj , for given rj’s depending on α, β.

Besides, Uε satisfies the Ito formula

dη(Uε) + ∂xH(Uε)dt

=ε∂2xη(Uε)dt+ ∂qη(Uε)Φ(Uε)dW (t)− εη′′(Uε) · (Uεx,U

εx)dt

+1

2|G(Uε)|2∂2qqη(Uε)dt. (3.6)

for all entropy - entropy flux couple (η,H) where η is of the form (2.7) witha convex function g of class C2 such that g′′ is bounded on R.

Note that (3.6) has the following meaning: for all test-function ϕ ∈ C∞(T),we have

〈η(Uε), ϕ〉(t) = 〈η(Uε0), ϕ〉(t) +

∫ t

0〈H(Uε(s)), ∂xϕ〉 − ε〈η(Uε(s)), ∂2xϕ〉ds

−∫ t

0ε〈η′′(Uε) · (Uε

x,Uεx), xϕ〉ds+

∑k≥1

∫ t

0〈∂qη(Uε)σk(U

ε), ϕ〉dW (s)

+1

2

∫ t

0〈|G(Uε)|2∂2qqη(Uε), ϕ〉ds,

for all t ∈ [0, T ], P-almost surely.

To prove the existence of such pathwise solutions, we will prove first theexistence of martingale solution and then use a Yamada-Watanabe argument(see [?]) to conclude. This means that we have to prove a result of pathwiseuniqueness, which is the content of the following theorem.

Theorem 3.3 (Uniqueness of pathwise solution to (3.1)). Let Uε0 ∈ (W 1,∞(T))2

satisfy ρε0 ≥ c0 a.e. in T, for a positive constant c0

Remark 3.4. ffff

11

3.1 Solution to the parabolic problem

3.1.1 Time splitting

Assume, without loss of generality, that ε = 1. To prove the existence of asolution to (3.1), we perform a splitting in time. Let τ > 0. Set tk = kτ ,k ∈ N. We solve alternatively the deterministic, parabolic part of (3.1) ontime intervals [t2k, t2k+1) and the stochastic part of (3.1) on time intervals[t2k+1, t2k+2), i.e.

• for t2k ≤ t < t2k+1,

∂tUτ + 2∂xA(Uτ ) = 2∂2xU

τ in Qt2k,t2k+1, (3.7a)

Uτ (t2k) = Uτ (t2k−) in T, (3.7b)

• for t2k+1 ≤ t < t2k+2,

dUτ =√

2Ψτ (Uτ )dW (t) in Qt2k+1,t2k+2, (3.8a)

Uτ (t2k+1) = Uτ (t2k+1−) in T. (3.8b)

Note that we took care to speed up the deterministic equation (3.7a) by afactor 2 and the stochastic equation (3.8a) by a factor

√2, this rescaling

procedure should yield a solution Uτ consistent with the solution Uε=1 to(3.1) when τ → 0. In (3.8) we have also regularized the coefficient Φ intoa coefficient Φτ . Precisely, we assume that, for a finite integer Kτ ≥ 1, foreach ρ ≥ 0, u ∈ R, we have

Φτ (ρ, u)ek = στk(·, ρ, u)1k≤Kτ = ρστ,∗k (·, ρ, u)1k≤Kτ , (3.9)

where στ,∗k (·, ρ, u) is a 1-periodic continuous function on R which is boundedtogether with all its derivatives. We also assume that the equivalent of thebounds (2.2)-(2.3) are satisfied uniformly in τ :

Gτ,∗(x, ρ, u) :=

( Kτ∑k=1

|στ,∗k (x, ρ, u)|2)1/2

≤ D0

[1 + u2 + ργ−1

]1/2, (3.10)

and A revoir

Gτ,∗j (x, ρ, u) :=

( Kτ∑k=1

|∇jρ,uστ,∗k (x, ρ, u)|2

)1/2

≤ ζ0(1 + ρ

γ−12−j), (3.11)

where x ∈ T, ρ ≥ 0, u ∈ R and j ∈ 1, 2. In what follows we often do as ifΦτ = Φ...

Let us define the following indicator functions

1det =∑k≥0

1[t2k,t2k+1), 1sto = 1− 1det, (3.12)

which will be used to localize various estimates below.

12

Definition 3.5 (Pathwise solution to the splitting approximation). LetU0 ∈ (W 1,2(T))2 and T > 0. A process (U(t))t∈[0,T ] with values in (L2(T))2

is said to be a pathwise solution to (3.7)-(3.8) if it is a predictable processsuch that

1. there exists an adapted random process (cτ (t)) with values in (0,+∞)such that, almost surely,

ρ, q ∈ C([0, T ];W 1,2(T)), ρ ∈ Lγ(QT ), ρ ≥ cτ a.e. in QT , (3.13)

2. for all test function ϕ ∈ C∞(T;R2), the following equation is satisfied:P-almost surely, for all t ∈ [0, T ],

⟨U(t), ϕ

⟩=⟨U0, ϕ

⟩−2

∫ t

01det(s)

⟨∂xA(U(s))− ∂2xU(s), ϕ

⟩ds

+√

2

∫ t

01sto(s)

⟨Ψτ (U(s)) dW (s), ϕ

⟩. (3.14)

Proposition 3.6 (Existence of pathwise solution to the splitting approx-imation). Let T > 0, let U0 ∈ (W 2,2(T))2 such that ρ0 ≥ c0 a.e. in Tfor a given constant c0 > 0. Then there exists a pathwise solution Uτ to(3.7)-(3.8), which has the following additional property:

1. almost surely, Uτ ∈ C([0, T ];W 2,1(T)),

2. for all p ≥ 1, E supt∈[0,T ] ‖Uτ (t)‖pW 1,2(T) < +∞,

3. Uτ satisfies the following Ito Formula.

Proof. The deterministic problem (3.7) is solved in [LPS96]: for Lipschitzcontinuous initial data (ρ0, q0) with an initial density ρ0 uniformly positive,say ρ0 ≥ c0 > 0 on T, the Problem (3.7) admits a unique solution U in theclass of functions

U ∈ L∞(0, τ,W 1,∞(T))2 ∩ C([0, τ ];L2(T))2; ρ ≥ c1 on T× [0, τ ].

Here c1 > 0 is a constant depending continuously on τ and on c0, ‖ρ0‖L∞(T),‖q0‖L∞(T) (see Theorem A.1 and Remark A.3 in this paper for more de-tail about this positivity result). By [DiP83b, Section 4], we have U ∈L∞(0, τ ;W 2,2(T)). It follows that ∂tU ∈ L∞(0, τ ;L2(T)) and, therefore,U ∈ C([0, τ ];W 1,2(T)). This gives the properties 1. and 2. of the Proposi-tion for T = t1. Note, besides, that, by Theorem 3.3 and Remark 3.4 below,the map U0 7→ U is Lipschitz continuous from L2(T)2 into C([0, τ ];L2(T)2).To finish with the first step of resolution of (3.7) for k = 0, let us set

cτ (t) = c1, t ∈ [0, t1).

13

On [t1, t2) we solve now in a second step the stochastic problem (3.8). It isan ordinary stochastic differential equation that does not act on the density.Recall that the coefficients of the noise in (3.9) are assumed to be functionswith bounded derivatives at all orders. Since x 7→ ρτ (t1, x) is smooth, wemay rewrite the second equation of (3.8) as

dq =Kτ∑k=1

gk(x, q)dβk(t), (3.15)

where gk ∈ C∞b (T×R). The existence of a solution to (3.15) on (t1, t2) withinitial datum q(t1, x) at t = t1 is ensured by a classical fixed point theorem,in the space of adapted functions

q ∈ C([t1, t2];L2(Ω× T)).

Since ρ is unchanged on the time interval [t1, t2), we set

cτ (t) = cτ (t1), t ∈ [t1, t2).

Note that, as such, (cτ (t))t∈[0,t2) is F0-measurable. In particular, this is anadapted process. By differentiating once with respect to x in (3.15), weobtain

d(∂xq) =Kτ∑k=1

(∂xgk(x, q) + ∂qgk(x, q)(∂xq)

)dβk(t).

By the Ito Formula and the Gronwall Lemma, it follows that

supt∈[t1,t2]

E‖∂xq‖pLp(T) ≤ CE‖∂xq(t1)‖pLp(T), p ≥ 2, (3.16)

where the constant C depends on the function gk’s, on p and on τ . Bydifferentiating again in (3.15), we have

d(∂2xq) =Kτ∑k=1

(∂2xgk(x, q) + 2∂2x,qgk(x, q)(∂xq) + ∂2qgk(x, q)|∂xq|2

+ ∂qgk(x, q)(∂2xq))dβk(t). (3.17)

Using (3.16) with p = 2 and p = 4, the Ito Formula and the GronwallLemma, we obtain

supt∈[t1,t2]

E‖∂2xq‖2L2(T) ≤ C(E‖∂2xq(t1)‖2L2(T)+E‖∂xq(t1)‖2L2(T)+E‖∂xq(t1)‖4L4(T)

),

(3.18)

14

where the constant C depends on the function gk’s and on τ . By the Doob’sMartingale Inequality, we have therefore

E supt∈[t1,t2]

∥∥∥∫ t

t1

∂qgk(x, q(s))∂2xq(s)dβk(s)

∥∥∥2L2(T)

≤2E∥∥∥∫ t2

t1

∂qgk(x, q(s))∂2xq(s)dβk(s)

∥∥∥2L2(T)

≤C(E‖∂2xq(t1)‖2L2(T) + E‖∂xq(t1)‖2L2(T) + E‖∂xq(t1)‖4L4(T)

).

Returning to (3.17), we deduce that

E supt∈[t1,t2]

‖∂2xq‖2L2(T) ≤ C(E‖∂2xq(t1)‖2L2(T)+E‖∂xq(t1)‖2L2(T)+E‖∂xq(t1)‖4L4(T)

).

(3.19)By a similar argument, using Doob’s Martingale Inequality, we can similarlyimprove (3.16) into

E supt∈[t1,t2]

‖∂xq‖pLp(T) ≤ CE‖∂xq(t1)‖pLp(T), p ≥ 2. (3.20)

Note that differentiation in (3.15) has to be justified. The argument is stan-dard: to obtain a solution to (3.15) which satisfies (3.20) and (3.19), wesimply prove existence by using a fixed-point in a smaller space, incorporat-ing the bounds (3.20) and (3.19). By (3.19), we have U(t2−) ∈ W 2,2(T).The initial datum Uτ (t2−) is therefore admissible with regard to the re-solution of the deterministic problem (3.7) on Qt2,t3 using identical ar-guments to those given at the beginning of the proof for the resolutionon the strip Q0,t1 . Furthermore, the function Uτ (t2−) is Ft2-measurable.Since Uτ (t2−) 7→ (Uτ (t))t∈(t2,t3) is Lipschitz continuous from L2(T)2 intoC([t2, t3];L

2(T)2), the random variable Uτ (t) is Ft2-measurable for everyt ∈ [t2, t3]. In particular, Uτ (t) is adapted on [t2, t3]. Moreover, there existsc2 > 0 depending continuously on τ and Uτ (t2−) such that ρτ ≥ c2 a.e. onQt2,t3 . In particular, c2 is Ft2-measurable and continuing cτ on [t2, t3) bysetting

cτ (t) = c2, t ∈ [t2, t3),

we obtain an adapted process. We can then go on with the resolution of theProblem 3.8 on Qt2k+1,t2k+2

, k = 1, as was done in the case k = 0, and soon. In this way we construct an adapted process (Uτ (t)) satisfying the firstitem of Definition 3.5. Besides Uτ ∈ C([0, T ];W 1,2(T)) almost surely andsatisfies item 2. of the Proposition. In particular, since Uτ has almost surecontinuous trajectories in L2(T)2, this is a predictible process. Eventually,to show that (3.14) is satisfied, observe that it is satisfied on each interval[tj , tj+1], [tK , t], where K is such that tK ≤ t < tK+1. By adding these mildformulations, we obtain (3.14). Ito.

15

3.1.2 Entropy bounds

If η ∈ C(R2) is an entropy and U : T→ R2, let

Γη(U) :=

∫Tη(U(x))dx

denote the total entropy of U.

Proposition 3.7 (Entropy bounds). Let m ∈ N. Let ηm be the entropygiven by (2.7) with g(ξ) = ξ2m. Then the solution Uτ to (3.7)-(3.8) satisfiesthe estimate

E supt∈[0,T ]

Γη(Uτ (t)) + 2E

∫∫QT

1detη′′(Uτ ) · (Uτ

x,Uτx)dxdt = O(1), (3.21)

where the quantity denoted by O(1) depends on T , γ, on the constant D0 in(2.2), on m and on the initial quantities EΓη(U0) for η ∈ η0, ηm, η2m.

Proof. To prove Proposition 3.7 we will use the following result.

Lemma 3.8. Let m,n ∈ N. We have

ρ(u2m + ρ2mγ−12 ) = O(1)ηm(U), ηm(U) = O(1)

[ρ(u2m + ρ2m

γ−12 )],

(3.22)where O(1) depends on m; we have

ηm(U) · ηn(U) = O(1) [ρηm+n(U)] , (3.23)

where O(1) depends on m and n and

ηn(U) = O(1) [η0(U) + ηm(U)] , (3.24)

where O(1) depends on m and n if 0 ≤ n ≤ m.

Lemma 3.8 is obtained simply by repeated applications of the Young Inequal-ity. Using an entropy identity for (3.7) and Ito Formula for the evolution by(3.8), we have

EΓηm(Uτ (t)) + 2E∫∫

Qt

1detη′′(Uτ ) · (Uτ

x,Uτx)dxds = EΓηm(Uτ

0) +ERηm(t),

where

Rηm(t) :=

∫∫Qt

1stoGτ (Uτ )2∂2qqηm(Uτ )dxds

is the Ito correction. If m = 0, then ∂2qqη = 0 and we obtain (note thedifference with (3.21) in the first term)

supt∈[0,T ]

EΓη0(Uτ (t)) + 2E∫∫

QT

1detη′′0(Uτ ) · (Uτ

x,Uτx)dxdt = O(1). (3.25)

16

To get an estimate on Rηm in the case m ≥ 1, we compute, by (2.10),

∂2qqηm(U) =1

ρcλ

∫ 1

−1g′′(u+ zρ(γ−1)/2

)(1− z2)λ+dz. (3.26)

Since g(ξ) = ξ2m, and (a + b)α ≤ 2α(aα + bα), a, b ∈ R, α = 2m − 2, weobtain, by (3.22),

∂2qqηm(U) = O(1)

[1

ρ· ηm−1(U)

ρ

].

Then we deduce from (2.13) and by (3.23) (note that for the energy ηE , wehave ηE = η1 with our notations), the estimate

Gτ (Uτ )2∂2qqηm(U) = O(1) [ηm−1(U) + ηm(U)] .

By (3.24), it follows that

Gτ (Uτ )2∂2qqηm(U) = O(1) [η0(U) + ηm(U)] .

We deduce that

ERηm(t) = O(1)

[∫ t

0E(Γηm(Uτ (s)) + Γη0(Uτ (s)))ds

](3.27)

and, by Gronwall’s Lemma and (3.25), we obtain

supt∈[0,T ]

EΓηm(Uτ (t)) + 2E∫∫

QT

1detη′′m(Uτ ) · (Uτ

x,Uτx)dxdt = O(1). (3.28)

To prove (3.21), we have to take into account the noise term: we have

0 ≤ Γηm(Uτ (t)) = Γηm(Uτ0) +Mηm(t) +Rηm(t)−Dηm(t) (3.29)

where

Mηm(t) =√

2∑k≥1

∫ t

01sto(s)〈στk(Uτ (s)), ∂qηm(Uτ (s))〉L2(T)dβk(s)

and

Dηm(t) = 2

∫∫Qt

1detηm′′(Uτ ) · (Uτ

x,Uτx)dxds.

Since Dηm ≥ 0, we obtain

0 ≤ Γηm(Uτ (t)) ≤ Γηm(Uτ0) +Mηm(t) +Rηm(t).

Similarly as for (3.27), we have

E supt∈[0,T ]

|Rηm(t)| = O(1)

[∫ T

0E(Γηm(Uτ (s)) + Γη0(Uτ (s)))ds

],

17

and therefore, by (3.28), the last term Rηm satisfies the bound

E supt∈[0,T ]

|Rηm(t)| = O(1).

By the Doob’s Martingale Inequality, we also have

E supt∈[0,T ]

|Mηm(t)| ≤ CE

∫ T

0

∑k≥1〈στk(Uτ (s)), ∂qηm(Uτ (s))〉2L2(T) ds

1/2

≤ CE(∫∫

QT

Gτ (Uτ )2|∂qηm(Uτ )|2 dxds)1/2

for a given constant C. Now we have

∂qηm(U) = 2mcλ

∫ 1

−1

(u+ zρ(γ−1)/2

)2m−1(1− z2)λ+dz.

From (3.22) follows |∂qηm(U)| = O(1)[η2m−1(U)

ρ2

]and by (3.23), (2.13) and

(3.24), we deduce

Gτ (U)2|∂qηm(U)|2 = O(1)[ρ−1(η0(U) + η1(U))η2m−1

]= O(1) [η0(U) + η2m(U)] .

Using (3.25), we obtain then

E supt∈[0,T ]

|Mη(t)| = O(1).

This concludes the proof of the proposition.

Corollary 3.9 (Bounds on the moments). Let m ∈ N. Let ηm be the entropygiven by (2.7) with g(ξ) = ξ2m. Then, the solution Uτ to (3.7)-(3.8) satisfies:

E supt∈[0,T ]

∫T

(|uτ |2m + |ρτ |m(γ−1)

)ρτdx = O(1), (3.30)

where O(1) depends on T , γ, on the constant D0 in (2.2), on m and on theinitial quantities EΓη(U0) for η ∈ η0, η2m.

If we introduce the measure

µτt : ϕ 7→∫Tϕ(x)ρτ (t, x)dx,

(which depend on the solution), we thus obtain estimates on

E∫T|uτ (t, x)|αdµτt (x) and E

∫T|ρτ (t, x)|βdµτt (x),

for all α, β ≥ 0. It is then straightforward, using Young’s inequality andHolder’s inequality to obtain some bounds on the moments of the entropyand entropy flux:

18

Proposition 3.10 (Bounds on the moments of the entropy). Let (η,H) begiven by (2.7)-(2.9), where g ∈ C2(R) is convex and subquadratic. Let ηmbe the entropy given by (2.7) with g(ξ) = ξ2m. Then, the solution Uτ to(3.7)-(3.8) satisfies the estimate

E supt∈[0,T ]

∫T

[|η(Uτ (t))|p + |H(Uτ (t))|p] dx = O(1), (3.31)

for all p ≥ 1, where O(1) depends on T , γ, on the constant D0 in (2.2), onp and on the initial quantities EΓη(U0) for η ∈ η0, ηp.

3.1.3 Lp estimates

The entropy estimates give Lp estimates on u and ρ with a weight ρ, seeEquation (3.30). The following estimates (Lp estimates without the weightρ) are an analog to the classical L∞ estimate for the deterministic equation(cf. [DiP83b, Section 4.]).

Proposition 3.11 (Lp estimates). Let Uτ be the solution to (3.7)-(3.8).Let p ≥ 1. Then

sup0≤t≤T

E‖uτ (t)‖Lp(T) + sup0≤t≤T

E∥∥∥|ρτ (t)|

γ−12

∥∥∥Lp(T)

= O(1), (3.32)

where O(1) depends on γ, on the constant D0 in (2.2), on p, ‖u0‖L∞, ‖ρ0‖L∞and T .

Proof. Let us introduce the Riemann invariants

zτ = uτ − |ρτ |θ, wτ = uτ + |ρτ |θ.

By [DiP83b, Section 4.], we have

zτ (t) ≥ z(t2k), wτ ≤ w(t2k), (3.33)

for each t ∈ [t2k, t2k+1] and k ≥ 0.Let m ∈ 2N, m ≥ 4. For a ∈ R, set g+(ξ) = ξm+ and

η+(U, a) =

∫Rg+(ξ − a)χ(ρ, ξ − u)dξ

= cλρ

∫ 1

−1(u+ zρ

γ−12 − a)m+ (1− z2)λ+dz.

Let Γ(t, a) be defined by

Γ(t, a) =

∫Tη+(Uτ (t, x), a)dx.

19

By the Ito formula, we have

Γ(t, a) = Γ(0, a) +3∑

α=1

Kτ∑k=1

∫ t

0hα(s)ζα,k(s, a)dY α,k(s),

almost surely, where the continuous semi-martingales Y α,k(s) are defined by

Y 1,k(t) = Y 2,k(t) = t, Y 3,k(t) = βk(t),

and the integrands ζα,k(s, a) are given by

ζ1,k(s, a) =− 2

∫TD2

Uη+(Uτ (s), a) · (Uτx(s),Uτ

x(s))dx δk,1

ζ2,k(s, a) =

∫Tστk(Uτ (s))2∂2qqη+(Uτ (s), a)dx,

and

ζ3,k(s, a) =√

2

∫Tστk(Uτ (s))∂qη+(Uτ (s), a)dx,

whileh1(s) = 1det(s), h2(s) = h3(s) = 1sto(s).

Lemma 3.12. Almost surely, the function Γ(t, a) is twice continuously dif-ferentiable with respect to a, with derivatives continuous in (t, a); the func-tions ζα,k(t, a) are twice continuously differentiable in a and continuous in(t, a). Besides, for each a, ζα,k(t, a) is an adapted process.

Proof. The lemma follows from the a.s. regularity Uτ ∈ C([0, T ];W 1,2(T)),cf. item 1. in Proposition (3.6).

Consider now the martingale

a(t) = a0 +

Kτ∑k=1

∫ t

01sto(s)a

k(s)dβk(s).

We assume that, for each k ≥ 1, ak is an adapted process with a.s. continu-ous trajectories. By the Ito-Wentzell formula [?, Theorem 8.1], we have

Γ(t, a(t)) =Γ0(a(0)) +

3∑α=1

∑k≥1

∫ t

0hα(s)ζα,k(s, a(s))dY α,k(s) (3.34)

+

∫ t

0

∂Γ

∂a(s, a(s))da(s) (3.35)

+∑k≥1

∫ t

0hα(s)

∂ζ3,k

∂a(s, a(s))ak(s)ds (3.36)

+1

2

∑k≥1

∫ t

0

∂2Γ

∂a2(t, a(t))|ak(s)|2ds, (3.37)

20

almost surely. Note that [?, Theorem 8.1] is stated for functions continuousin t instead of piecewise continuous; we obtain our result by using the Ito-Wentzell formula on each interval [tj , tj+1], tj ≤ t, and by summing theresult over j. Now, we use the following formula

∂qη+(U, a) =1

ρ∂uη+(U, a), στk(U) = ρσ∗,τk (U), ∂uη+(U, a) = −∂aη+(U, a)

and we remark that ζ1,k(s, a) is non-positive to obtain

Γ(t, a(t)) ≤Γ0(a(0)) +

Kτ∑k=1

∫ t

0

∫T

1sto(s)∂2uuη+(Uτ (s, x), a(s))

×[στ,∗k (Uτ (s, x))2 +

1

2|ak(s)|2 −

√2στ,∗k (Uτ (s, x))ak(s)

]dxds

(3.38)

+

Kτ∑k=1

∫ t

0

∫T

1sto(s)∂uη+(Uτ (s, x), a(s))

×[√

2στ,∗k (Uτ (s, x))− ak(s)]dxdβk(s) (3.39)

almost surely. The term (3.38) is deduced from the terms (3.34) (the partof the sum for α = 2), (3.36) and (3.37). The term (3.39) is deduced from(3.34) (the part of the sum for α = 3) and (3.35). Let us set now

ak(s) =√

2σ∗τk (Uτ (y, s)), a0 = ‖u+0 ‖L∞ + ‖ρ+0 ‖γ−12

L∞ , (3.40)

where y ∈ T is fixed and let us denote by

a+(t, y) = ‖u+0 ‖L∞ + ‖ρ+0 ‖γ−12

L∞ +√

2

Kτ∑k=1

∫ t

01sto(s)σ

∗τk (Uτ (y, s))dβk(s)

the resulting martingale. Let also (ρε) be an approximation of the unit onT. Note that, for all y ∈ T, ak is an adapted process with a.s. continuoustrajectories (cf. item 1. in Proposition (3.6)). We apply (3.38)-(3.39) withsuch values of ak and a0, multiply the result by ρε(x − y) and sum overy ∈ T; we have then∫

TΓ(t, a+(t, y))ρε(x− y)dy ≤ Aε(t) +Bε(t), (3.41)

almost surely, where

Aε(t) =Kτ∑k=1

∫ t

0

∫∫T×T

1sto(s)∂2uuη+(Uτ (s, x), a+(s, y))

×∣∣στ,∗k (Uτ (s, x))− στ,∗k (Uτ (s, y))

∣∣2 ρε(x− y)dxdyds,

21

and

Bε(t) =√

2Kτ∑k=1

∫ t

0

∫∫T×T

1sto(s)∂uη+(Uτ (s, x), a+(s, y))

×[στ,∗k (Uτ (s, x))− στ,∗k (Uτ (s, y))

]ρε(x− y)dxdydβk(s).

Indeed, with a0 given by (3.40), the first term Γ(0, a(0)) in (3.38)-(3.39)vanishes. Note that we have used the Fubini Theorem to rewrite Aε(t) andthe stochastic Fubini Theorem to rewrite Bε(t). Let us give some details forthe latter: we apply [DPZ92, Theorem 4.18]. Note that

Bε(T ) =

∫T

(∫ T

0b(t, y)dW (t)

)dy,

where

b(t, y)ek =√

21k≤Kτ

∫T

1sto(t)∂uη+(Uτ (t, x), a+(t, y))

×[στ,∗k (Uτ (t, x))− στ,∗k (Uτ (t, y))

]ρε(x− y)dx.

Then b is a measurable process from (Ω × (0, T ) × T,PT × B(T)) into(L0

2,B(L02)), where L0

2 is the space of Hilbert-Schmidt operators on U. In-deed, since only a finite number of the quantities b(t, y)ek do not vanish,it is sufficient to prove that each of them is a measurable process from(Ω× (0, T )×T,PT ×B(T)) into (R,B(R)). This follows from, first, the factthat

(U, U, a, y) 7→∫T∂uη+(U(x), a(x))

[στ,∗k (U(x))− στ,∗k (U)

]ρε(x− y)dx

is continuous from X := C2 × U × C(R)× T into R, where, we recall,

U =

U =

(ρq

)∈ R2; ρ > 0

,

and where we have set

C2 = U : T→ U ; ρ, u ∈ C(T) .

Secondly, by Proposition (3.6), the map

(t, y) 7→ (Uτ (t),Uτ (t, y), a+(t, y), y)

is measurable from (Ω× (0, T )×T,PT ×B(T) into (X,B(X)). To apply thestochastic Fubini Theorem, we have also to check the integrability condition∫

T|||b(·, ·, y)|||T dy < +∞, (3.42)

22

where

|||b(·, ·, y)|||2T =2

Kτ∑k=1

E∫ T

01sto(t)

∣∣∣∣∫T∂uη+(Uτ (t, x), a+(t, y))

×[στ,∗k (Uτ (t, x))− στ,∗k (Uτ (t, y))

]ρε(x− y)dx

∣∣2 dt.Condition (3.42) follows from the estimate∫T|||b(·, ·, y)|||2T dy ≤4

Kτ∑k=1

E∫ T

01sto(t)

∫∫T×T|∂uη+(Uτ (t, x), a+(t, y))|2

×[|στ,∗k (Uτ (t, x))|2 + |στ,∗k (Uτ (t, y))|2

]ρ2ε(x− y)dxdt.

(3.43)

Indeed, we have

|∂uη+(U, a)| ≤ mcλρ

∫ 1

−1(|u|+ |z|ρ

γ−12 )m−1(1− z2)λ+dz,

and therefore∫T|||b(·, ·, y)|||2T dy = O(1)

∫∫T×T

ρ2ε(x− y)dxdy = ε−1O(1),

where the term O(1) depends on Kτ , ‖σ∗,τk ‖L∞ (finite by hypothesis) andE‖(ρτ , uτ )‖L∞(QT ). Now we want to pass to the limit [ε→ 0] in (3.41). Wewill show that, up to extraction of subsequence in ε,

limε→0

Aε(t) = limε→0

Bε(t) = 0,

almost surely. Since Uτ is almost surely bounded and continuous on T ×(0, T )(cf. item 1. in Proposition (3.6)), we have,

|∂2uuη+(Uτ , a)| ≤ m(m− 1) cλρ

∫ 1

−1(|u|+ |z|ρ

γ−12 )m−2(1− z2)λ+dz = O(1),

almost surely, where O(1) depends on particular on ‖(ρτ , uτ )‖L∞(QT ). Itfollows that

Aε(t) = O(1)Kτ∑k=1

∫ t

0

∫∫T×T

∣∣στ,∗k (Uτ (s, x))− στ,∗k (Uτ (s, y))∣∣2 ρε(x− y)dxdyds

= o(1).

Furthermore, the Ito isometry formula gives

E|Bε(t)|2 =

Kτ∑k=1

E∫ t

01sto(s)c

kε(s)ds,

23

with

ckε(s) =

∣∣∣∣∫∫T×T

∂uη+(Uτ (s, x), a+(s, y))

×[στ,∗k (Uτ (s, x))− στ,∗k (Uτ (s, y))

]ρε(x− y)dxdy

∣∣2 .Since ‖σ∗,τk ‖L∞ = O(1) and∫∫

T×Tρε(x− y)dxdy = 1,

we have ckε(s) = O(1)(1 + ‖(ρτ , uτ )‖mL∞(QT )), a quantity that is integrable

with respect to (ω, s) by item 2. in Proposition (3.6). At fixed (ω, s), wealso have

limε→0

ckε(s) = 0

since ∂uη+(Uτ (t, x), a+(s, y)) is bounded and U τ continuous. By the domi-nated convergence theorem, we deduce that lim

ε→0E|Bε(t)|2 = 0. In particular,

up to a subsequence, Bε(t) → 0 almost surely. The convergence of the lefthand-side in (3.41) can be analysed in a similar way. In conclusion, at thelimit [ε→ 0] in (3.41), we obtain∫

Tη+(Uτ (t, x), a+(t, x))dx ≤ 0

almost surely, that is to say

uτ (t, x) + zρτ (t, x)γ−12 ≤ a+(t, x)

almost surely, for all (x, t) ∈ T × [0, T ] and all z ∈ [−1, 1]. In particular,summing over z ∈ [0, 1] gives

uτ (t, x) +1

2ρτ (t, x)

γ−12 ≤ a+(t, x)

almost surely, for all (x, t) ∈ T× [0, T ]. A similar work based on the entropy

η−(U, a) = cλρ

∫ 1

−1(u+ zρ

γ−12 − a)m− (1− z2)λ+dz

then gives

|uτ (t, x)|+ 1

2ρτ (t, x)

γ−12 ≤ a(t, x) (3.44)

almost surely, for all (x, t) ∈ T× [0, T ], where

a(t, x) = ‖u0‖L∞ + ‖ρ0‖γ−12

L∞ +√

2

Kτ∑k=1

∫ t

01sto(s)σ

τ,∗k (Uτ (s, x))dβk(s).

24

Let h(t, x) = |uτ (t, x)|+ 12ρ

τ (t, x)γ−12 denote the left hand-side of (3.44) and

let p ≥ 2. Note that (3.10) implies

Kτ∑k=1

|στ,∗k (Uτ (t, x))|2 = O(1)(1 + |h(t, x)|2), (3.45)

where O(1) depends on D0. By the Burkholder-Davis-Gundy Inequality, wededuce from (3.44) and (3.45) the following estimate:

E∫T|h(t, x)|pdx = O(1)

1 + E∫T

(Kτ∑k=1

∫ t

0|στ,∗k (Uτ (s, x))|2ds

)p/2dx

= O(1)

[1 + E

∫T

(∫ t

0|h(s, x)|2ds

)p/2dx

]

= O(1)

[1 + E

∫T

∫ t

0|h(s, x)|pdsdx

], (3.46)

where O(1) depends on D0, p, ‖u0‖L∞ , ‖ρ0‖L∞ and T . Using the GronwallLemma, (3.32) then follows from (3.46).

3.1.4 Gradient estimates

In Proposition 3.7 above, we have obtained an estimate on Uτx. In the

case where η = ηE is the energy (this corresponds to g(ξ) = 12ξ

2), somecomputations show that

η′′E(U) · (Ux,Ux) = κγ|ρ|γ−2|ρx|2 + ρ|ux|2. (3.47)

More generally, we have the following weighted estimates (see Corollary 3.14below).

Proposition 3.13 (Gradient bounds). Let m ∈ N. Let ηm be the entropygiven by (2.7) with g(ξ) = ξ2m. Then, the solution Uτ to (3.7)-(3.8) satisfiesthe estimate

E∫∫

QT

1det(t)G[2](ρτ , uτ )

[θ2|ρτ |γ−2|∂xρτ |2 + ρτ |∂xuτ |2

]dxdt

≤E∫∫

QT

1det(t)G[1](ρτ , uτ )

[2θ|ρτ |

γ−22 |∂xρτ | · |ρτ |1/2|∂xuτ |

]dxdt+O(1),

(3.48)

where

G[2](ρ, u) = cλ

∫ 1

−1g′′(u+ zρ

γ−12 )(1− z2)λ+dz,

G[1](ρ, u) = cλ

∫ 1

−1|z|g′′(u+ zρ

γ−12 )(1− z2)λ+dz,

25

and O(1) depends on T , γ, on the constant D0 in (2.2) and on the initialquantities EΓη(U0) for η ∈ η0, ηm, η2m.

Proof. We introduce the probability measure

dmλ(z) = cλ(1− z2)λ+dz

and the 2× 2 matrix

S =

(1 0u 1

),

which satisfies

∂xU = SW, W :=

(∂xρρ∂xu

). (3.49)

By (3.21), we then have∫ T

0E∫T

1det(t) 〈S∗η′′(Uτ )SW,W〉dxdt = O(1), (3.50)

where 〈·, ·〉 is the canonical scalar product on R2 and S∗ is the adjoint of Sfor this scalar product. We compute (recall that θ = γ−1

2 )

η′′(U) =1

ρ

∫R

[A(z)g′

(u+ zρθ

)+B(z)g′′

(u+ zρθ

)]dmλ(z),

where

A(z) =

(γ2−14 zρθ 0

0 0

), B(z) =

((−u+ θzρθ

)2 −u+ θzρθ

−u+ θzρθ 1

).

In particular

S∗AS(z) =

(γ2−14 zρθ 0

0 0

), S∗BS(z) =

(θ2z2ρ2θ θzρθ

θzρθ 1

),

and (3.49)-(3.50) give

E∫∫

QT

1det(t)(I|∂xρτ |2 + J∂xρ

τ · |ρτ |1/2∂xuτ + Kρτ |∂xuτ |2)dxdt = O(1),

(3.51)where

I = |ρτ |2θ−1∫Rθ2z2g′′

(uτ + z|ρτ |θ

)dmλ(z)

+ |ρτ |θ−1∫R

γ2 − 1

4zg′(uτ + z|ρτ |θ

)dmλ(z),

26

and

J =2|ρτ |θ−12

∫Rθzg′′

(uτ + z|ρτ |θ

)dmλ(z),

K =

∫Rg′′(uτ + z|ρτ |θ

)dmλ(z).

We observe that 2zdmλ(z) = − cλλ+1d(1 − z2)λ+1

+ . By integration by parts,the second term in I can therefore be written

1

λ+ 1|ρτ |2θ−1

∫R

γ2 − 1

8(1− z2)g′′

(u+ z|ρτ |

γ−12

)dmλ(z).

Since γ2−18

1λ+1 = θ2, we have

I = |ρτ |2θ−1∫Rθ2g′′

(uτ + z|ρτ |

γ−12

)dmλ(z).

This gives (3.48).

We apply (3.48) with g(ξ) := |ξ|2m+2 and η = ηm+1 given by (2.7). Then

|u|2m + ρm(γ−1) = O(1)G[2](ρ, u)

and since the higher order terms in G[1] are (strictly) dominated by thehigher order terms in G[2], our estimate (3.48) gives

E∫∫

QT

1det(t)[|uτ |2m + |ρτ |m(γ−1)

] [|ρτ |γ−2|∂xρτ |2 + ρτ |∂xuτ |2

]dxdt

= O(1).

By letting m vary, we obtain the following

Corollary 3.14. The solution Uτ to (3.7)-(3.8) satisfies the estimates

E∫∫

QT

1det(t)(|uτ |α|ρτ |γ−2 + |ρτ |γ−2+β

)|∂xρτ |2dxdt = O(1), (3.52)

and

E∫∫

QT

1det(t)(|uτ |αρτ + |ρτ |1+β

)|∂xuτ |2dxdt = O(1), (3.53)

for all α, β ≥ 0, where O(1) depends on T , γ, on the constant D0 in (2.2)and on a finite number of quantities EΓηj (U0), for entropies ηj associatedto various power-like functions |ξ|rj , for given rj’s depending on α, β.

27

3.1.5 Positivity of the density

Proposition 3.15 (Positivity). Let Uτ be the solution to (3.7)-(3.8) withinitial datum U0 = (ρ0, q0) and assume that ρ0 is uniformly positive: thereexists c0 > 0 such that ρ0 ≥ c0 a.e. on T. Let m > 3. Then, a.s., thereexists c > 0 depending on c0, T ,∫∫

QT

1det(t)ρτ |∂xuτ |2dxdt and

∫∫QT

|uτ |mdxdt (3.54)

only, such thatρτ ≥ c (3.55)

a.e. in T× [0, T ].

Proof. We apply Theorem (A.1) (and Remark A.2) proved in Appendix A.

Remark 3.16. Note that the constant c in (3.55) depends on ω, but onlythrough the values of the random variables involved in (3.54).

3.1.6 Bounds on higher derivatives of U τ

Assume, by adjusting τ if necessary, that T = t2K , K ∈ N∗. Let R > 0 andlet us introduce the stopping time TR defined as the infimum of the timest2k such that

supt∈[0,t2k]

Γη(Uτ (t)) + 2

∫∫Qt2k

1detη′′(Uτ ) · (Uτ

x,Uτx)dxdt ≤ R, (3.56)

where the infimum over the empty set is taken to be t2K = T by definition.Here we take for η the energy, i.e. the entropy associated to g(ξ) = 1

2 |ξ|2.

In that case, we have by (3.47) the control∫∫QTR

1det(t)ρτ |∂xuτ |2dxdt ≤ R.

Let then UτR denote the process deduced from Uτ by following a determinist

evolution after the time TR: UτR is defined like Uτ by (3.7) where Φτ is

replaced with Φτ1t<TR . Since Γη(Uτ (TR)) ≤ R, we can adapt the proof of

Proposition 3.7 (it is now purely deterministic) to obtain∫ T

TR

∫T

1det(t)ρτR|∂xuτR|2dxdt ≤M(R),

where the deterministic constant M(R) depends on T , γ, D0 and R only.In particular, we have∫∫

QT

1det(t)ρτR|∂xuτR|2dxdt ≤M(R) +R.

28

Proposition 3.15 in the previous section then asserts that there exists c(R) >0 depending on c0, T and R such that

ρτR ≥ c(R) a.e. on T× [0, T ]. (3.57)

In this section we will use this uniform estimate (3.57) and the entropyestimates from Section 3.1.2 to obtain some bounds independent on τ onthe higher derivatives of Uτ

R. Note in particular that the proof of Corol-lary 3.9 and Proposition 3.13 apply readily to Uτ

R as well so that we havethe following

Remark 3.17. The estimates (3.30), (3.52), (3.53) are satisfied by UτR.

We will prove the following result.

Proposition 3.18 (Bounds on higher derivatives). Let U0 = (ρ0, q0) ∈W 1,∞(T)2, U0 deterministic. Let Uτ be the solution to (3.7)-(3.8) withinitial datum U0 = (ρ0, q0). Assume that ρ0 ≥ c0 a.e. on T where c0 > 0.Let p > 2 and α ∈ (0, 1/2). Then for all R > 0, we have

E‖ρτR‖Cα([0,T ];Lp(T)),E‖ρτR‖Lp(0,T ;W 3/2,p(T)) = O(1), (3.58)

andE‖uτR‖Cα([0,T ];Lp(T)),E‖uτR‖Lp(0,T ;W 3/2,p(T)) = O(1), (3.59)

where O(1) depends on R, on c0, on p, α, γ, T , on the constants D0 andD1 in (2.2), (2.3) and on the moments of U0 and ∂xU0 up to a given orderdepending on p and α.

Proof. For 0 ≤ s < t ≤ T , let Qs,t denote the cylinder T×(s, t). We work onthe variables ρ, u = q/ρ and rewrite (3.7) as a system of two heat equationswith source terms in the domain Qt2k,t2k+1

:(1

2∂t − ∂2x

)ρτ = −uτ∂xρτ − ρτ∂xuτ =: f τ1 , (3.60)

and(1

2∂t − ∂2x

)uτ = −uτ∂xuτ − κγ|ρτ |γ−2∂xρτ + 2

∂xρτ

ρτ∂xu

τ =: f τ2 . (3.61)

Let us extend f τ1 and f τ2 by 0 in the strips Qt2k+1,t2k+2, and set

Fτ =

(f τ1f τ2

),

and let S(t) denote the semi-group associated to −∂2x on Lp(T). For f ∈L1(QT ), z0 ∈ L1(T), the solution to the equation(

1

2∂t − ∂2x

)z(t, x) = f(t, x), x ∈ T, t > 0,

29

with initial datum z0 is

z(t) = S(2t)z0 + 2

∫ t

0S(2(t− s))f(s)ds.

By iteration, it follows that Uτ is given by

Uτ (t) = S(t])Uτ (0) +

∫ t]

0S(t] − s)Fτ (s[)ds

+∑k≥1

√2

∫ t]

0S(t] − s])1sto(s)ψk(U

τ (s))dβk(s),

where we have defined

t] = min(2t− t2n, t2n+2), t[ =t+ t2n

2, t2n ≤ t < t2n+2.

For the “stopped” process UτR, we have, similarly, a decomposition

UτR = Uτ

0,R + Uτdet,R + Uτ

sto,R, (3.62)

where

Uτ0,R(t) = S(t])U

τ (0),

Uτdet,R(t) =

∫ t]

0S(t] − s)Fτ

R(s[)ds,

Uτsto,R(t) =

∑k≥1

√2

∫ t]∧TR

0S(t] − s])1sto(s)ψk(U

τ (s))dβk(s),

and where FτR is defined by similarity with Fτ , i.e.

FτR =

f τ1,Rf τ2,R

=

−uτR∂xρτR − ρτR∂xuτR

−uτR∂xuτR − κγ|ρτR|γ−2∂xρτR + 2∂xρτRρτR

∂xuτR

.

Higher derivatives in x: to obtain some estimates on the higher deriva-tives in x of ρτR, uτR, we need to use a bootstrap method. Actually we will useimproved estimates in Equation (3.62) three times. For a better readibilityof this part of the proof, we will adopt the following convention of notation:a constant CR, which may vary from lines to lines, is a constant dependingon T , γ, on the constant D0 in (2.2) and on R. A number ε > 0 being given,we use the notation ε for quantities small with ε but possibly different fromε. Therefore in what follows, Lr−ε may well be actually Lr−

ε10 . Also, Lmε

is Lm for a m (quite high) depending on ε, which we do not specify andagain, the value of mε may vary from line to line. Eventually, we denote byMε(U0) a constant depending only on the moments∫

T(umε0 + ρ

mε(γ−1)0 )ρ0dx,

∫T|∂xu0|mε + |∂xρ0|mεdx,

30

where the exponent mε depends on ε.

We begin by showing that

‖∂xρτR‖L6−ε(QT ) ≤Mε(U0). (3.63)

Rewriting

f τ1,R = −|ρτR|2−γ2 · |ρτR|

γ−22 uτR∂xρ

τR − ρτR∂xuτR,

we deduce from Remark 3.17, from the moment estimate (3.30), and fromthe gradient estimates (3.52), (3.53) and from the Holder inequality that

‖f τ1,R‖L2−ε(QT ) ≤Mε(U0).

By (B.7) this implies ‖∂xρτdet,R‖L6−ε(QT ) ≤Mε(U0). We have besides ρτsto,R =0 and thus (3.63).Then, in the equation satisfied by uτR, we use the expression

f τ2,R = − 1

|ρτR|1/2· |ρτR|1/2uτR∂xuτR − κγ

1

|ρτR|2−γ2

· |ρτR|γ−22 ∂xρ

τR

+ 21

|ρτR|2· ∂xρτR · ρτR∂xuτR.

We deduce from Remark 3.17, from (3.30), (3.52), (3.53), (3.63), from (3.57)and from the Holder inequality that ‖f τ2,R‖L3/2−ε(QT )

≤ CRMε(U0), and thus,by (B.7),

‖∂xuτdet,R‖3−εL3−ε(QT )≤ CRMε(U0). (3.64)

To get a bound on uτsto,R, we use the following lemma.

Lemma 3.19. Let q ≥ 2. For k ∈ N∗, let hk ∈ L2(Ω × (0, T );Lq(T)) besome predictable processes such that

H :=

(∑k≥1|hk|2

)1/2

(3.65)

satisfies H ∈ Lq(Ω× (0, T )× T). Then we have the estimate

E∥∥∥∥∑k≥1

∫ t]

0S(t] − s])hk(s)dβk(s)

∥∥∥∥qLq(T)

≤ CBDG(q)tq−12

] E‖H‖qLq(Qt] ), (3.66)

for a given constant CBDG(q) depending on q only which is bounded forbounded values of q ∈ [2,+∞).

31

Proof of Lemma 3.19: we apply the Burkholder-Davis-Gundy Inequality inthe 2-smooth Banach space Lq(T) to obtain

E sup0≤σ≤t]

∥∥∥∥∑k≥1

∫ σ

0S(t] − s])hk(s)dβk(s)

∥∥∥∥qLq(T)

≤ CBDG(q)E(∫ t]

0‖S(t] − s])H(s)‖2L2(T)ds

)q/2.

Since S(t) is a contraction from L2(T) into L2(T), we obtain (3.66) byJensen’s inequality.

We apply Lemma 3.19 with

hk(t) = ∂x(1sto(t)σk(U

τR(t))

)1t<TR

and q = 3 − ε. By the growth hypothesis (2.3) and (3.57) we obtain, forT ≥ 1,

E‖∂xuτsto,R(t)‖3−εL3−ε(T) ≤ CTCR

(E‖∂xuτR‖3−εL3−ε(Qt] )

+ E‖∂xρτR‖3−εL3−ε(Qt] )

).

(3.67)Here C := sup2≤q≤3CBDG(q) is an absolute constant.From the identity (3.62), from (3.63) and the estimates (3.64), (3.67) andthe Gronwall Lemma, we obtain

E‖∂xuτR‖3−εL3−ε(QT )≤ CRMε(U0). (3.68)

Now, Remark 3.17, the estimates (3.30), (3.57), (3.63), (3.68) and the iden-tity

f τ1,R = −|ρτR|−16 · |ρτR|

16uτR · ∂xρτR − ρτR · ∂xuτR,

giveE‖f τ1,R‖3−εL3−ε(QT )

≤ CRMε(U0).

Notice that for example, a bound for the term ρτR · ∂xuτR in Lp is given by‖ρτR‖Lm‖∂xuτR‖Lq with q = 3 − ε, m = (3 − ε)(3 − 2ε)/ε, and the obtainedspace is Lp with p = 3 − 2ε which is also denoting by L3−ε. By (B.7) and(B.4b), it follows that

E‖∂xρτR‖mεLmε (QT )

≤ CRMε(U0). (3.69)

Eventually, it follows from the decomposition

f τ2,R = − 1

|ρτR|1/(2m)· |ρτR|1/(2m)uτR · ∂xuτR−κγ|ρτR|γ−2 · ∂xρτR

+ 21

|ρτR|· ∂xρτR · ∂xuτR,

32

from Remark 3.17 and from estimates (3.30), (3.57), (3.68) and (3.69) that

E‖f τ2,R‖3−εL3−ε(QT )≤ CRMε(U0),

which gives by (B.4a) the estimate

E‖∂xuτdet,R‖mεLmε (QT )

≤ CRMε(U0). (3.70)

We also apply Lemma 3.19 to the martingale part in (3.62). Let m > 3 befixed (quite large). If 2 ≤ mε ≤ m, we have

E‖∂xuτsto,R(t)‖mεLmε (T) ≤ CTE‖∂xuτR‖

mεLmε (Qt] )

, (3.71)

where C = sup2≤q≤mCBDG(q). By Gronwall Lemma it follows that

E‖∂xuτR‖mεLmε (QT )

≤ CRMε(U0). (3.72)

Now the “near L∞” estimates (3.69), (3.72) together with the positivityestimate (3.57) give f τ2,R ∈ Lp(QT ) where p is arbitrary. We use the estimate∥∥∥∥|D|3/2 ∫ t]

0S(t]− s)f(s[)ds

∥∥∥∥Lq(QT )

≤ C‖f‖Lp(QT ) if1

q>

1

p− 1

6, (3.73)

which can be derived similarly to (B.7) by using the estimate

‖|D|3/2S(t)‖Lpx→Lqx ≤ Ct− 1

2

(1p− 1q

)− 3

4 . (3.74)

Here, |D| denotes the square-root of the Laplacian: |D| = (−∂2x)1/2. Wealso deduce from (3.74) that

‖|D|3/2S(t])u0‖Lp(QT ) ≤ C‖u0‖Lp(QT ).

Therefore (3.62) and (3.73) give

E‖|D|3/2(uτ0,R + uτdet,R)‖pLp(QT ) = O(1). (3.75)

By (2.3) and by interpolation, we have

‖H‖Lp(QT ) ≤ CR(1 + ‖|D|3/2uτR‖Lp(QT )),

where H is defined by (3.65) with hk = |D3/2|(1sto(t)σk(U

τR(t))

)1t<TR . By

Lemma 3.19, we deduce that

E∥∥|D|3/2uτsto,R∥∥pLp(T) ≤ C(1 + E‖|D|3/2uτR‖

pLp(QT )

).

33

By the Gronwall Lemma and (3.75) it follows that E‖uτR‖Lp(0,T ;W 3/2,p(T)) =O(1). To obtain time estimates on uτR now, we use the Kolmogorov criterionand the a priori bounds of Section 3.1.2. Let p > 2. Set

hk(s) = 1sto(s)ρτR(s)σk(U

τR(s)).

By the Burkholder-Davis-Gundy Inequality in the 2-smooth Banach spaceLp(T) we have, for 0 ≤ t ≤ t′ ≤ T ,

E∥∥∥∥∑k≥1

∫ t′]

t]

S(t′] − s])hk(s)dβk(s)∥∥∥∥pLp(T)

≤ CBDG(p)E(∫ t′]

t]

‖S(t′] − s])H(s)‖2L2(T)dxds

)p/2,

where H is defined as in (3.65). Since S(t) is a contraction from L2(T) intoL2(T), (2.2), Remark 3.17 and the moment estimate (3.30) show that

E

∥∥∥∥∑k≥1

∫ t′]

t]

S(t′] − s])hk(s)dβk(s)∥∥∥∥pLp(T)

≤ CRMε(U0)|t′] − t]|p/2

≤ CRMε(U0)|t′ − t|p/2.

The Kolmogorov continuity Theorem ([DPZ92], Theorem 3.3) therefore gives

E‖uτsto,R‖Cα([0,T ];Lp(T)) = O(1),

where 0 < α < 12 −

1p . We can use a similar argument to deal with the term

uτdet,R: we have

E∥∥∥∥∫ t′]

t]

S(t′] − s])f τ2,R(s)dβk(s)

∥∥∥∥pLp(T)

≤ CRMε(U0)|t′] − t]|p−1,

and thusE‖uτdet,R‖Cα([0,T ];Lp(T)) = O(1),

where 0 < α < 1− 2p . By the identity

S(t′])u0 − S(t])u0 =

∫ t′]

t]

∂xS(t)∂xu0dt

and (B.3), we obtain also ‖uτ0,R‖C1/2([0,T ];Lp(T)) = O(1). This concludes theproof of (3.59). The proof of (3.58) is similar and simpler since ρτsto,R = 0.

34

3.1.7 Compactness argument

For k ∈ N∗, we introduce the independent processes

Xk(t) =√

2

∫ t

01sto(s)dβk(s)

and set W τ (t) =∑

k≥1Xk(t)ek. When τ → 0, each Xk converges in law toβk. Let

XW = C([0, T ];U0

).

Since the embedding U → U0 is Hilbert-Schmidt, the XW -valued processW τ converges in law to W :

µW τ → µW , (3.76)

where µW τ denote the the law of W τ and µW is the the law of W on XW .

Let us fix p > 2. Define the path space X = XU ×XW , where

XU =[C([0, T ];W 1,p(T)

)]2.

Let us denote by µτU the law of Uτ on XU. The joint law of Uτ and W τ onX is denoted by µτ .

Proposition 3.20. The set µτ ; τ ∈ (0, 1) is tight and therefore relativelyweakly compact in X .

Proof. First, we prove tightness of µτU; τ ∈ (0, 1) in XU. Let α ∈ (0, 1/2)and M > 0. Then

KM :=U ∈ XU; ‖U‖Cα([0,T ];Lp(T)) + ‖U‖Lp([0,T ];W 3/2,p(T)) ≤M

is compact in XU. Recall that the stopping time TR is defined by (3.56). Bythe Markov inequality and the entropy estimate (3.21), we have

P(TR < T ) ≤ C

R,

where the constant C depends on γ, T , D0 and on the moments of U0 and∂xU0 up to a given order. We have therefore, by Proposition 3.18 and theMarkov inequality,

P(Uτ /∈ KM ) ≤ C

R+ P(Uτ

R /∈ KM ) ≤ C

R+CRM

,

by possibly augmenting the constant C. Therefore, given η > 0 there existsR,M > 0 such that

µτU(KM ) ≥ 1− η.

Besides, the law µW τ is tight by (3.76). Consequently the set of the jointlaws µτ ; τ ∈ (0, 1) is tight. By Prokhorov’s theorem, it is relatively weaklycompact.

35

Let now (τn) be a sequence decreasing to 0. For simplicity we will keep thenotation τ for (τn) and possible subsequences of (τn). Let µ be an adherencevalue, for the weak convergence, of µτ . By the Skorokhod Theorem we canassume a.e. convergence of the random variables by changing the probabilityspace.

Proposition 3.21. There exists a probability space (Ω, F , P) with a sequenceof X -valued random variables (Uτ , W τ ), τ ∈ (0, 1), and (U, W ) such that

1. the laws of (Uτ , W τ ) and (U, W ) under P coincide with µτ and µrespectively,

2. (Uτ , W τ ) converges P-almost surely to (U, W ) in the topology of X .

3.1.8 Identification of the limit

Our aim in this section is to identify the limit (U, W ) given by Proposi-tion 3.21. Let (Ft) be the P-augmented canonical filtration of the process(U, W ), i.e.

Ft = σ(σ(%tU, %tW

)∪N ∈ F ; P(N) = 0

), t ∈ [0, T ],

where %t is the operator of restriction to the interval [0, t] defined as follows:if E is a Banach space and t ∈ [0, T ], then

%t : C([0, T ];E) −→ C([0, t];E)

k 7−→ k|[0,t].(3.77)

Clearly, %t is a continuous mapping. The following statement is given forUε (we reintroduce the factor ε here), which is built like U.

Proposition 3.22 (Martingale solution to (3.1)). The sextuplet(Ω, F , (Ft), P, W , Uε

)is a weak martingale solution to (3.1).

The proof of Proposition 3.22 uses a method of construction of martingalesolutions of SPDEs that avoids in part the use of representation Theorem.This technique has been developed in Ondrejat [Ond10], Brzezniak, On-drejat [BO11] and used in particular in Hofmanova, Seidler [HS12] and in[Hof13b, DHV13].

Recall that A (the flux of Equation (1.1)) is defined by (2.4). Let us definefor all t ∈ [0, T ] and a test function ϕ = (ϕ1, ϕ2) ∈ C∞(T;R2),

M τ (t) =⟨Uτ (t), ϕ

⟩−⟨U0, ϕ

⟩+ 2

∫ t

01det(s)

⟨∂xA(Uτ )− ∂2xUτ , ϕ

⟩ds,

M τ (t) =⟨Uτ (t), ϕ

⟩−⟨U0, ϕ

⟩+ 2

∫ t

01det(s)

⟨∂xA(Uτ )− ∂2xUτ , ϕ

⟩ds,

M(t) =⟨U(t), ϕ

⟩−⟨U0, ϕ

⟩+

∫ t

0

⟨∂xA(U)− ∂2xU, ϕ

⟩ds.

36

In what follows, we fix some times s, t ∈ [0, T ], s ≤ t, and a continuousfunction

γ : C([0, s];Lp(T)

)× C

([0, s];U0

)−→ [0, 1].

The proof of Proposition 3.22 will be a consequence of the following twolemmas.

Lemma 3.23. The process W is a (Ft)-cylindrical Wiener process, i.e.there exists a collection of mutually independent real-valued (Ft)-Wienerprocesses βkk≥1 such that W =

∑k≥1 βkek.

Proof. By (3.76), W is a U0-valued cylindrical Wiener process and is (Ft)-adapted. According to the Levy martingale characterization theorem, itremains to show that it is also a (Ft)-martingale. We have

E γ(%sU

τ , %sWτ)[W τ (t)−W τ (s)

]= E γ

(%sU

τ , %sWτ)[W τ (t)−W τ (s)

]= 0

since W τ is a martingale and the laws of (Uτ , W τ ) and (Uτ ,W τ ) coincide.Next, the uniform estimate

supτ∈(0,1)

E‖W τ (t)‖2U0= sup

τ∈(0,1)E‖W τ (t)‖2U0

<∞

and the Vitali convergence theorem yield

E γ(%sU, %sW

)[W (t)− W (s)

]= 0

which finishes the proof.

Lemma 3.24. The processes

M, M2 −∑k≥1

∫ ·0

⟨ψk(U), ϕ

⟩2dr, M βk −

∫ ·0

⟨ψk(U), ϕ

⟩dr (3.78)

are (Ft)-martingales.

Proof. Here, we use the same approach as in the previous lemma. Let usdenote by Xτ

k , k ≥ 1 the real-valued Wiener processes corresponding to W τ ,that is W τ =

∑k≥1 X

τk ek. For all τ ∈ (0, 1), the process

M τ =∑k≥1

∫ ·0

⟨στk(Uτ ), ϕ2

⟩dXτ

k (r)

is a square integrable (Ft)-martingale and therefore

M τ2 := (M τ )2 −

∑k≥1

∫ ·0

⟨στk(Uτ ), ϕ2

⟩2d〈〈Xτ 〉〉(r),

37

and

M τ3 := M τβk −

∫ ·0

⟨στk(Uτ ), ϕ2

⟩d〈〈Xτ 〉〉(r)

are (Ft)-martingales, where we have denoted by

〈〈Xτ 〉〉(t) = 2

∫ t

01sto(r)dr

the quadratic variation of Xτk . Besides, it follows from the equality of laws

that

E γ(%sU

τ , %sWτ)[M τ (t)− M τ (s)

]= E γ

(%sU

τ , %sWτ)[M τ (t)−M τ (s)

].

hence E γ(%sU

τ , %sWτ)[M τ (t) − M τ (s)

]= 0. We can pass to the limit in

this equation, due to the moment estimates (3.30) and the Vitali convergencetheorem. We obtain

E γ(%sU, %sW

)[M(t)− M(s)

]= 0,

i.e. M is a (Ft)-martingale. We proceed similarly to show that

M2 := M2 −∑k≥1

∫ ·0

⟨ψk(U), ϕ

⟩2dr

is a (Ft)-martingale, by passing to the limit in the identity

E γ(%sU

τ , %sWτ)[M τ

2 (t)− M τ2 (s)

]= 0,

and again similarly for

M3 := Mβk −∫ ·0

⟨ψk(U), ϕ

⟩dr.

Proof of Proposition 3.22. Once the above lemmas are established, we inferthat ⟨⟨

M −∑k≥1

∫ ·0

⟨σk(U) dβk, ϕ2

⟩⟩⟩= 0,

where 〈〈 · 〉〉 denotes the quadratic variation process. Accordingly, we have

⟨U(t), ϕ

⟩=⟨U0, ϕ

⟩−∫ t

0

⟨∂xA(U)− ∂2xU, ϕ

⟩ds

+∑k≥1

∫ t

0

⟨σk(U) dβk, ϕ2

⟩, t ∈ [0, T ], P-a.s.,

and the proof is complete.

38

Proof of Theorem ??. We have just obtained the existence of a martingalesolution (

Ω, F , (Ft), P, W , Uε)

to (3.1). By passing to the limit on the approximation Uτ , taking care tothe occurrence of the factor ε now, we also obtain the various estimates ofTheorem ?? and the Ito formula (3.6).

4 Probabilistic Young measures

Let(Ω,F , (Ft),P,W,Uε

)be a martingale solution to (3.1). Our aim is to

prove the convergence of Uε. The standard tool for this is the notion ofmeasure-valued solution (introduced by Di Perna, [DiP83a]). In this sectionwe give some precisions about it in our context of random solutions. Moreprecisely, we know that, almost surely, (Uε) defines a Young measure νε onR+ × R by the formula

〈νεx,t, ϕ〉 := 〈δUε(t,x), ϕ〉 = ϕ(Uε(t, x)), ∀ϕ ∈ Cb(R+ × R). (4.1)

Our aim is to show that νε ν (in the sense to be specified), where νhas some specific properties. To that purpose, we will use the probabilisticcompensated compactness method developed in the Appendix of [FN08]and some results on the convergence of probabilistic Young measures thatwe introduce here.

4.1 Young measures embedded in a space of Probability mea-sures

Let (Q,A, λ) be a finite measure space. Without loss of generality, we willassume λ(Q) = 1. A Young measure on Q (with state space E) is a mea-surable map Q → P1(E), where E is a topological space endowed with theσ-algebra of Borel sets, P1(E) is the set of probability measures on E, itselfendowed with the σ-algebra of Borel sets corresponding to the topologydefined by the weak3 convergence of measures, i.e. µn → µ in P1(E) if

〈µn, ϕ〉 → 〈µ, ϕ〉, ∀ϕ ∈ Cb(E).

As in (4.1), any measurable map w : Q → E can be viewed as a Youngmeasure ν defined by

〈νz, ϕ〉 = 〈δw(z), ϕ〉 = ϕ(w(z)), ∀ϕ ∈ Cb(E), ∀z ∈ Q.3actually, weak convergence of probability measures, also corresponding to the tight

convergence of finite measures

39

A Young measure ν on Q can itself be seen as a probability measure onQ× E defined by

〈ν, ψ〉 =

∫Q

∫Eψ(p, z)dνz(p)dλ(z), ∀ψ ∈ Cb(E ×Q).

We then have, for all ψ ∈ Cb(Q) (ψ independent on p ∈ E), 〈ν, ψ〉 = 〈λ, ψ〉,that is to say

π∗ν = λ, (4.2)

where π is the projection E ×Q→ Q and the push-forward of ν by π is de-fined by π∗ν(A) = ν(π−1(A)), for all Borel subset A of Q. Assume now thatQ is a subset of Rs and E is a closed subset of Rm, m, s ∈ N∗, and, conversely,let µ is a probability measure on E × Q such that π∗µ = λ: by the SlicingTheorem (cf. Attouch, Buttazzo, Michaille [ABM06, Theorem 4.2.4]), wehave: for λ-a.e. z ∈ Q, there exists µz ∈ P1(E) such that,

z 7→ 〈µz, ϕ〉

is measurable from Q to R for every ϕ ∈ Cb(Q), and

〈µ, ψ〉 =

∫Q

∫Eψ(p, z)dµz(p)dλ(z),

for all ψ ∈ Cb(E ×Q), which precisely means that µ is a Young measure onQ. We therefore denote by

Y = ν ∈ P1(E ×Q);π∗ν = λ

the set of Young measures on Q.

We use now the Prohorov’s Theorem, cf. Billingsley [Bil99], to give a com-pactness criteria in Y. We assume that Q is a compact subset of Rs and Eis a closed subset of Rm. We also assume that the σ-algebra A of Q is theσ-algebra of Borel sets of Q.

Proposition 4.1 (Bound against a Lyapunov functional). Let η : E → R+

satisfy the growth condition

limp∈E,|p|→+∞

η(p) = +∞.

Let C > 0 be a positive constant. Then the set

KC =

ν ∈ Y;

∫Q×E

η(p)dν(z, p) ≤ C

(4.3)

is a compact subset of Y.

40

Proof. The condition π∗ν = λ being stable by weak convergence, Y is closedin P1(E ×Q). By Prohorov’s Theorem, [Bil99], KC is relatively compact inY if, and only if it is tight. Besides, KC is closed since∫

Q×Eη(p)dν(z, p) ≤ lim inf

n→+∞

∫Q×E

η(p)dνn(z, p)

if (νn) converges weakly to ν. It is therefore sufficient to prove that KC istight, which is classical: let ε > 0. For R ≥ 0, let

V (R) = inf|p|>R

η(p).

Then V (R)→ +∞ as R→ +∞ by hypothesis and, setting MR = [B(0, R)∩E]×Q, we have

V (R)ν(M cR) ≤

∫Q×E

η(p)dν(z, p) ≤ C,

for all ν ∈ K, whence supν∈K ν(M cR) < ε for R large enough.

4.2 A compactness criterion for probabilistic Young mea-sures

As above, we assume that Q is a compact subset of Rs and E is a closedsubset of Rm.

Definition 4.2. A random Young measure is a Y-valued random variable.

Proposition 4.3. Let η : E → R+ satisfy the growth condition

limp∈E,|p|→+∞

η(p) = +∞.

Let M > 0 be a positive constant. If (νn) is a sequence of random Youngmeasures on Q satisfying the bound

E∫Q×E

η(p)dνn(z, p) ≤M,

then, up to a subsequence, (νn) is converging in law.

Proof. We endow P1(E×Q) with the Prohorov’s metric d. Then P1(E×Q)is a complete, separable metric space, weak convergence coincides with d-convergence, and a subset A is relatively compact if, and only if it is tight,[Bil99]. Let L(νn) ∈ P1(Y) denote the law of νn. To prove that it is tight,we use the Prohorov’s Theorem. Let ε > 0. For C > 0, let KC be thecompact set defined by (4.3). For ν ∈ Y, we have

P(ν /∈ KC) = P(

1 <1

C

∫Q×E

η(p)dν(z, p)

)≤ 1

CE∫Q×E

η(p)dν(z, p),

41

hence

supn∈NL(νn)(Y \KC) = sup

n∈NP(νn /∈ KC) ≤ M

C< ε,

for C large enough, which proves the result.

Let XW = C([0, T ];U0

)be the path space for the Wiener process W . If (νn)

is as in Proposition 4.3, then the law of (νn,W ) is tight in Y ×XW since thelaw of W is tight in XW . We obtain the following extension:

Proposition 4.4. Under the hypothesis of Proposition 4.3, there exists asubsequence of (νn,W ) converging in law on Y × XW .

Remark 4.5 (Almost-sure convergence). Assume that (νn,W ) is conver-ging in law on Y × XW . Then we can apply the Skorokhod Theorem [Bil99]to (νn,W ): there exists a probability space (Ω, F , P), some random Youngmeasures νn, ν : Ω → Y, some random variable W : Ω → XW , such thatLaw(νn) = Law(νn), Law(ν) = Law(ν), Law(W ) = Law(W ) and νn → νa.s. in P1(E × Q). Let q > 1. If νn is issued from a sequence of randomvariables bounded in Lq(Q;E) (as this will be the case in our context), say

νn = δun , un ∈ Lq(Q;E), (4.4)

then νn is itself a Dirac mass.

The proof of the above statement is as follows: using the Jensen’s Formula,(4.4) can be characterized by

E∫Q×E

ψ(p)dνn,z(p)dλ(z) = E∫Qψ

(∫Epdνn,z(p)

)dλ(z),

where ψ : Rm → R is a strictly convex function satisfying the growth condi-tion |ψ(p)| ≤ C(1 + |p|q). In other words,

Eϕ(νn) = Eθ(νn), (4.5)

where

ϕ : µ 7→∫Q×E

ψ(p)dµz(p)dλ(z), θ : µ 7→∫Qψ

(∫Epdµz(p)

)dλ(z).

Of course, ϕ is continuous on Y and, by the Lebesgue dominated convergencetheorem, θ is continuous on the subset

µ ∈ Y;

∫Q×E

|p|qdµz(p)dλ(z) ≤ R.

Consequently, by identity of the laws, (4.5) is satisfied by νn, i.e.

νn = δun , un(z) :=

∫Epdνn,z

almost surely.

42

4.3 Convergence to a random Young measure

We apply the results of paragraphs 4.1-4.2 to the case νε := δUε , Q = QT ,E = R+ × R. Strictly speaking we will consider a sequence (εn) decreasingto 0 and will obtain some limits along some subsequences of (εn). However,for the simplicity of notations we will keep the notation νε instead of νεn inwhat follows.

Proposition 4.6. Let Uε0 ∈ W 1,∞(T) be bounded in L∞(T). Let Uε be a

martingale solution to (3.1) satisfying (??) and let νε := δUε be the associ-ated random Young measure. Then there exists a probability space (Ω, F , P),some random variables (νε, W ) and (ν, W ) with values in Y ×XW such that

1. the law of (νε, W ) under P coincide with the law of (νε,W ),

2. (νε, W ) converges P-almost surely to (ν, W ) in the topology of Y×XW .

Proof. we apply the Proposition 4.4 to (νε,W ) taking for η the energy, i.e.the entropy given by (2.7) with g(ξ) = |ξ|2 (which is infinite at infinity by(3.22)). We use the estimate (??) for such an η.

5 Reduction of the Young measure

Proposition 4.6 above gives the existence of a random young measure νsuch that νε := δUε converges in law in the sense of Young measures toν. We will now apply the compensated compactness method to prove thata.s., for a.e. (x, t) ∈ QT , either νx,t is a Dirac mass or νx,t is concentratedon the vacuum region ρ = 0. To do this, we will use the probabilisticcompensated compactness method of [FN08] to obtain a set of functionalequations satisfied by ν. Then we conclude by adapting the arguments of[LPS96].

5.1 Compensated compactness

In this section, we fix an entropy - entropy flux couple (η,H) of the form(2.7)-(2.9) where g ∈ C2(R) is convex, with g sub-quadratic and g′ sub-linear:

|g(ξ)| ≤ C(1 + |ξ|2), |g′(ξ)| ≤ C(1 + |ξ|), (5.1)

for all ξ ∈ R, for a given C ≥ 0.

5.1.1 Murat’s Lemma

For p ∈ [1,+∞], we denote by W 1,p0 (QT ) the set of function u in the Sobolev

space W 1,p(QT ) such that u = 0 on T × 0 and T × T. We denote by

W−1,p(QT ) the dual of W 1,p′

0 (QT ), where p′ is the conjugate exponent to p.

43

Proposition 5.1. Let q ∈ (1, 2). Let η be an entropy of the form (2.7) withg convex satisfying (5.1). Let Uε

0 ∈ L∞(T). Then the sequence of randomvariables (εη(Uε)xx)ε>0 is tight in W−1,q(QT ).

Proof. Let q ∈ (q, 2). Let m > 1,

m = max

(1γ≤2

2(2− γ)(2− q)(γ − 1)q

,2qq

q − q,

q

q − 1

).

Let R > 0. Assume that ω is fixed such that

X1ε := ‖|ρε|

γ−12 ‖Lm(QT ) ≤ R, X2

ε := ‖uε‖Lm(QT ) ≤ R, (5.2)

and

X3ε :=ε

∫∫QT

|ρε|γ−2

[|ρε|γ + |uε|4

]|∂xρε|2

+ ρε[1 + |ρε|max(2,γ) + |uε|4

]|∂xuε|2

dxdt ≤ R. (5.3)

Then we have the convergence

limε→0

εη(Uε)x = 0 in Lq(QT ), (5.4)

which we admit for the moment. This convergence (5.4) implies in particularthat (εη(Uε)xx) is in a compact KR of W−1,q(QT ). Consequently, we have

P(εη(Uε)xx /∈ KR) ≤3∑i=1

P(Xiε > R).

By the estimates (3.3), (3.4), (3.5) and the Markov Inequality, we deducethat, given η > 0, we may choose R > 0 such that

P(εη(Uε)xx /∈ KR) < η,

uniformly in ε. This shows that (εη(Uε)xx) is tight in W−1,q(QT ). Theproof of (5.4) is similar to the analysis in [LPS96], pp 627-629, with theminor difference that we have here only Lr estimates with r arbitrary largeon the unknown, instead of the L∞ estimates of the deterministic case. Wenote first that, by (5.1), we have

|∂ρη(U)| ≤ C(

1 + |u|2 + ρ(γ−1)),

and|∂uη(U)| ≤ Cρ

(1 + |u|+ ρ

γ−12

),

44

for a given non-negative constant that we still denote by C. By the YoungInequality, it follows that

|η(Uε)x|q ≤C(

1 + |uε|2q + |ρε|q(γ−1))|∂xρε|q

+ C|ρε|q(

1 + |uε|q + |ρε|qγ−12

)|∂xuε|q

≤C + C(1 + |uε|2q

)|∂xρε|q + C|ρε|2(γ−1)|∂xρε|2

C + Cρε[1 + |ρε|max(2,γ) + |uε|4

]|∂xuε|2,

where C depends on q. By (5.2), (5.3), we obtain

εq∫∫

QT

|η(Uε)x|q dxdt ≤ CRεq−1 + Cεq∫∫

QT

(1 + |uε|2q

)|∂xρε|q dxdt,

(5.5)where the constant CR depends on R. If γ ≤ 2, we have furthermore(

1 + |uε|2q)|∂xρε|q = |ρε|

q2(2−γ) (1 + |uε|2q

)|ρε|

q2(γ−2)|∂xρε|q

≤ C|ρε|2−qq

(2−γ)+ C

(1 + |uε|4

)|ρε|(γ−2)|∂xρε|2.

By (5.2), (5.3), we obtain

εq∫∫

QT

|η(Uε)x|q dxdt ≤ CRεq−1. (5.6)

If γ > 2, we decompose(1 + |uε|2q

)|∂xρε|q =

(1 + |uε|2q

)|∂xρε|q1ρε>δ +

(1 + |uε|2q

)|∂xρε|q1ρε≤δ

The first term is bounded as follows:(1 + |uε|2q

)|∂xρε|q1ρε>δ ≤ Cδ−

q2(γ−2)

(1 +

(1 + |uε|4

)|ρε|(γ−2)|∂xρε|2

).

In particular, by (5.2) and (5.3), we have

εq∫∫

QT

(1 + |uε|2q

)|∂xρε|q1ρε>δ ≤ CRδ−

q2(γ−2)εq−1. (5.7)

To estimate the part ρε ≤ δ, we first use the Holder Inequality and theestimate (5.2) to obtain

εq∫∫

QT

(1 + |uε|2q

)|∂xρε|q1ρε≤δ

≤(∫∫

QT

(1 + |uε|2q)qq−q) q−q

q(εq∫∫

QT

|∂xρε|q1ρε≤δ) qq

≤CR(εq∫∫

QT

|∂xρε|q1ρε≤δ) qq. (5.8)

45

Then, we multiply the first Equation of the system (3.1a), the equation

∂tρε + ∂x(ρεuε) = ε∂2xρ

ε,

by min(ρε, δ), and then sum the result over QT . This gives

ε

∫∫QT

|∂xρε|21ρε≤δ ≤ Cδ2 + C(∫∫

QT

ρε|uε||∂xρε|1ρε≤δ)

≤ Cδ2 + Cδ‖uε‖L

qq−1

(QT )(∫∫

QT

|∂xρε|q1ρε≤δ) 1q

≤ Cδ2 + CRδ(∫∫

QT

|∂xρε|q1ρε≤δ) 1q,

from which we deduce(εq∫∫

QT

|∂xρε|q1ρε≤δ) qq ≤ CRδq.

Reporting this result in (5.8) and using (5.7) and (5.5), and also recallingthe estimate (5.6), we obtain

εq∫∫

QT

|η(Uε)x|q dxdt ≤ CR(εq−1 + δ−

q2(γ−2)εq−1 + δq

),

whatever the value of γ > 1. This concludes the proof of (5.4) and ofProposition 5.1.

The next Proposition is similar to Lemma 4.20 in [FN08].

Proposition 5.2. Let η be an entropy of the form (2.7) with g convexsatisfying (5.1). Let Uε

0 ∈ L∞(T). Let

M ε(t) =

∫ t

0∂qη(Uε)(s)Φ(Uε)(s)dW (s).

Then ∂tMε is tight in H−1(QT ).

Proof. The proof is in essential the proof of Lemma 4.19 in [FN08]. However,we will proceed slightly differently (instead of using Marchaud fractionalderivative we work directly with fractional Sobolev spaces and an Aubin-Simon compactness lemma). Let m ≥ 2. Let 0 ≤ s ≤ t ≤ T . In whatfollows we denote by C any constant, that may vary from line to line, whichdepends on the data only and independent on ε. By the Burholder-Davis-Gundy Inequality, we have

E‖M ε(t)−M ε(s)‖mLm(T) ≤ C∫TE∣∣∣∣∫ t

s|∂qη(Uε)|2G2(Uε)dσ

∣∣∣∣m/2 dx,46

and, using the Holder Inequality,

E‖M ε(t)−M ε(s)‖mLm(T) ≤ C|t−s|m2−1∫ t

sE∫T

[|∂qη(Uε)|2G2(Uε)

]m/2dσdx.

By (2.2) and (5.1), we have |∂qη(Uε)|2G2(Uε) ≤ R(ρε, |uε|), where R is apolynomial. In particular, by (3.3), we obtain

E‖M ε(t)−M ε(s)‖mLm(T) ≤ C|t− s|m2 .

and, by integration with respect to t and s,

E∫ T

0

∫ T

0

‖M ε(t)−M ε(s)‖mLm(T)

|t− s|1+νmdtds ≤ C, (5.9)

as soon as ν < 1/2. The left-hand side in this inequality (5.9) is the norm ofM ε in Lm(Ω;W ν,m(0, T ;Lm(T))). Since Lm(T) → H−1(T), it follows that

E‖M ε‖W ν,m(0,T ;H−1(T)) ≤ C. (5.10)

Besides, by the growth inequalities (2.2) and the Lp estimate (3.3), M ε alsosatisfies the bound

E‖M ε‖L2(QT ) ≤ C. (5.11)

Let us assume m > 2 now. We then have the continuous injection

W ν,m(0, T ;H−1(T)) → C0,µ([0, T ];H−1(T))

for every 0 < µ < ν− 1m . By the Aubin-Simon compactness Lemma (Simon,

[Sim87]), the set

AR :=M ∈ L2(QT ); ‖M ε‖W ν,m(0,T ;H−1(T)) ≤ R, ‖M‖L2(QT ) ≤ R

is compact in C([0, T ];H−1(T)), hence compact in L2(0, T ;H−1(T)). Con-sequently (5.10) and (5.11) show that (M ε) is tight as a L2(0, T ;H−1(T))-random variable, and we conclude that (∂tM

ε) is tight as a H−1(QT )-random variable.

5.1.2 Functional equation

Let us recall that a sequence of random variables (an) with values in anormed space X is said to be stochastically bounded if, for all η > 0, thereexists M > 0 such that P(‖an‖X ≥ M) ≤ η for all n. By the Markovinequality, E‖an‖X ≤ C implies (an) stochastically bounded.

We consider the Ito Formula (3.6). Let p > 2. By the Lp estimates (3.3),which gives bounds on η(Uε), H(Uε) in Lp(QT ), the left-hand side of (3.6)is stochastically bounded in W−1,p(QT ). By Proposition 5.2 above, the term

47

∂tMε in the right-hand side of (3.6) is tight in H−1(QT ). The two remaining

terms

εη′′(Uε) · (Uεx,U

εx) and

1

2|G(Uε)|2∂2qqη(Uε)

are stochastically bounded in measure on QT by (3.4)-(3.5) and (2.2)-(3.3)respectively. Eventually, by Proposition 5.1, the term ε(η(Uε))xx is tightin W−1,q(QT ) for any q ∈ (1, 2). We want now to apply the stochasticversion of the Murat’s Lemma, Lemma A.3 in [FN08]. If we refer strictlyto the statement of Lemma A.3 in [FN08], there is an obstacle here, due tothe fact that ε(η(Uε))xx is neither tight in H−1(QT ), neither stochasticallybounded in measure on QT . However, in the proof of Lemma A.3 in [FN08],the property which is used regarding the term that is stochastically boundedin measure on QT is only the fact that it is tight in W−1,q(QT ) for 1 < q < 2due to the compact injection W 1,r

0 (QT ) C(QT ) for r > 2. The argumentabout interpolation theory which combines this compactness result with thestochastic bound in W−1,p(QT ) can therefore be directly applied here: wededuce that the sequence of H−1(QT ) random variables

divt,x(η(Uε), H(Uε)) = η(Uε)t +H(Uε)x

is tight. Let now (η, H) be a second entropy - entropy flux couple associatedby (2.7)-(2.9) to a convex C2 function g satisfying (5.1). Similarly, thesequence of random variables

curlt,x(−H(Uε), η(Uε)) = η(Uε)t + H(Uε)x

is tight. By Theorem A.2 in [FN08] and Proposition 4.6, we obtain, for a.e.(x, t) ∈ QT ,

〈η〉〈H〉 − 〈η〉〈H〉 = 〈ηH − ηH〉, (5.12)

where 〈φ〉 := 〈φ, νx,t〉 and where the identity in (5.12) is the identity betweenthe laws of the processes involved. To obtain an almost sure equality, weapply Skorokhod Theorem: by Remark 4.5, there exists a new probabilityspace (Ω, F , P), a new C([0, T ];W 1,2(T)) random variable Uε, ν a randomYoung measure, such that Law(δUε) = Law(δUε), Law(ν) = Law(ν) and,a.s.,

δUε → ν.

Then, by a slight adaptation of the probabilistic div-curl Lemma, Theorem 2in [FN08], we obtain (5.12) almost everywhere in Ω, and with 〈φ〉 := 〈φ, νx,t〉.

5.2 Reduction of the Young measure

We now follow [LPS96] to conclude. We switch from the variables (ρ, u) or(ρ, q) to (w, z), where

z = u− ρθ, w = u+ ρθ.

48

By (5.12), we have, a.s., for a.e. (x, t) ∈ QT ,

2λ(〈χ(v)u〉〈χ(v′)〉 − 〈χ(v)〉〈χ(v′)u〉

)= (v − v′)

(〈χ(v)χ(v′)〉 − 〈χ(v)〉〈χ(v′)〉

), (5.13)

where

〈χ(v)〉 =

∫χ(w, z, v) dνx,t(w, z), χ(w, z, v) := (v − z)λ+(w − v)λ+.

Let us fix (ω, x, t) such that (5.13) is satisfied. Let C denote the set

C = v ∈ R ; 〈χ(v)〉 > 0 =⋃

(w,z)∈suppνx,t

v ; z < v < w.

LetV = (ρ, u) ∈ R+ × R|ρ = 0 = (w, z) ∈ R2|w = z

denote the vacuum region. If C is empty, then νx,t is concentrated on V .Assume C not empty. By Lemma I.2 in [LPS96] then, C is an open intervalin R, say C =]a, b[, where −∞ ≤ a < b ≤ +∞ (we use here the frenchnotation for open intervals to avoid the confusion with the point (a, b) ofR2). Furthermore all the computations of [LPS96] apply here, and thus, asin Section I.6 of [LPS96], we obtain

〈ρ2λθ〈χ πi〉φ πi〉 = 0, (5.14)

for any continuous function φ with compact support in C, where πi : R2 → Rdenote the projection on the first coordinate w if i = 1, and the projectionon the second coordinate z if i = 2. If we assume that there exists Q ∈ R2

satisfyingQ ∈ supp(νx,t) \ V, πi(Q) ∈ C, (5.15)

for i ∈ 1, 2, then there exists a neighbourhood K of Q such that K∩V = ∅,νx,t(K) > 0, πi(K) ⊂ C. But then 〈χ πi〉 > 0 on K, ρ > 0 on K and,choosing a continuous function φ compactly supported in C such that φ > 0on K we obtain a contradiction to (5.14). Consequently (5.15) cannot besatisfied. This implies that there cannot exists two distinct points P,Qin supp(νx,t) \ V . Indeed, if two such points exists, then either π1(Q) <π1(P ), and then Q satisfies (5.15) with i = 1, or π1(Q) = π1(P ) and, say,π2(P ) < π2(Q) and then Q also satisfies (5.15). The other cases are similarby symmetry of P and Q.Therefore if C 6= ∅, then the support of the restriction of νx,t to C is reducedto a point. In particular, a and b are finite. Then, by Lemma I.2 in [LPS96],P := (a, b) ∈ supp(νx,t) and νx,t = µx,t+αδU(t,x), where µx,t = νx,t|V . Using

(5.13), we obtain

0 = (v − v′)χ(b, a, v)χ(b, a, v′)(α− α2),

and thus α = 0 or 1. We have therefore proved the following result.

49

Proposition 5.3 (Reduction of the Young measure). Either νx,t is concen-trated on the vacuum region V , or νx,t is reduced to a Dirac mass δU(t,x).

5.3 Martingale solution

By Proposition 5.3, we have∫∫QT

S(Uε(t, x))ϕ(t, x)dxdt→∫∫

QT

S(U(t, x))ϕ(t, x)dxdt (5.16)

almost surely for every continuous and bounded function S on R+×R whichvanishes on the vacuum region 0×R and every ϕ ∈ L1(QT ). There is alsostrong convergence of S(Uε) to S(U) in L2(QT ), almost-surely. This can bededuced from (5.16), which gives directly the weak convergence in L2(QT )by taking ϕ ∈ L2(QT ), but also the convergence of the norms by taking S2

instead of S and ϕ ≡ 1.

For the moment we have only supposed that Uε0 ∈W 1,∞(T) and that (Uε

0)is bounded in L∞(T). Assume furthermore

Uε0 → U0 in L1(T). (5.17)

The entropy and entropy flux as defined in (2.10), (2.11) are examples offunctions S as above: we can therefore pass to the limit in (3.6). We will pro-ceed as follows (along the lines of the proof of Theorem 3.4.12 in [Hof13b]):let ϕ ∈ C1(T) be fixed. Since η(Uε) → η(U) in L1(Ω × QT ) we have, forpossibly a further subsequence,

〈η(Uε), ϕ〉(t)→ 〈η(U), ϕ〉(t), ∀t ∈ D, almost surely, (5.18)

where D is a set of full measure in [0, T ] which contains the point t = 0. Set

eε = εη′′(Uε) · (Uεx,U

εx), eε = εη′′(Uε) · (Uε

x, Uεx).

LetMb(QT ) denote the set of bounded Borel measures on QT andM+b (QT )

denote the subset of nonnegative measures. By Remark ??, there exists arandom variable

e ∈ L2w(Ω;Mb(QT )), e ∈M+

b (QT ) P-almost surely,

such that, for all ψ ∈ C(QT ), for all Y ∈ L2(Ω), and up to a subsequence,

E(〈eε, ψ〉Y

)→ E

(〈e, ψ〉Y

). (5.19)

In (5.19), 〈·, ·〉 denotes the duality pairing between Mb(QT ) and C(QT )(which we extend below in the right-hand side of (5.20) to the pairing be-tween Mb(QT ) and Bb(QT ), the set of Borel bounded functions over QT ).

50

The subscript w in L2w(Ω;Mb(QT )) indicates that we consider weak-star

measurable mappings e from Ω into Mb(QT ), i.e. maps e such that 〈e, ψ〉is F-measurable for every ψ ∈ C(QT ). Then L2

w(Ω;Mb(QT )) is the dual ofthe space L2(Ω;C(QT )) (Edwards [Edw65, Theorem 8.20.3]), hence (5.19)follows from the Banach-Alaoglu Theorem and from the estimate (??) withI = (0, T ). Besides (??) with arbitrary I implies that, almost surely, e hasno atom in time, i.e.

E e(T× t) = 0, ∀t ∈ [0, T ].

This shows in particular that (5.19) holds true when ψ = 1[0,t)ϕ where

ϕ ∈ C(T): for all Y ∈ L2(Ω),

E(∫ t

0〈eε, ϕ〉dsY

)→ E

(〈e,1[0,t)ϕ〉Y

). (5.20)

Let now (Ft) be the P-augmented canonical filtration of the process (U, W ),i.e.

Ft = σ(σ(%tU, %tW

)∪N ∈ F ; P(N) = 0

), t ∈ [0, T ],

where the restriction operator %t is defined in (3.77). Then the sextuplet(Ω, F , (Ft), P, W , U

)is a weak martingale solution to (1.1). To show this, we use a reasoninganalogous to the one followed in Section 3.1.8. Let ϕ ∈ C1(T). Define

M ε(t) =⟨η(Uε)(t), ϕ

⟩−⟨η(Uε

0), ϕ⟩

+

∫ t

0

⟨∂xH(Uε)− ε∂2xη(Uε), ϕ

⟩ds

−∫ t

0〈eε(s), ϕ〉ds,

M ε(t) =⟨η(Uε)(t), ϕ

⟩−⟨η(Uε

0), ϕ⟩

+

∫ t

0

⟨∂xH(Uε)− ε∂2xη(Uε), ϕ

⟩ds

−∫ t

0〈eε(s), ϕ〉ds,

M(t) =⟨η(U)(t), ϕ

⟩−⟨η(U0), ϕ

⟩+

∫ t

0

⟨∂xH(U), ϕ

⟩ds− 〈e,1[0,t)ϕ〉.

Then M ε and M ε have same laws. By the convergence (5.16), (5.18) and(5.20) we show, as in Section 3.1.8 that the processes

M, M2 −∑k≥1

∫ ·0

⟨σk(U)∂qη(U), ϕ

⟩2dr, M βk −

∫ ·0

⟨σk(U)∂qη(U), ϕ

⟩dr

(5.21)

51

are (Ft)-martingales. There is however a notable difference between theresult of Lemma 3.24 and the result (5.21) here, in the fact that the martin-gales in (5.21) are indexed by D ⊂ [0, T ] since we have used the convergence(5.18). If all the processes in (5.21) were continuous martingales indexed by[0, T ], we would infer, as in the proof of Proposition 3.22, that⟨

η(U)(t), ϕ⟩−⟨η(U0), ϕ

⟩−∫ t

0

⟨H(U), ∂xϕ

⟩ds

= −〈e,1[0,t)ϕ〉+∑k≥1

∫ t

0

⟨σk(U)∂qη(U), ϕ

⟩dβk(s), (5.22)

for all t ∈ [0, T ], P-almost surely. Nevertheless, D contains 0 and is densein [0, T ] since it is of full measure, and it turns out, by the Proposition A.1in [Hof13b] on densely defined martingales, that this is sufficient to obtain(5.22) for all t ∈ D, P-almost surely. Then we conclude as in the proof ofTheorem 4.13 of [Hof13b]: let N(t) denote the continuous semi-martingaledefined by

N(t) =

∫ t

0

⟨H(U), ∂xϕ

⟩ds+

∑k≥1

∫ t

0

⟨σk(U)∂qη(U), ϕ

⟩dβk(s).

Let t ∈ (0, T ] be fixed and let α ∈ C1c ([0, t)). By the Ito Formula we compute

the stochastic differential of N(s)α(s) to get

0 =

∫ t

0N(s)α′(s)ds+

∫ t

0

⟨H(U), ∂xϕ

⟩α(s) ds

+∑k≥1

∫ t

0

⟨σk(U)∂qη(U), ϕ

⟩α(s) dβk(s). (5.23)

By (5.22), we have

N(t) =⟨η(U)(t), ϕ

⟩−⟨η(U0), ϕ

⟩+ 〈e,1[0,t)ϕ〉,

for all t ∈ D, P-almost surely. In particular, by the Fubini Theorem,∫ t

0N(s)α′(s)ds =

∫ t

0

⟨η(U)(s), ϕ

⟩α′(s) ds

+⟨η(U0), ϕ

⟩α(0)−

∫ t

0α(σ)dρ(σ), (5.24)

P-almost surely, where we have defined the measure ρ by ρ(B) = 〈e,1Bϕ〉,for B a Borel subset of [0, T ]. If α,ϕ ≥ 0, then∫ t

0α(σ)dρ(σ) ≥ 0, P− almost surely

and (2.16) follows from (5.23), (5.24). This concludes the proof of Theo-rem 2.7.

52

6 Conclusion

We want to discuss in this concluding section some open questions related tothe qualitative behaviour of solutions to (1.1). The first one is the question ofuniqueness. Uniqueness of weak entropy solutions is an unsolved questionin the deterministic setting. For the stochastic problem (1.1), one mayconsider the question of uniqueness of martingale solutions, i.e. uniquenessin law. Is it a promising question? Would it be easier to answer thanthe question of uniqueness in the deterministic setting, as, for example,uniqueness in law manifests itself in stochastic differential equations withcontinuous coefficients, which represent therefore stochastic perturbationsof o.d.e’s displaying some indeterminacy [SV79]? It may well be the casethat this issue of uniqueness is simpler for a noise of the form (ρuΦW )x, butfor such a noise occurring in the flux term, we do not know for the momenthow to prove existence (see [LPS13] for the equivalent problem in the scalarcase).

An other problem concerns the long-time behaviour of solutions to (1.1). Itis known that for scalar stochastic conservation laws with additive noise,and for non-degenerate fluxes, there is a unique ergodic invariant measure,cf. [EKMS00, DV13]. Since both fields of (1.1) are genuinely non-linear,a form of non-degeneracy condition is clearly satisfied in (1.1). Actually,in the deterministic case Φ ≡ 0, the solution converges to the constantstate determined by the conservation of mass [CF99, Theorem 5.4], whichindicates that some kind of dissipation effects (via interaction of waves, cf.also [GL70]) occur in the Euler system for isentropic gas dynamics. However,in a system there is in a way more room for waves to evolve than in a scalarconservation law, and the long-time behaviour in (1.1) may be different fromthe one described in [EKMS00, DV13].

Specifically, consider the case γ = 2. For such a value the system of Eulerequations for isentropic gas dynamics is equivalent to the following Saint-Venant system of equations for shallow water:

ht + (hu)xdt = 0, in QT , (6.1a)

(hu)t + (hu2 + gh2

2)x + ghZx = 0, in QT , (6.1b)

with Z(t, x) = Φ∗(x)dWdt and QT = T × (0, T ). When Z = Z(x), (6.1) is amodel for the one-dimensional flow of a fluid of height h and speed u over aground described by the curve z = Z(x) (u(x) is the speed of the column ofwater over the abscissa x)4. For a random Z as in (6.1b), the system (6.1)describes the evolution of the fluid in terms of (h, u) when its behaviour is

4the fact that u is independent on the altitude z is admissible as long as h is smallcompared to the longitudinal length L of the channel, L = 1 here, cf. [GP01]

53

forced by the moving topography, the question being thus to determine ifan equilibrium in law (and which kind of equilibrium) for such a randomprocess can be reached when time goes to +∞. An hint for the existence ofa unique, ergodic, invariant measure is the “loss of memory in the system”given by the ergodic theorem: if f is a bounded, continuous functional ofthe solution U(t), then

limT→+∞

1

T

∫ T

0f(U(t))dt→ 〈f, µ〉 a.s. (6.2)

where µ is the invariant measure. Before testing the ergodic convergence(6.2), one has first to restrict the evolution to the right manifold. Indeed,in the scalar case [EKMS00, DV13], say for the equation

dv + (A(v))x = ∂xφ(x)dW (t), x ∈ T, t > 0,

there is a unique invariant measure µλ indexed by the constant parameter

λ =

∫Tv(x)dx ∈ R.

For (6.1), the entropy solution is evolving on the manifold∫Th(x)dx = cst.

Is there or is there not an other quantity preserved in the evolution? Thisis currently not clear. A way to enforce preservation of a second quantity,is to use symmetry, by choosing Z to be an odd function of x, as well as u,while h is an even function of x. Then q = hu is an odd function of x andthe identity ∫

Tq(t, x)dx = 0

is maintained in the evolution. In that case the convergence (6.2) is clearlyobserved on our numerical test. Actually, it is even not necessary to startwith an h and a u having a particular symmetry: as long as Z is symmetric,it will enforce the symmetry on h and u. This is illustrated by Figure ?? onpage ??, where are depicted, respectively, the evolution of

t 7→ 1

t

∫ t

0E(U(s))ds and t 7→

∫Tq(s, x)ds,

where

E(U) :=

∫T

1

2hu2 +

1

2gh2 dx

denotes the energy of the system. For the four test cases considered, thequantity

∫T hdx is the same of course. Note also that the numerical simula-

tions for each test case are done simultaneously, therefore the same noise is

54

drawned for each test. In the non-symmetric case, we are for the momentnot able to conclude, cf. Figure ?? on page ??, where no significant conver-gence of the energy is observed. The question of the large-time behaviourof solutions will be addressed in a future work.

A A parabolic uniformization effect

The following result, a phenomenon of uniformization from below for a non-negative solution for a parabolic equation with a drift-term with quite lowregularity, will be proved with techniques similar to those used in [MV09].not really, explain why

Theorem A.1 (Positivity). Let τ > 0. Let 1det be the step function definedby (3.12). Let ρ, u ∈ L2(0, T ;H1(T)) satisfy

ρ|∂xu|2 ∈ L1(T× (0, T )), ∂tρ ∈ L2(0, T ;H−1(T)). (A.1)

Assume that ρ is a non-negative solution in which sense? to the equation

1

2∂tρ+ 1det

[∂x(ρu)− ∂2xρ

]= 0 (A.2)

and that ρ0 := ρ(0) satisfy 1ρ0∈ Lq(T) where q > 3. Then, for all τ ∈ (0, T ]

and m > 3, there exists a constant c > 0 depending on ‖ρ−10 ‖Lq(T), q, T , τ ,m and ∫∫

QT

ρ|∂xu|2dxdt and ‖u‖Lm(QT ) (A.3)

only, such thatρ ≥ c (A.4)

a.e. in T× [τ, T ].

Proof. Note first that the L2tH

1x regularity of ρ, together with the L2

tH−1x

regularity of ∂tρ implies the existence of a representative still denoted ρwhich is continuous in time with values in L2. This is the value of thisrepresentative at time t = 0 which is denoted by ρ0. Now, to prove theresult, we will show a bound from above on w = 1

ρ . Actually, w is notyet well-defined and we shall more rigorously prove a bound from above onwε = 1

ρ+ε that is uniform with respect to the parameter ε > 0. For simplicitywe will work directly on w, the main lines of the proof are easily adapted forwε. By a chain-rule formula (cf. Lemma 1.4 in Carrillo, Wittbold [CW99]for example) we derive the following equation for w:

1

2∂tw − 1det∂

2xw = 1det(−8|∂xw1/2|2 + w∂xu− u∂xw). (A.5)

55

Similarly, we have, for p ≥ 2,

1

2∂tz2

p− 1det∂

2x

z2

p= 1det

(−4(p+ 1)

p2|∂xz|2 + z2∂xu−

u

p∂xz

2

), (A.6)

where z := wp/2. We will use (A.6) to obtain an estimate on ‖w‖L∞t Lpx infunction of ‖u‖Lr(QT ) and ‖w(0)‖Lp(T) only (see Equation A.9 below). Letus sum (A.6) on T: we obtain

1

2

d

dt

∫Tz2dx+

4(p+ 1)

p1det

∫T|∂xz|2dx = −(2p+ 1)1det

∫Tuz∂xzdx.

Consequently, we have

1

2

d

dt

∫Tz2dx+

2(p+ 1)

p1det

∫T|∂xz|2dx =

p(p+ 1)

21det

∫Tu2z2dx.

Integrating then over t ∈ [0, σ] where σ ≤ T , we obtain

Uσ ≤p(p+ 1)

2

∫∫Qσ

1detu2z2dxdt+

1

2‖z(0)‖2L2(T),

where

Uσ :=1

2supt∈[0,σ]

‖z(t)‖2L2(T) +2(p+ 1)

p‖1det∂xz‖2L2(Qσ)

.

By the Holder Inequality, it follows that

Uσ ≤p(p+ 1)

2‖u‖2L3(Qσ)

‖1detz‖2L6(Qσ)+

1

2‖z(0)‖2L2(T). (A.7)

To obtain an estimate on the right hand-side of (A.7), we apply the followinginequality (proven below)

‖1detz‖L6(Qσ) ≤ C

(supt∈[0,σ]

‖z(t)‖L2(T)

)2/3

‖1det∂xz‖1/3L2(Qσ)

, (A.8)

where C is a numerical constant. This gives

Uσ ≤ C2 p(p+ 1)

2‖u‖2L3(Qσ)

Uσ +1

2‖z(0)‖2L2(T),

and then, since m > 3,

Uσ ≤ C2 p(p+ 1)

2σe‖u‖2Lm(QT )

Uσ +1

2‖z(0)‖2L2(T), e :=

2

3− 2

m.

Consequently, for σ < σ0, we obtain Uσ ≤ C‖wp/2(0)‖2L2(T), where σ0 > 0

and C ≥ 0 are some constant depending only on ‖u‖Lm(QT ) and p. Since an

56

estimate on Uσ gives in turn an estimate on ‖z(σ)‖2L2(T) = ‖wp/2(σ)‖2L2(T),we can iterate our procedure to deduce the following bound:

supt∈[0,T ]

‖w(t)‖pLp(T) + ‖1det∂xwp/2‖2L2(QT )

≤ C‖w(0)‖pLp(T), (A.9)

where C is a constant depending on p, m, T , ‖u‖Lm(QT ). This estimate isthe first step in the proof of Theorem A.1. To achieve the proof of (A.9), westill have to show (A.8). We use the injection Hδ(T) ⊂ Lr(T), δ ∈ [0, 1/2),1r := 1

2−δ, an interpolation inequality and the Poincare Inequality to obtain

‖z(t)‖rLr(T) ≤ CP ‖z(t)‖r(1−δ)L2(T) ‖∂xz(t)‖

rδL2(T), (A.10)

for a given numerical constant CP . Then we multiply the result by 1det(t)and we sum over t ∈ [0, σ]. For rδ = 2 (an equality which sets the value of(δ, r) to (1/3, 6)), we obtain (A.8).

In the second step of the proof, we will derive the following L∞ estimate onw:

‖w‖L∞(Qτ,T ) ≤ C(τ, T, ‖u‖Lm(QT ), ‖ρ

1/2∂xu‖L2(QT ), ‖w‖Lp0 (QT )), (A.11)

for a p0 > 3 that will be specified later. To prove (A.11), we use the equation(A.5), which we rewrite in the mild form justify

w(t) = S(t])w(0) +

∫ t]

0S(t] − s)f(s[)ds,

where we have set

t] := min(2t− t2n, t2n+2), t[ :=t+ t2n

2, t2n ≤ t ≤ t2n+2,

and where f is the right hand-side of (A.5). Since

f ≤ 1det(w∂xu− u∂xw),

we obtain0 ≤ w(t) ≤ S(t])w(0) +W1(t) +W2(t),

with

W1(t) =

∫ t]

0S(t] − s)(1detw∂xu)(s[)ds,

W2(t) = −∫ t]

0S(t] − s)(1detu∂xw)(s[)ds.

Let us set g = ρ1/2∂xu and rewrite W1(t) as

W1(t) =

∫ t]

0S(t] − s)(1detw

3/2g)(s[)ds.

57

Fix σ ∈ (0, T ). Let pk ∈ [1,+∞), rk ∈ [1, 2) be given. By (B.7) with j = 0,we have

‖W1‖Lpk+1 (Qσ,T ) ≤ C‖w3/2g‖Lrk (Qσ,T ),

1

pk+1<

1

rk<

1

pk+1+

2

3, (A.12)

≤ C‖w‖3/2Lpk (Qσ,T )‖g‖L2(QT ), (A.13)

where Qσ,T = T× (σ, T ), and provided pk and rk satisfy the relation

1

2+

3

2pk=

1

rk. (A.14)

Furthermore, by integration by parts, W2(t) = W1(t)−W3(t),

W3(t) = −∫ t]

0∂xS(t] − s)(1detuw)(s[)ds.

By (B.7) with j = 1, we have

‖W3‖Lpk+1 (Qσ,T ) ≤ C‖wu‖Lqk (Qσ,T ),1

pk+1<

1

qk<

1

pk+1+

1

3, (A.15)

≤ C‖w‖Lpk (Qσ,T )‖u‖Lm(QT ), (A.16)

provided pk and qk satisfy the relation

1

pk+

1

m=

1

qk. (A.17)

Finally, the estimate (B.2) shows that

‖S(t])w(0)‖L∞(Qσ,T ) ≤ Cσ− 1

2p0 ‖w(0)‖Lp0 (T), 1 ≤ p0 ≤ +∞, (A.18)

for a constant C depending on T only. Let

a = −1

6+ δ, b =

1

m− 1

3+ δ,

δ small, and let1

pk+1= min

(3

2pk+ a,

1

pk+ b

). (A.19)

Then rk and qk defined by (A.14), (A.17) satisfy the constraints (A.12),(A.15) respectively. It follows then from (A.13), (A.16), (A.18) that ‖w‖Lpk (Qσ,T )satisfies the recursion formula

‖w‖Lpk+1 (Qσ,T ) ≤ Cσ− 1

2p0 ‖w(0)‖Lp0 (T) + C(1 + ‖w‖3/2Lpk (Qσ,T )), k ≥ 0,

where the constant C depends on T , ‖u‖Lm(QT ), ‖ρ1/2∂xu‖L2(QT ) only. We

choose p0 > 31−6δ . Then there exists a finite K > 0 such that 1

pK> 0

while 1pK+1

given by (A.19) is negative, which means that we may as well

take pK+1 = +∞. We deduce the estimate (A.11). Using then the estimate(A.9) for p = p0, we obtain (A.4), which concludes the proof of the Theorem.

58

Remark A.2. Note that, if ρ0 is positive, say ρ0 ≥ c0 a.e. in T, wherec0 > 0, then w0 ∈ L∞(T). We may therefore replace take σ = 0 in theestimates (A.13), (A.16) above and replace (A.18) by the estimate

‖S(t])w(0)‖L∞(QT ) ≤ T‖w(0)‖L∞(T)

to conclude that (A.4) holds on [0, T ].

Remark A.3. Note also that it is possible to give some precisions on thebound from below (I.57) in [LPS96], regarding the positivity of the density ρin the deterministic parabolic approximation of the isentropic Euler system.Since, for such a system, the terms in (A.3) are bounded, respectively, bythe initial entropy∫

TηE(U0(x))dx ≤ C(‖ρ0‖L∞(T), ‖u0‖L∞(T))

and the L∞ norm

‖u‖L∞(QT ) ≤ TC(‖ρ0‖L∞(T), ‖u0‖L∞(T)),

where here C is a continuous function of its arguments, we obtain ρ ≥ c1a.e. in QT , where c1 depends continuously on T , ‖ρ0‖L∞(T), ‖u0‖L∞(T), c0,where c0 = infx∈T ρ0(x).

B Regularizing effects of the one-dimensional heatequation

Let T > 0, let z ∈ C([0, T ];L2(T)) satisfy

z(t) = S(t)z0 +

∫ t

0S(t− s)f(s)ds, (B.1)

for some given data z0 and f , where S(t) is the semi-group associated to theheat operator ∂t − ∂2x on T. The function z is a mild solution to the heatequation

(∂t − ∂2x)z = f in QT ,

with initial condition z(0) = z0. If f ∈ Lp(QT ), p > 65 and z0 ∈ L2(T)

then, by the regularizing properties of S(t), (B.1) gives a z which is indeedin C([0, T ];L2(T)). More precisely, we have

‖S(t)‖Lpx→Lqx ≤ Ct− 1

2

(1p− 1q

), (B.2)

for 1 < p ≤ q ≤ +∞, for a given constant C and therefore, for possibly adifferent constant C,

‖S(t− s)f(s)‖L2(T) ≤ C(t− s)−µ‖f(s)‖Lp(T), µ :=1

2

(1

p− 1

2

).

59

By the Young inequality for the convolutions of functions, Lp′ ∗ Lp embeds

in the space of continuous functions (here p′ is the conjugate exponent to p),hence z defined by (B.1) is indeed continuous in time with values in L2(T)if t 7→ t−µ is in Lp

′, which is equivalent to the condition p > 6

5 . Using moregenerally the regularizing properties

‖∂jxS(t)‖Lpx→Lqx ≤ Ct− 1

2

(1p− 1q

)− j

2 (B.3)

for j ∈ N, we obtain that, for a given constant C ≥ 0,∥∥∥∥∂jx ∫ t

0S(t− s)f(s)ds

∥∥∥∥Lq(QT )

≤ C‖f‖Lp(QT ) if1

q<

1

p<

1

q+

2− j3

,

(B.4a)∥∥∂jxS(t)z0∥∥Lq(QT )

≤ C‖z0‖Lp(T) if1

q<

1

p<

3

q− j, (B.4b)

for j ∈ 0, 1.

Remark B.1 (Time regularity). Let ν > 0, m ≥ 1. If t 7→ t−ν is inLm(0, T ) then it is also in W σ,m(0, T ) for a certain σ > 0. This shows thatwe have actually∥∥∥∥∂x ∫ t

0S(t−s)f(s)ds

∥∥∥∥Wσ,q(0,T ;Lq(T))

≤ C‖f‖Lp(QT ) if1

q>

1

p− 1

3, (B.5)

for a given σ > 0 depending on p and q.

We end this section with a variation over (B.4a) for the solution of a splitevolution equation (this is used in Section 3.1.6): let τ > 0, set tn = nτ ,n ∈ N, set

t] := min(2t− t2n, t2n+2), t[ :=t+ t2n

2, t2n ≤ t ≤ t2n+2,

and let

z(t) =

∫ t]

0S(t] − s)f(s[)ds.

The function z is the solution to

1

2∂tz − ∂2xz = f, in T× (t2n, t2n+1), ∂tz = 0, in T× (t2n+1, t2n+2),

for n = 0, 1, . . .. If T = t2K , K ∈ N∗, we have, for h ∈ L1(0, T ),∫ T

0h(t])dt =

1

2

∫ T

0h(t)dt+

K−1∑n=0

τh(t2n+2)

=

∫ T

0h(t)dt+

1

2

K−1∑n=0

∫ t2n+2

t2n

(h(t2n+2)− h(t))dt. (B.6)

60

If there exists a σ ∈ (0, 1) such that h ∈ W σ,1(0, T ), we can estimate theremainder in (B.6) and obtain∣∣∣∣ ∫ T

0h(t])dt−

∫ T

0h(t)dt

∣∣∣∣ ≤ C‖h‖Wσ,1(0,T )τσ.

By Remark B.1 and (B.4), we deduce in particular that, for τ ≤ 1 and j = 0or 1,∥∥∥∥∂jx ∫ t]

0S(t] − s)f(s[)ds

∥∥∥∥Lq(QT )

≤ C‖f‖Lp(QT ) if1

q<

1

p<

1

q+

2− j3

,

(B.7)where f is the extension by 0 of f to the strips T × (t2n+1, t2n+2), n =0, . . . ,K−1, and C ≥ 0 a numerical constant. Similarly, by (B.4b), we have∥∥∂jxS(t])z0

∥∥Lq(QT )

≤ C‖z0‖Lp(T) if1

q<

1

p<

3

q− j, (B.8)

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