STOCHASTIC NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLUIDS

Transcription

1 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS DOMINIC BREIT AND MARTINA HOFMANOVÁ Abstract. We study the Navier-Stokes equations governing the motion of an isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density. We establish existence of a so-called finite energy weak martingale solution under the condition that the adiabatic constant satisfies γ > 3/. The proof is based on a four layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure. 1. Introduction We consider the Navier-Stokes system for isentropic compressible viscous fluids driven by a multiplicative stochastic forcing and prove existence of a solution that is weak in both PDE and probabilistic sense. To be more precise, let = [, 1 3 denote the three-dimensional torus, let T > and set =, T. We study the following system which governs the time evolution of density ϱ and velocity u of a compressible viscous fluid: 1.1a 1.1b dϱ + divϱudt =, dϱu + [ divϱu u ν u λ + ν div u + pϱ dt = Φϱ, ϱu dw. These equations describe the balance of mass and momentum of the flow. Here pϱ is the pressure which is posed to follow the γ-law, i.e. pϱ = aϱ γ where a > and a is the squared reciprocal of the Mach-number ratio of flow velocity and speed of sound. For the adiabatic exponent γ also called isentropic expansion factor we pose γ > 3. Finally, the viscosity coefficients ν, λ satisfy ν >, λ + 3 ν. The driving process W is a cylindrical Wiener process defined on some probability space Ω, F, P and the coefficient Φ is generally nonlinear and satisfies suitable growth conditions. The precise description of the problem setting will be given in the next section. The literature devoted to deterministic case is very extensive see for instance Feireisl [1, Feireisl, Novotný and Petzeltová [14, Lions [1, Novotný and Straškraba [7 and the references therein. The existence of weak solutions in the non-stationary setting is well-known provided γ > 3 in three dimensions, in two dimensions γ > 1 suffices instead. This might not be optimal but already covers important examples like mono-atomic gases where γ = 5 3. In the stationary situation the results have been recently extended to γ > 1, see [17, 31. The theory for the stochastic counterpart still remains underdeveloped. The only available results see Feireisl, Maslowski and Novotný [13 for d = 3 and [35 in the case d = concern the Navier-Stokes system for compressible barotropic fluids under a stochastic perturbation of the form ϱ dw. This particular case of a multiplicative noise permits reduction of the problem. After applying some transformation it can be solved pathwise and therefore existence of a weak solution was established using deterministic arguments. This method has the drawback that the constructed solutions do not necessarily satisfy an energy inequality and are not Date: December 18, Mathematics Subject Classification. 6H15, 35R6, 76N1, 353. Key words and phrases. Compressible fluids, stochastic Navier-Stokes equations, weak solution, martingale solution. 1

2 DOMINIC BREIT AND MARTINA HOFMANOVÁ progressively measurable hence the stochastic integral is not defined. We are not aware of any results concerning the Navier-Stokes equations for compressible fluids driven by a general multiplicative noise. Nevertheless, study of such models is of essential interest as they were proposed as models for turbulence, see Mikulevicius and Rozovskii [5. In case of a more general noise, the simplification mentioned before is no longer possible and methods from infinite-dimensional stochastic analysis are required. There is a bulk of literature available concerning stochastic versions of the incompressible Navier-Stokes equations. Let us mention the pioneering paper by Bensoussan and Temam [ and for an overview of the known results, recent developments, as well as further references, we refer to [8, [15 and [3. The literature concerning other fluid types is rather rare. Just very recently first results on stochastic models for Non-Newtonian fluids appeared see [4, [34 and [36. Incompressible non-homogenous fluids with stochastic forcing were studied in [18 and more recently in [33; one-dimensional stochastic isentropic Euler equations in [3. We aim at a systematic study of compressible fluids under random perturbations. Our main result is the existence of a weak martingale solution to 1.1 in the sense of Definition.1, see Theorem.4. Our solution satisfies an energy inequality which shows the time evolution of the energy compared to the initial energy. The setting includes in particular the case of Φϱ, ϱu dw = Φ 1 ϱ dw 1 + Φ ϱu dw with two independent cylindrical Wiener processes W 1 and W and suitable growth assumptions on Φ 1 and Φ, which is the main example we have in mind. Here the first term describes some external force; the case Φ 1 ϱ = ϱ studied in [13 is included but we could also allow nonlinear dependence in ϱ the case Φϱ, ϱu dw = ϱ dw corresponds to the forcing ϱ f from deterministic models. The second term is a friction term; the model case is Φ ϱu being proportional to the momentum ϱu but the dependence can be nonlinear as well. The solution is understood weakly in space-time in the sense of distributions and also weakly in the probabilistic sense the underlying probability space is part of the solution. Such a concept of solution is very common in the theory of stochastic partial differential equations SPDEs, in particular in fluid dynamics when the corresponding uniqueness is often not known. We refer the reader to Subsection.1 for a detailed discussion of this issue. The proof of Theorem.4 relies on a four layer approximation scheme that is motivated by the technique developed by Feireisl, Novotný and Petzeltová [14 in order to deal with the corresponding deterministic counterpart. In each step we are confronted with the limit procedure in several nonlinear terms and in the stochastic integral. There is one significant difference in comparison to the deterministic situation leading to the concept of martingale solution: In general it is not possible to get any compactness in ω as no topological structure on the sample space Ω is assumed. To overcome this difficulty, it is classical to rather concentrate on compactness of the set of laws of the approximations and apply the Skorokhod representation theorem. It yields existence of a new probability space with a sequence of random variables that have the same laws as the original ones and that in addition converges almost surely. However, a major drawback is that the Skorokhod representation Theorem is restricted to metric spaces. The structure of the compressible Navier-Stokes equations naturally leads to weakly converging sequences. On account of this we work with the Jakubowski-Skorokhod Theorem which is valid on a large class of topological spaces including separable Banach spaces with weak topology. Further discussion of the key ideas of the proof is postponed to Subsection.. The exposition is organized as follows. In Section we continue with the introductory part: we introduce the basic set-up, the concept of solution and state the main result, Theorem.4. Once the notation is fixed we present also a short outline of the proof, Subsection.. The remainder of the paper is devoted to the detailed proof of Theorem.4 that proceeds in several steps.. Mathematical framework and the main result To begin with, let us set up the precise conditions on the random perturbation of the system 1.1. Let Ω, F, F t t, P be a stochastic basis with a complete, right-continuous filtration.

3 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 3 The process W is a cylindrical Wiener process, that is, W t = β kte k with β k being mutually independent real-valued standard Wiener processes relative to F t t. Here e k denotes a complete orthonormal system in a separable Hilbert space U e.g. U = L would be a natural choice. To give the precise definition of the diffusion coefficient Φ, consider ρ L γ, ρ, and v L such that ρv L. We recall that we assume γ > 3. Denote q = ρv and let Φρ, q : U L 1 be defined as follows Φρ, qe k = g k, ρ, q. The coefficients g k : R R 3 R 3 are C 1 -functions that satisfy uniformly in x.1 g k x, ρ, q C ρ + ρ γ+1 + q,. ρ,q g k x, ρ, q C 1 + ρ γ 1. Remark that in this setting L 1 is the natural space for values of the operator Φρ, ρv. Indeed, due to lack of a priori estimates for 1.1 it is not possible to consider Φρ, ρv as a mapping with values in a space with higher integrability. This fact brings difficulties concerning the definition of the stochastic integral in 1.1. In fact, the space L 1 does neither belong to the class -smooth Banach spaces nor to the class of UMD Banach spaces where the theory of stochastic Itô-integration is well-established see e.g. [5, [6, [9. However, since we expect the momentum equation 1.1b to be satisfied only in the sense of distributions anyway, we make use of the embedding L 1 W b, which is true provided b > 3. Hence we understand the stochastic integral as a process in the Hilbert space W b,. To be more precise, it is easy to check that under the above assumptions on ρ and v, the mapping Φρ, ρv belongs to L U; W b,, the space of Hilbert-Schmidt operators from U to W b,. Indeed, due to.1 there holds.3 Φρ, ρv L U;Wx b, g k ρ, ρv C g Wx b, k ρ, ρv L 1 x Cρ ρ 1 g k x, ρ, ρv dx = Cρ ρ + ρ γ + ρ v dx <, where ρ denotes the mean value of ρ over. Consequently, if 1 ρ L γ Ω, T, P, dp dt; L γ, ρv L Ω, T, P, dp dt; L, and the mean value ρt is essentially bounded then the stochastic integral Φρ, ρv dw is a well-defined F t -martingale taking values in W b,. Note that the continuity equation 1.1a implies that the mean value ϱt of the density ϱ is constant in time but in general depends on ω. Finally, we define the auxiliary space U U via { U = v = α k e k ; α } k k <, endowed with the norm v U = αk k, v = α k e k. Note that the embedding U U is Hilbert-Schmidt. Moreover, trajectories of W are P-a.s. in C[, T ; U see [7. 1 Here P denotes the predictable σ-algebra associated to Ft.

4 4 DOMINIC BREIT AND MARTINA HOFMANOVÁ.1. The concept of solution and the main result. We aim at establishing existence of a solution to 1.1 that is weak in both probabilistic and PDEs sense. Let us devote this subsection to the introduction of these two notions. From the point of view of the theory of PDEs, we follow the approach of [14 and consider the so-called finite energy weak solutions. In particular, the system 1.1 is satisfied in the sense of distributions, the corresponding energy inequality holds true and, moreover, the continuum equation 1.1a is satisfied in the renormalized sense. From the probabilistic point of view, two concepts of solution are typically considered in the theory of stochastic evolution equations, namely, pathwise or strong solutions and martingale or weak solutions. In the former notion the underlying probability space as well as the driving process is fixed in advance while in the latter case these stochastic elements become part of the solution of the problem. Clearly, existence of a pathwise solution is stronger and implies existence of a martingale solution. In the present work we are only able to establish existence of a martingale solution to 1.1. Due to classical Yamada-Watanabe-type argument see e.g. [19, [3, existence of a pathwise solution would then follow if pathwise uniqueness held true. However, uniqueness for the Navier Stokes equations for compressible fluids is an open problem even in the deterministic setting. In hand with this issue goes the way how the initial condition is posed: there is given a probability measure on L γ L γ γ+1, hereafter denoted by Λ. It fulfills some further assumptions specified in Theorem.4 and plays the role of an initial law for the system 1.1. That is, we require that the law of ϱ, ϱu coincides with Λ. Let us summarize the above in the following definition. Definition.1 Solution. Let Λ be a Borel probability measure on L γ L γ γ+1. Then Ω, F, Ft, P, ϱ, u, W is called a finite energy weak martingale solution to 1.1 with the initial data Λ provided a Ω, F, F t, P is a stochastic basis with a complete right-continuous filtration, b W is an F t -cylindrical Wiener process, c the density ϱ satisfies ϱ, t ϱt,, ψ C[, T for any ψ C P-a.s., the function t ϱt,, ψ is progressively measurable, and [ E ϱt, p L γ < for all 1 p < ; t [,T d the velocity field u is adapted, u L Ω, T ; W 1,, [ T p E u W 1, dt < for all 1 p < ; e the momentum ϱu satisfies t ϱu, ϕ C[, T for any ϕ C P-a.s., the function t ϱu, ϕ is progressively measurable, [ E ϱu p t [,T L γ γ+1 < for all 1 p < ; f Λ = P ϱ, ϱu 1. g Φϱ, ϱu L Ω [, T, P, dp dt; L U; W l, for some l > 3, h for all ψ C and ϕ C and all t [, T there holds P-a.s. t ϱt, ψ = ϱ, ψ + ϱu, ψ ds, t t ϱut, ϕ = ϱu, ϕ + ϱu u, ϕ ds ν u, ϕ ds λ + ν t t div u, div ϕ ds + a ϱ γ, div ϕ ds

5 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 5 + t Φϱ, ϱu dw, ϕ, i for all p [1, the following energy inequality holds true [ 1 E t T T ϱt ut a p + γ 1 ϱγ t dx [ 3 T + E ν u + λ + ν div u p.4 dx ds T [ 3 1 ϱu p a Cp E + T ϱ γ 1 ϱγ dx j Let b C 1 R such that b z = for all z M b. Then for all ψ C and all t [, T there holds P-a.s. t t bϱt, ψ = bϱ, ψ + bϱu, ψ ds b ϱϱ bϱu div u, ψ ds. Remark.. In j above, the continuity equation is assumed to hold in the renormalized sense. This concept was introduced in [1. It is an essential tool to pass to the limit in the nonlinear pressure and therefore common in compressible fluid mechanics. Remark.3. The condition g was included in order to point out that the stochastic integral in 1.1 is a well-defined stochastic process with values in W b,, in particular, the integrand is progressively measurable. Nevertheless, the conditions on ϱ and u together with the energy inequality.4 already imply that Φϱ, ϱu takes values in L U; W b,. To conclude this subsection we state our main result. Theorem.4. Let γ > 3/. Assume that for the initial law Λ there exists M, such that } Λ {ρ, q L γ L γ γ+1 ; ρ, ρ M, qx = whenever ρx = = 1, and that for all p [1, the following moment estimate holds true.5 1 q γ γ+1 ρ + a p γ 1 ργ dλρ, q <. L γ x Lx Then there exists a finite energy weak martingale solution to 1.1 with the initial data Λ. Remark.5. Note that the condition.5 is directly connected to the energy inequality.4. More precisely, 1 q γ γ+1 ρ + a p [ γ 1 ργ 1 ϱu dλρ, q = E + a p T ϱ γ 1 ϱγ dx 3 L γ x Lx L 1 x which is the quantity that appears on the right hand side of.4 cf. Proposition 3.1. It follows from our proof that C does not depend on a, γ, λ or ν. Remark.6. In order to simplify the computations we only study the case of periodic boundary conditions note that the density does not require any boundary assumptions in the weak formulation. However, with a bit of additional work our theory can also be applied to the case of no-slip boundary conditions. Furthermore, the reader might observe that the assumption upon the initial law Λ that implies ϱ M a.s. can be weakened to E ϱ p < p [,. Furthermore, the total mass remains constant in time, i.e. L 1 x ϱt = ϱ t [, T. Remark.7. In dimension two the result of Theorem.4 even holds under the weaker assumption γ > 1.

6 6 DOMINIC BREIT AND MARTINA HOFMANOVÁ.. Outline of the proof. Our proof relies on a four layer approximation scheme whose core follows the technique developed by Feireisl, Novotný and Petzeltová [14 in order to deal with the corresponding deterministic counterpart. To be more precise, we regularize the continuum equation by a second order term and modify correspondingly the momentum equation so that the energy inequality is preserved. In addition, we consider an artificial pressure term that allows to weaken the hypothesis upon the adiabatic constant γ. Thus we are led to study the following approximate system.6a.6b dϱ + divϱudt = ε ϱ dt, dϱu + [ divϱu u ν u λ + ν div u +a ϱ γ + δ ϱ β + ε u ϱ dt = Φϱ, ϱu dw, where β > max{ 9, γ}. The term ε u ϱ is added to the momentum equation to maintain the energy balance. In order to ensure its convergence to in the vanishing viscosity limit the artificial pressure δϱ β is needed it implies higher integrability of ϱ. It yields an estimate for ε ϱ which is uniformly in ε by.6a. The aim is to pass to the limit first in ε and subsequently in δ, however, in order to solve.6 for ε > and δ > fixed we need two additional approximation layers. In particular, we employ a stopping time technique to establish the existence of a unique solution to a finitedimensional approximation of.6. We gain so-called Faedo-Galerkin approximation, on each random time interval [, τ R where the stopping time τ R is defined as τ R = inf { t [, T ; u L R } { t inf t [, T ; Φ N ϱ, ϱu } dw R L with the convention inf = T, where Φ N is a suitable finite-dimensional approximation of Φ. It is then showed that the blow up cannot occur in a finite time. So letting R gives a unique solution to the Faedo-Galerkin approximation on the whole time interval [, T. The passage to the limit as N yields existence of a solution to.6. Except for the first passage to the limit, i.e. as R, we always employ the stochastic compactness method. Let us discuss briefly its main features. The compactness method is widely used for solving various PDEs: one approximates the model problem, finds suitable uniform estimates proving that the set of approximate solutions is relatively compact in some path space and this leads to convergence of a subsequence whose limit is shown to fulfill the target equation. The situation is more involved in the stochastic setting due to presence of the additional variable ω. Indeed, generally it is not possible to get any compactness in ω as no topological structure on Ω is assumed. To overcome this issue, one concentrates rather on compactness of the set of laws of the approximations and then the Skorokhod representation theorem comes into play. It gives existence of a new probability space with a sequence of random variables that have the same laws as the original ones so they can be shown to satisfy the same approximate problems though with different Wiener processes and that in addition converge almost surely. Powerful as it sounds there is one drawback of the classical Skorokhod representation theorem see e.g. [11, Theorem 11.7.: it is restricted to random variables taking values in separable metric spaces. Nevertheless, Jakubowski [ gave a suitable generalization of this result that holds true in the class of so-called quasi-polish spaces. That is, topological spaces that are not metrizable but retain several important properties of Polish spaces see [3, Section 3 for further discussion. Namely, separable Banach spaces equipped with weak topology or spaces of weakly continuous functions with values in a separable Banach space belong to this class which perfectly covers the needs of our paper. Another important ingredient of the proof is then the identification of the limit procedure. To be more precise, the difficulties arise in the passage of the limit in the stochastic integral as one now deals with a sequence of stochastic integrals driven by a sequence of Wiener processes. One possibility is to pass to the limit directly and such technical convergence results appeared in several works see [1 or [19, a detailed proof can be found in [9. Another way is to show

7 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 7 that the limit process is a martingale, identify its quadratic variation and apply an integral representation theorem for martingales, if available see [7. Our proof relies on neither of those and follows a rather new general and elementary method that was introduced in [6 and already generalized to different settings see [8 for the application to quasi-polish spaces. The keystone is to identify not only the quadratic variation of the corresponding martingale but also its cross variation with the limit Wiener process obtained through compactness. This permits to conclude directly without use of any further difficult results. 3. The Faedo-Galerkin approximation In this section, we present the first part of our proof of Theorem.4. In particular, we prove existence of a unique solution to a Faedo-Galerkin approximation of the following viscous problem.6 where ε >, δ > and β > max { 9, γ}. To be more precise, let us consider a suitable orthogonal system formed by a family of smooth functions ψ n. We choose ψ n such that it is an orthonormal system with respect to the L inner product which is orthogonal with respect to the the W l, inner product where l > 5 is fixed. Now, let us define the finite dimensional spaces X N = span{ψ 1,..., ψ N }, N N, and let P N : L X N be the projection onto X N which also acts as a linear projection P N : W l, X N. The aim of this section is to find a unique solution to the finite-dimensional approximation of.6. Namely, we consider 3.1a 3.1b 3.1c dϱ + divϱudt = ε ϱ dt, dϱu + [ divϱu u ν u λ + ν div u + a ϱ γ + δ ϱ β + ε u ϱ dt = Φ N ϱ, ϱu dw, ϱ = ϱ, ϱu = q. The equation 3.1b is to be understood in the dual space XN. The coefficient in the stochastic term is defined as follows: Φ N ρ, qe k = gk N ρ, q, gk N ρ, q = M 1 gk ρ, q 3. [ρpn, ρ where for ϱ L 1 with ϱ a.e. 3.3 M[ϱ : X N XN, M[ϱv, w = ρ v w dx, v, w X N. Note that we can identify XN with X N via the natural embedding such that M[ρ is a positive symmetric semidefinite operator on a Hilbert space having a unique square root in the same class. It follows from the definition of M[ρ that M[ρv = P N ρv. Note further that we can extend M[ρ to L in case of bounded ρ or to W l, if ρ L by setting M[ρv = P N ρ P N v. More details on the properties of M can be found in [14, Section. and in Appendix A. The initial condition ϱ, q is a random variable with the law Γ, where Γ is a Borel probability measure on C +κ C, with κ >, satisfying { } Γ ρ, q C +κ C ; < ρ ρ ρ = 1, and for all p [, 3.4 Cx +κ Cx 1 q ρ + a γ 1 ργ + δ β 1 ρβ p L 1 x dγρ, q C.

8 8 DOMINIC BREIT AND MARTINA HOFMANOVÁ As in [14, Section, the system 3.1 can be equivalently rewritten as a fixed point problem 3.5 ut = M 1[ Sut q + + t t Φ N Su, Suu dw N [ Su, u ds. In the brackets the stochastic integral is interpreted as an element of XN. Here Su is a unique classical solution to 3.1a with a strictly positive initial condition ϱ C +κ, i.e. < ϱ ϱ ϱ. This classical solution exists and belongs to C[, T ; C +κ provided u C[, T, C. A maximum principle applies in this case such that for all x 3.6 ϱ exp t t div u dσ Sut, x ϱ exp div u dσ. For the properties of S we refer to [14, Lemma.. The operators M[ϱ are invertible provided ϱ is strictly positive. We further define N [ϱ, u, ψ = [ ν u divϱu u + λ + ν div u aϱ γ δϱ β ε u ϱ ψ dx for all ψ X N. Note that for ϱ and u satisfying the conditions above N [ϱ, u is well-defined. In order to study 3.5, we shall fix some notation. For v = N i=1 α iψ i X N and R N let us define the following truncation operators v R = N θ R α i α i ψ i. i=1 Here θ R is a smooth cut-off function with port in [ R, R such that θz = 1 on [ R, R. Note that by construction the mapping Θ R : v v R satisfies 3.7 Θ R : X N X N, Θ R v Θ R u XN CN v u XN, for all u, v X N. Let N N, R N be fixed. In the first step, we will solve the following problem 3.8 by using the Banach fixed point theorem in the Banach space B = L Ω; C[, T ; X N with T sufficiently small. Repeating the same technique shows existence and uniqueness on the whole time interval [, T. Finally we pass to the limit as R. Consider 3.8 ut = M 1[ S u R t [ ϱ u R t [ + N S u R, u R ds t + Θ R Φ N S u R, S u R u R dw with u = M 1 [ϱ q. Note that now we have u = u R. Let T : B B be the operator defined by the above right hand side. We will show that it is a contraction. The deterministic part T det can be estimated using the approach of [14, Section.3 and there holds T det u T det v B T CN, R, T u v B, where the constant does not depend on the initial condition. In several points one needs the fact that we are working on a finite dimensional space: equivalence of norms is used and also Lipschitz continuity of M 1 in ϱ see [14,.1. Let us focus on the stochastic part T sto.

9 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 9 We have T sto u T sto v B = E M 1[ S u R t t Θ R Φ N S u R, S u R u R dw t T M 1[ S v R t t Θ R Φ N S v R, S v R v R dw M 1 CE [ S u R t M 1[ S v R t t T LXN,X N t Θ R Φ N S u R, S u R u R dw X N M 1 + CE [ S v R t t t T LXN,X Θ R Φ N S u R, S u R u R dw N t Θ R Φ N S v R, S v R v R dw = S 1 + S. As a consequence of the assumption ρ > we have by the definition of M, 3.6 and equivalence of norms a.s. M 1[ S v R t inf S v R 1 t LXn,Xn x T 1 ρ exp div v R ds CN, R. Hence we gain by Burgholder-Davis-Gundy inequality t S CN, R E Φ N S u R, S u R u R Φ N S v R, S v R v R dw t T T CN, R E gk N S u R, S u R u R gk S N v R, S v R v R ds. X N Due to the construction of gk N in 3. we have 3.9 I = gk N S u R, S u R u R gk S N v R, S v R v R X N [ CM 1 S u R [ M 1 S v R g k S u R, S u R u R LX N,X N S u R + C M 1 = I 1 + I. [ S v R LX N,X N g k S u R, S u R u R S u R X N g k L X N S v R, S v R v R S v R Concerning the first term on the above right hand side, we apply Lemma A.1,.1, 3.6 and [14, Lemma. and obtain I 1 CN, R S u R S v R L T CN, R, T t T u R v R X N. For the second term on the right hand side of 3.9 we make use of A.1 and conclude S I CN, R u R S v R + L u R v R L T CN, R, T u R v R, X t T N X N L

10 1 DOMINIC BREIT AND MARTINA HOFMANOVÁ where we applied [14, Lemma. and the Lipschitz continuity of ρ, q g k ρ, q ρ. The latter follows from.1 and. since we only consider ρ CN, R >. Consequently, S T CN, R, T u v B. For S 1 we have by [14,.1,.1 S S 1 CN, R E u R t S v R t t T T CN, R, T E u R v R = T X CN, R, T u v t T N B hence plugging all together we have shown that L 1 T sto u T sto v B T CN, R, T u v B. Since we know that also the deterministic part in 3.8 is a contraction if T is sufficiently small, we obtain T u T v B κ u v B with κ, 1. This allows us to apply Banach s fixed point theorem and we obtain a unique solution to 3.8 on the interval [, T. Extension of this existence and uniqueness result to the whole interval [, T can be done by considering kt, k N, as the new times of origin and solving 3.8 on each subinterval [kt, k + 1T. Note that the time T chosen above does not depend on the initial datum Passage to the limit as R. It follows from the previous section that for every N N and R N there exists a unique solution to 3.8. As the next step, we keep N fixed, denote the solution to 3.8 by ũ R and we pass to the limit as R to obtain the existence of a unique solution to 3.1. Towards this end, let us define { τ R = inf t [, T ; ũ R t } { t L R inf t [, T ; Φ N } L Sũ R, Sũ R ũ R dw R with the convention inf = T. Note that τ R defines an F t -stopping time and let ϱ R = Sũ R. Then ϱ R, ũ R is the unique solution to 3.1 on [, τ R. Besides, due to uniqueness, if R > R then τ R τ R and ϱ R, ũ R = ϱ R, ũ R on [, τ R. Therefore, one can define ϱ, ũ by ϱ, ũ := ϱ R, ũ R on [, τ R. In order to make sure that ϱ, ũ is defined on the whole time interval [, T, i.e. the blow up cannot occur in a finite time, we proceed with the basic energy estimate that will be used several times throughout the paper. Proposition 3.1. Let p [1,. Then the following estimate holds true 3.1 [ E + t T T C 1 ϱ R ũ R + a γ 1 ϱγ R + δ β 1 ϱβ R dx ν ũ R + λ + ν div ũ R dx ds + ε [ 1 ϱ u + a γ 1 ϱγ E with a constant independent of R, N, ε and δ. δ β 1 ϱβ T dx aγ ϱ γ R T 3 p p + δβ ϱβ ϱr dx ds Proof. In order to obtain this a priori estimate we observe that restricting ourselves to [, τ R the two equations 3.8 and 3.1 coincide and we apply Itô s formula to the functional f : L X N R, ρ, q 1 q, M 1 [ρq, R

11 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 11 where ρ = ϱ R and q = ϱ R ũ R. This corresponds exactly to testing by ũ R in the deterministic case. Indeed, there holds and q fρ, q = M 1 [ρq X N, qfρ, q = M 1 [ρ LX N, X N ρ fρ, q = 1 q, M 1 [ρm[ M 1 [ρq LL, R, and therefore f ϱ 1 R, ϱ R ũ R = ϱ R ũ R dx, q f ϱ R, ϱ R ũ R = ũr, qf ϱ R, ϱ R ũ R = M 1 [ ϱ R, ρ f ϱ 1 R, ϱ R ũ R = ũ R. We obtain 1 ϱt τ R ũt τ R dx = 1 t τr ϱ u dx ν ũ dx dσ T 3 T 3 t τr λ + ν div ũ dx dσ T 3 t τr t τr + ϱũ ũ : ũ dx dσ ε ũ ϱ ũ dx dσ T 3 t τr t τr + a ϱ γ div ũ dx dσ + δ ϱ β div ũ dx dσ + t τr ũ gk N ϱ, ϱũ dx dβ k σ + ε t τr ũ ϱ dx dσ T 3 1 t τr ũ ϱũ dx dσ + 1 t τr M 1 [ ϱg k N ϱ, ϱũ, gk N ϱ, ϱũ dσ T 3 = J J 11. Now, we observe that J 5 + J 9 =, J 4 + J 1 =, J 6 = a t τr t τr t ϱ γ dx dσ εaγ ϱ γ ϱ dx dσ, γ 1 similarly for J 7. Due to definitions of g N and M 1 we have M 1 [ ϱg k N ϱ, ϱũ, gk N ϱ, ϱũ = M 1 [ ϱg N k ϱ, ϱũ, M 1 [ ϱg N k ϱ, ϱũ = gk ϱ, ϱũ P N dx T ϱ g k ϱ, ϱũ dx 3 T ϱ 3 C ϱ + ϱ γ + ϱ ũ dx. Here we also used continuity of P N on L and.1. We get t τr J 11 C 1 + ϱ γ + ϱ ũ dx dσ. Hence according to the Gronwall lemma we can write 1 E T ϱt τ R ũt τ R + a γ 1 ϱγ t τ R + δ β 1 ϱβ t τ R dx 3 [ t τr + E ν ũ + λ + ν div ũ + ε aγ ϱ γ + δβ ϱ β ϱ dx ds T 3 1 C 1 + E ϱ u + a γ 1 ϱγ + δ β 1 ϱβ dx.

12 1 DOMINIC BREIT AND MARTINA HOFMANOVÁ Let us now take remum in time, p-th power and expectation. For the stochastic integral J 8 we make use of the Burkholder-Davis-Gundy inequality and the assumption.1 to obtain, for all t [, T, [ t τr E J 8 p C E ũ g N p k ϱ, ϱũ dx ds s t τ R [ t τr = C E [ t τr C E M 1 [ ϱũ PN gk ϱ, ϱũ M 1 [ ϱũ dx ϱ dx ds p g k ϱ, ϱũ p dx ds ϱ [ t τr C E M[ ϱũ ũ dx ϱ + ϱ γ + ϱ ũ p dx ds T 3 p t τr κ E ϱ ũ dx ds + Cκ E ϱ + ϱ γ + ϱ ũ p dx ds. t τ R Finally, taking κ small enough and using the Gronwall lemma completes the proof. Corollary 3.. It holds that P R N τ R = T = 1 and as a consequence the process ϱ, ũ is the unique solution to 3.1 on [, T. Proof. Since 3.11 P R N τ R < T P τ R < T P + P t T t T ũ R t L t R Φ N L Sũ R, Sũ R ũ R dw R for all R, it is enough to show that the right hand side converges to zero as R. To this end, we recall the maximum principle for ϱ R 3.6 and gain t t ϱ exp div ũ R ds ϱ R t, x ϱ exp div ũ R ds. Since ũ R B = L Ω; C[, T ; X N and all the norms on X N are equivalent, the above left hand side can be further estimated from below by T ϱ exp T c ũ R L ds ϱ R t, x. Plugging this into 3.1 we infer that [ T 3.1 E exp c ũ R L ds t T ũ R L c. Next, let us fix two increasing sequences a R and b R such that a R, b R and a R e b R = R for each R N. As in [16, we introduce the following events [ T A = exp c ũ R L ds ũ R L a R [ B = c [ C = T ũ R L dt b R ũ R L a R e b R t T. t T

13 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 13 Then A B C because on A B there holds that t T ũ R L = eb R e b R e b R exp c t T T ũ R L ũ R L ds t T Furthermore, according to 3.1, 3.1 and the Chebyshev inequality PA 1 C a R, PB 1 C b R. ũ R L eb R a R. Due to the general inequality for probabilities PC PA + PB 1 we deduce that PC 1 C a R C b R 1, R. This yields the desired convergence of the first term on the right hand side of For the second term, we have due to equivalence of norms on X N and Burkholder-Davis- Gundy inequality t t E Φ N ϱ R, ϱ R ũ R dw C E Φ N ϱ R, ϱ R ũ R dw t T L t T W l, T C E M 1 g k ϱ R, ϱ R ũ R [ ϱr P N dr. ϱr W l, Next, there holds 3.13 M 1 g k ϱ R, ϱ R ũ R [ ϱr P N ϱr W l, = M 1 g k ϱ R, ϱ R ũ R [ ϱr P N, ψ ψ W ϱr l, ψ W l, 1 = g k ϱ R, ϱ R ũ R P N, M ψ W ϱr 1 [ϱn ψ l, ψ W l, 1 g k ϱ R, ϱ R ũ R ϱr We further estimate using.1 g k ϱ R, ϱ R ũ R C 1 + ϱ R ϱr γ L L + ϱ γ R ũ R 3.14 L and 3.15 ψ W l, M 1 [ ϱr ψ = L ψ W l, ψ W l, 1 ψ W l, ψ W l, 1 ψ W l, ψ W l, 1 C ψ W l, ψ W l, 1 ψ W l, 1 M[ ϱr ψ, ψ M[ ϱr ψ L ψ W l, ψ W l, 1 L ϱr L P N ψ L C M 1 [ ϱr ψ. L ψ W l, ψ W l, 1 ϱr P N ψ L ψ W l, ψ W l, 1 ϱr L ψ W l, C ϱ R L. ϱr L P N ψ W l,

14 14 DOMINIC BREIT AND MARTINA HOFMANOVÁ Altogether we deduce t E Φ N ϱ R, ϱ R ũ R dw t T C L T C E ϱ R L 1 + ϱr γ L + ϱ γ R ũ N L dr 1 + E ϱ R L + E ϱ R γ L + E ϱ γ R ũ N 4 L C, t T t T t T where we used 3.1 with p =. Finally, the convergence of the second term on the right hand side of 3.11 follows from Chebyshev s inequality and the proof is complete. 4. The viscous approximation In this section, we continue with our proof of Theorem.4 and prove existence of a martingale solution to the viscous approximation.6 with the initial law Γ see the beginning of Section 3 for its definition, where ε, δ are fixed. In particular, we justify the passage to the limit in 3.1 as N. Let ϱ N, u N denote the solution to 3.1 and observe that by the same approach as in Proposition 3.1 it can be shown that it satisfies the corresponding a priori estimate uniformly in N. In fact, there holds for any p < 4.1 [ E t T T 1 ϱ N u N + a γ 1 ϱγ N + δ β 1 ϱβ N + ν u N + λ + ν div u N dx ds + ε [ 1 C p 1 + E ϱ u + a γ 1 ϱγ + dx T δ β 1 ϱβ aγϱ γ N p dx + p δβϱβ N ϱn dx ds uniformly in N, ε and ϱ. Thus we obtain uniform bounds in the following spaces u N L p Ω; L, T ; W 1,, ϱn u N L p Ω; L, T ; L, ϱ N L p Ω; L, T ; L β, εδ ϱn β/ L p Ω; L, T ; W 1,. We recall that β > max { 9, γ}. Here p [1, is arbitrary due to 3.4 and the estimate of u N L p Ω; L, T ; L is obtained as in [, Remark 5.1, page 4. Besides, testing 3.1a by ϱ N yields t t ϱ N dx + ε ϱ N dx dσ = ϱ dx div u N ϱ N dx dσ. And therefore since β > max{ 9, γ}, 4. and 4.4 imply for any p [1, [ T p T T p E ε ϱ N dx dσ C E[ 1 + u N dx dσ + ϱ N 4 dx dσ C. This yields the uniform bound 4.6 εϱn L p Ω; L, T ; W 1,. Moreover, from 4.4 and 4.5 we obtain by interpolation that [ T p [ p [ T p E ϱ β N L dt E ϱ β N L 1 T 3 + E ϱ β N 3 L 3 dt C. t T In particular we obtain a uniform bound 4.7 ϱ N L p Ω; L β+1, =, T, for all p [1, as β > max{ 9, γ}.

15 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS Compactness and identification of the limit. Let us now prepare the setup for our compactness method. We define the path space X = X ϱ X u X ϱu X ϱ X W where X ϱ = C w [, T ; L β L 4, T ; L 4 L, T ; W 1,, w, X u = L, T ; W 1,, w, X ϱu = C w [, T ; W 1,, X ϱ = L, X W = C[, T ; U. Let us denote by µ ϱn, µ un, µ PN ϱ N u N and µ ϱ, respectively, the law of ϱ N, u N, P N ϱ N u N and ϱ N = ϱ on the corresponding path space. By µ W we denote the law of W on X W. The joint law of all variables on X is denoted by µ N. Proposition 4.1. The set {µ un ; N N} is tight on X u. Proof. The proof follows directly from 4.. Indeed, for any R > the set B R = { u L, T ; W 1, ; u L,T ;W 1, R } is relatively compact in X u and µ un B c R = P u N L,T ;W 1, R 1 R E u N L,T ;W 1, C R which yields the claim. Proposition 4.. The set {µ ϱn ; N N} is tight on X ϱ. Proof. Due to 4.3 and 4.4 we obtain that 4.8 {ϱ N u N } is bounded in L p Ω; L, T ; L β β+1 hence {divϱ N u N } is bounded in L p Ω; L β 1,, T ; W β+1 and similarly {ε ϱ N } is bounded in L p Ω; L, T ; W,. As a consequence, E ϱ N p C,1 [,T ;W, β β+1 due the continuity equation 3.1a. Now, the required tightness in C w [, T ; L β follows by a similar reasoning as in Proposition 4.1 together with the compact embedding see [8, Corollary B. C L, T ; L β C,1 β, [, T ; W β+1 c C w [, T ; L β. Next, observe that by applying interpolation to 4.4 and 4.6 we obtain T E ϱ N 4 W 1, 4β β+ [ T dt E ϱ N 4 L + E ϱ β N W 1,dt C. t T Since W κ,q is compactly embedded into L 4 we make use of the Aubin-Lions compact embedding L 4, T ; W κ,q C,1 β, [, T ; W β+1 c L 4 and conclude as in Proposition 4.1. Tightness in L, T ; W 1,, w follows directly from 4.6 which completes the proof. Proposition 4.3. The set {µ PN ϱ N u N ; N N} is tight on X ϱu. If a topological space X is equipped with the weak topology we write X, w.

16 16 DOMINIC BREIT AND MARTINA HOFMANOVÁ Proof. First, we shall study time regularity of P N ϱ N u N. Towards this end, let us decompose P N ϱ N u N into two parts, namely, P N ϱ N u N t = Y N t + Z N t, where Y N t = P N q Z N t = ε t t P N [ divϱn u N u N + ν u N + λ + ν div u N P N [ un ϱ N ds, a ϱ γ N t δ ϱβ N ds + Φ N ϱ N, ϱ N u N dw s, and consider them separately. Hölder continuity of Z N. We show that there exists κ, 1 such that 4.9 E Z N Cκ [,T ;W l, C. To this end, we observe that according to 4., 4.4 and the embedding W 1, L 6 there holds E ϱ N u N p C E ϱ N p + C E u L β+6 t T L β N p C. x L t L6 x 6β t L x By interpolation with 4.8 and noticing that β > 4 there exists r > such that we have a uniform bound in ϱ N u N L p Ω; L r, T ; L. Now we have all in hand to apply maximal regularity estimates to 3.1a with divϱ N u N L p Ω; L r, T ; W 1, as a right hand side and deduce a uniform estimate in 4.1 ϱ N L p Ω; L r, T ; W 1,. Finally, we combine this with 4. and the continuity of P N on W l, and 4.9 follows. Hölder continuity of Y N. As the next step, we prove that there exist ϑ > and m > 5/ such that 4.11 E Y N Cϑ [,T ;W m, C. Let us now estimate the stochastic integral. Due to Burkholder-Davis-Gundy inequality we obtain for any θ t θ t E Φ N ϱ N, ϱ N u N dw C E M 1 g k ϱ N, ϱ N u N θ/ [ϱn P N dr. ϱn W l, s W l, s Here, we can apply the estimates established in and deduce t θ t E Φ N ϱ N, ϱ N u N dw C E ϱ N L 1 + ϱn γ L + ϱ γ N u N θ/ L dr s W l, s [ C t s θ/ E 1 + ϱn γ L + ϱ γ N u N θ/ L ϱ N L t T C t s 1 θ/ + E ϱ N θ L + E ϱ N θγ L + E ϱ γ N u N θ L t T t T t T C t s θ/. By the Kolmogorov continuity criterion we conclude that for any σ [, 1/ t E Φ N ϱ N, ϱ N u N dw C. C σ [,T ;W l, Besides, from 4. and 4.8 we get 4.1 ϱ N u N u N L q Ω; L, T ; L 6β 4β+3

17 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 17 uniformly in N and therefore 4.13 {divϱ N u N u N } is bounded in L p Ω; L, T ; W 1, 6β 4β+3. As a consequence of 4. and Since l > 5 {ν u N + λ + ν div u N } is bounded in L p Ω; L, T ; W 1,, there holds W 1, {a ϱ γ N + δ ϱβ N } is bounded in Lp Ω; L β+1 β, T ; W 1, β+1 β. 6β 4β+3 W l,, W 1, β+1 β W l, and thanks to uniform boundedness of P N on W l, and W 1, it follows that { PN divϱ N u N u N } is bounded in L p Ω; L, T ; W l,, { PN ν un + λ + ν div u N } is bounded in L p Ω; L, T ; W 1,, { PN a ϱ γ N + } δ ϱβ N is bounded in L p Ω; L β+1 β, T ; W l,. Finally, 4.11 follows for some m > l. Conclusion. Collecting the above results we obtain that E P N ϱ N u N C τ [,T ;W m, C for some τ, 1 and m > 5. This implies the desired tightness by making use of 4.8, uniform boundedness of P N on W 1,, the embedding L β β+1 W 1, together with the compact embedding see [8, Corollary B. L, T ; L β β+1 C τ [, T ; W m, c C w [, T ; L β β+1. Since also the laws µ ϱ and µ W, respectively, are tight as being Radon measures on the Polish spaces X ϱ and X W, respectively, we can deduce tightness of the joint laws µ N. Corollary 4.4. The set {µ N ; N N} is tight on X. The path space X is not a Polish space and so our compactness argument is based on the Jakubowski-Skorokhod representation theorem instead of the classical Skorokhod representation theorem, see [. To be more precise, passing to a weakly convergent subsequence µ N and denoting by µ the limit law we infer the following result. Proposition 4.5. There exists a probability space Ω, F, P with X -valued Borel measurable random variables ϱ N, ũ N, q N, ϱ,n, W N, N N, and ϱ, ũ, q, ϱ, W such that a the law of ϱ N, ũ N, q N, ϱ,n, W N is given by µ N, n N, b the law of ϱ, ũ, q, ϱ, W, denoted by µ, is a Radon measure, c ϱ N, ũ N, q N, ϱ,n, W N converges P-almost surely to ϱ, ũ, q, ϱ, W in the topology of X. We are immediately able to identify ϱ,n, q N, N N, and ϱ, q. Lemma 4.6. There holds P-a.s. that ϱ,n, q N = ϱ N, P N ϱ N ũ N, ϱ, q = ϱ, ϱũ. Proof. The first statement follows from the equality of joint laws of ϱ N, u N, P N ϱ N u N, ϱ N and ϱ N, ũ N, q N, ϱ,n. Identification of ϱ follows from the a.s. convergence and in order to identify the limit q, note that ϱ N ϱ in C w [, T ; L β ϱ N ũ N ϱũ in L 1, T ; L 1 P-a.s.

18 18 DOMINIC BREIT AND MARTINA HOFMANOVÁ as a consequence of the convergence of ϱ N and ũ N in X ϱ and X u, respectively. Clearly, this also identifies the limit of P N ϱ N ũ N with ϱũ. From 4.1 and equality of joint laws we deduce [ 1 Ẽ t T T ϱ N ũ N + a γ 1 ϱγ N + δ 4.16 β 1 ϱβ N dx 3 T + ν ũ N + λ + ν div ũ N dx ds [ 1 C p 1 + Ẽ q N + a T ϱ 3,N γ 1 ϱγ,n + δ p β 1 ϱβ,n dx = C p q β β+1 ρ + a γ 1 ργ + δ p β 1 ρβ dγρ, q Cp, Γ L β x Lx uniformly in N, ε and ϱ. Based on Proposition 4.5 and 4.16 we are going to achieve a series of further convergences after taking not relabelled subsequences. Corollary 4.7. The following convergence holds true P-a.s. L 1 x 4.17 ϱ N ũ N ũ N ϱũ ũ in L 1, T ; L 1. Proof. From Proposition 4.5 and Lemma 4.6, we gain ϱ T N ũ N = P N ϱ N ũ N ũ N dx dt L t L x T T ϱũ ũ dx dt = ϱũ In the last step we used the compact embedding L β β+1 c W 1, which implies together with Proposition 4.5 and Lemma 4.6 that L t L x P N ϱ N ũ N ϱũ in L, T ; W 1, P-a.s. P-a.s. According to 4.18, we infer that for almost every ω, the sequence ϱ N ũ N ω is bounded in L, T ; L. Hence combining weak and strong convergence from Proposition 4.5 implies ϱn ũ N ϱũ 4.19 in L, T ; L P-a.s. So 4.17 follows by combining 4.18 and Let us now fix some notation that will be used in the sequel. We denote by r t the operator of restriction to the interval [, t acting on various path spaces. In particular, if X stands for one of the path spaces X ϱ, X u, X ϱu or X W and t [, T, we define 4. r t : X X [,t, f f [,t. Clearly, r t is a continuous mapping. Let F t be the P-augmented canonical filtration of the process ϱ, ũ, W, respectively, that is 4.1 F t = σ σ { } r t ϱ, r t ũ, r t W N F ; PN =, t [, T. Finally, we have all in hand to conclude this Section by the following existence result. Proposition 4.8. Ω, F, Ft, P, ϱ, ũ, W is a weak martingale solution to.6 with the initial law Γ.

19 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 19 We divide the proof into two parts. First, we prove that the equation.6a holds true and establish strong convergence of ϱ N in L, T ; L a.s. Second, we focus on the momentum equation.6b and employ a new general method of constructing martingale solutions to SPDEs, that does not rely on any kind of martingale representation theorem and therefore holds independent interest especially in situations where these representation theorems are no longer available. Lemma 4.9. ϱ, ũ is a weak solution to.6a, i.e. for all ψ C and and all t [, T there holds P-a.s. t t ϱt, ψ = ϱ, ψ + ϱũ, ψ ds ε ϱ, ψ ds. Furthermore, P-a.s. ϱ N ϱ in L, T ; L. Proof. Let us we define, for all t [, T and ψ C, the functional Lρ, q t ψ = ρt, ψ ρ, ψ t q, ψ ds + ε t ρ, ψ ds. For notational simplicity we neglect the ψ-dependence in the following. The mapping ρ, q Lρ, q t is continuous on X ϱ X ϱu. Hence the laws of Lϱ N, ϱ N u N t and L ϱ N, ϱ N ũ N t coincide and since ϱ N, ϱ N u N solves 3.1a we deduce that Ẽ L ϱn, ϱ N ũ N t = E LϱN, ϱ N u N t =. Next, we pass to the limit on the left hand side by 4.4, 4.8 and the Vitali convergence theorem which verifies.6a. In order to prove the strong convergence of ϱ N, we recall that due to Proposition 4.5 there holds P-a.s. ϱ N ϱ in L, T ; L. Hence in order to prove strong convergence it is sufficient to establish convergence of the norms in L, T ; L. Since both ϱ N, ũ N and ϱ, ũ solve.6a, we shall test by ϱ N and ϱ, respectively, to obtain P-a.s. t t ϱ N t L + ε ϱ N L ds = ϱ N L div ũ N ϱ N dx ds, T 3 t t ϱt L + ε ϱ L ds = ϱ L div ũ ϱ dx ds. Due to Proposition 4.5 we pass to the limit in the first term on the left hand side after taking a subsequence as well as in both terms on the right hand side. This implies P-a.s. and completes the proof. ϱ N L t,x ϱ L t,x Lemma 4.1. We have for all 1 q < β β+1 where β > max { 9, γ} ϱ N ũ N ϱũ in L q Ω. Proof. Similar to the proof of 4.17 we have P-a.s. ϱ N ũ N dx dt = ϱ N ũ N ũ N dx dt ϱũ ũ dx dt, so ϱ N ũ N ϱũ in L. Combining this with Proposition 4.5 and taking a subsequence yields P-a.s. ϱ N ũ N ϱũ in L 1. The higher integrability from 4.8 implies the claim.

20 DOMINIC BREIT AND MARTINA HOFMANOVÁ Proposition The process W is a F t -cylindrical Wiener process, where the filtration F t was defined in 4.1. Besides, Ω, F, Ft, P, ϱ, ũ, W is a finite energy weak martingale solution to.6. 3 Proof. The first part of the claim follows immediately from the fact that W N has the same law as W. As a consequence, there exists a collection of mutually independent real-valued F t -Wiener processes β k N such that W N = β k N e k, i.e. there exists a collection of mutually independent real-valued F t -Wiener processes β k such that W = β k e k. Let us now define for all t [, T and ϕ N N X N the functionals we neglect the dependence on the fixed test-function in the notation Mρ, v, q t = qt, ϕ q, ϕ + λ + ν ε t t t t t q v, ϕ dr ν v, ϕ dr t div v, div ϕ dr + a ρ γ, div ϕ t dr + δ ρ β, div ϕ dr v ρ, ϕ dr, N N ρ, q t = g N k ρ, q, ϕ dr, Nρ, qt = gk ρ, q, ϕ dr t Nk N ρ, q t = g N k ρ, q, ϕ t dr, N k ρ, q t = gk ρ, q, ϕ dr. Let Mρ, v, q s,t denote the increment Mρ, v, q t Mρ, v, q s and similarly for the other processes. Note that the proof will be complete once we show that the process M ϱ, ũ, ϱũ is a F t -martingale and its quadratic and cross variations satisfy, respectively, 4. M ϱ, ũ, ϱũ = N ϱ, ϱũ, t M ϱ, ũ, ϱũ, β k = N k ϱ, ϱũ. Indeed, in that case we have due to bilinearity of the cross-variation M ϱ, ũ, ϱũ Φ ϱ, ϱũ d W, ϕ = M ϱ, ũ, ϱũ = g k ϱ, ϱũ, ϕ dm ϱ, ũ, ϱũ, β k + Φ ϱ, ϱũ d W, ϕ and.6b is satisfied. Let us verify 4.. To this end, we claim that with the above uniform estimates in hand, the mappings ρ, v, q Mρ, v, q t, ρ, v, q N N ρ, q t, ρ, v, q N N k ρ, q t and ρ, v, q Nρ, q t, ρ, v, q N k ρ, q t are well-defined and measurable on a subspace of X ϱ X u X ϱu where the joint law of ϱ, ũ, q is ported, i.e. where all the uniform estimates hold true and ϱt is bounded in t and ω. Indeed, in the case of N N ρ, q t we have similarly to t g N k ρ, q, ϕ t 4.3 ds C ρ L 1 + ρ γ + q dx ds, ρ 3 This has to be understood in the sense of Definition.1 via an obvious modification by adding the artificial viscosity and artificial pressure terms.

21 STOCHASTIC NAVIER-STOKES EUATIONS FOR COMPRESSIBLE FLUIDS 1 for Nρ, q t by.1 similarly to.3 t gk ρ, q, ϕ ds C C t t g k ρ, q L 1 ds ρ + ρ γ + q dx ds ρ Both are finite due to 4.4 and 4.1. Mρ, v, q, N N k ρ, v t and N k ρ, v t can be handled similarly and therefore, the following random variables have the same laws Mϱ N, u N, ϱ N u N d M ϱ N, ũ N, ϱ N ũ N, N N ϱ N, ϱ N u N d N N ϱ N, ϱ N ũ N, N N k ϱ N, ϱ N u N d N N k ϱ N, ϱ N ũ N. Let us now fix times s, t [, T such that s < t and let h : X ϱ [,s X u [,s X W [,s [, 1 be a continuous function. Since t Mϱ N, u N, ϱ N u N t = Φ N ϱ N, ϱ N u N dw, ϕ = t g N k ϱ N, ϱ N u N, ϕ dβ k is a square integrable F t -martingale, we infer that [ MϱN, u N, ϱ N u N N N ϱ N, ϱ N u N, Mϱ N, u N, ϱ N u N β k N N k ϱ N, ϱ N u N are F t -martingales. Besides, it follows from the equality of laws that recall that the restriction operator r s was defined in 4. Ẽ h [ r s ϱ N, r s ũ N, r s WN M ϱn, ũ N, ϱ N ũ N s,t 4.4 = E h [ r s ϱ N, r s u N, r s W N MϱN, u N, ϱ N u N s,t =, Ẽ h [ r s ϱ N, r s ũ N, r s WN [M ϱ N, ũ N, ϱ N ũ N s,t N N ϱ N, ϱ N ũ N s,t 4.5 = E h [ r s ϱ N, r s u N, r s W N [Mϱ N, u N, ϱ N u N s,t N N ϱ N, ϱ N u N s,t =, Ẽ h [ r s ϱ N, r s ũ N, r s WN [M ϱ N, ũ N, ϱ N ũ N β k N s,t Nk N ϱ N, ϱ N ũ N s,t 4.6 = E h [ r s ϱ N, r s u N, r s W N [Mϱ N, u N, ϱ N u N β k s,t Nk N ϱ N, ϱ N u N s,t =. As the next step, we employ the assumptions.1 and. and the estimates 4., 4.4, 4.8, 4.1, 4.1 together with Proposition 4.5, Corollary 4.7, Lemma 4.9, Lemma 4.1 and the Vitali convergence theorem, pass to the limit in 4.4, 4.5 and 4.6 and establish the following identities that justify 4. Ẽ h r s ϱ, r s ũ, r s W [ M ϱ, ũ, ϱũs,t =, Ẽ h r s ϱ, r s ũ, r s W [ [M ϱ, ũ, ϱũ s,t N ϱ, ϱũ s,t =, Ẽ h r s ϱ, r s ũ, r s W [ [M ϱ, ũ, ϱũ β k s,t N k ϱ, ϱũ s,t =. Let us comment on the passage to the limit in the terms coming from the stochastic integral, i.e. N N ϱ N, ϱ N ũ N and Nk N ϱ N, ϱ N ũ N. The convergence in 4.6 being easier, let us only focus on 4.5 in detail. As a first step we aim to show for all k N that g N k ϱ N, ϱ N ũ N, ϕ g k ϱ, ϱũ, ϕ 4.7 P L-a.e.

22 DOMINIC BREIT AND MARTINA HOFMANOVÁ We first remark that by definition and the symmetry of M N [ϱ we have g N k ϱ N, ϱ N ũ N, ϕ = M 1 gk ϱ N, ϱ N ũ N gk ϱ N, ϱ N ũ N N [ ϱ N P N, ϕ =, M 1 N [ ϱ N ϕ. ϱn ϱn As a consequence of the strong convergences in Proposition 4.5 and Lemma 4.1 we have at least after taking a subsequence 4.8 g k ϱ N, ϱ N ũ N ϱn g k ϱ, ϱũ ϱ in L P L-a.e. where we also used.1,. and the a priori estimates. Moreover, for every v W l,, using again strong convergence of ϱ N, the embedding W l, L and continuity of P N on L and W l,, we have M N [ ϱ N v ϱv L P N ϱn ϱ P N v L + P N ϱ P N v P N ϱv L + P N ϱv ϱv L c ϱ N ϱ L v W l, + c ϱ L P N v v W l, + P N ϱv ϱv L P L-a.e. Hence M N [ ϱ N ϱ pointwise as an operator from W l, L. From a formal point of view it should follow that M 1 N [ ϱ N ϱ P L-a.e. in the same sense recalling that the square root of a positive semidefinite operator is unique. In order to make this argument rigorous we extend M N [ρ to an operator W l, W l, thus we stay in the same space. So we set M N [ ϱ N : W l, W l,, M N [ ϱ N Ψv = M N [ ϱ N Φ, v, Ψw = Φ, w l,, where w, v, Φ W l,. Now we have M N [ ϱ N Ψ ϱ Φ, P L-a.e. pointwise as an operator from W l, W l, and hence M 1 N [ ϱ N Ψ ϱ Φ, P L-a.e. The latter can be easily justified by using the series expansion A 1 = c k I A k, c k R, c k <, k= which holds for every symmetric positive semidefinite operator A on some real Hilbert space H with z H 1Az, z H 1. Finally, we gain 4.9 M 1 N [ ϱ N ϱ k= P L-a.e. pointwise as an operator from W l, L. Plugging 4.8 and 4.9 together we have shown 4.7. The convergence g N k ϱ N, ϱ N ũ N, ϕ gk ϱ, ϱũ, ϕ P L-a.e. follows once we show that 4.3 Φ N ϱ N, ϱ N ũ N, ϕ Φ ϱ, ϱũ, ϕ in L U; R P L-a.e. To this end, we estimate I = Φ N ϱ N, ϱ N ũ N, ϕ Φ ϱ, ϱũ, ϕ LU;R