The stochastic Navier–Stokes equations for heat-conducting, compressible fluids: global existence of weak solutions

We investigate the well posedness of the stochastic Navier–Stokes equations for viscous, compressible, non-isentropic fluids. The global existence of finite-energy weak martingale solutions for large initial data within a bounded domain of Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} is established under the condition that the adiabatic exponent γ>d/2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma > d/2.$$\end{document} The flow is driven by a stochastic forcing of multiplicative type, white in time and colored in space. This work extends recent results on the isentropic case, the main contribution being to address the issues which arise from coupling with the temperature equation. The notion of solution and corresponding compactness analysis can be viewed as a stochastic counterpart to the work of Feireisl (Dynamics of viscous compressible fluids, vol 26. Oxford University Press, Oxford, 2004).


Governing equations
This paper is devoted to the analysis of the initial boundary value problem for the stochastic Navier-Stokes equations for non-isentropic, compressible fluids. This system of stochastic partial differential equations governs the evolution of a viscous, compressible fluid (or gas) subject to random perturbations by noise within a connected, bounded domain D which is an open subset of R d with smooth boundary.

Notion of solution
Next we introduce our notion of weak solution to (1.1). The definition includes both analytic and probabilistic aspects. On the probabilistic side, there is the stochastic basis and the accompanying notion of measurability. This ensures a form of time consistency between the noise and the solution and allows the stochastic integrals to be properly defined. On the analytic side, a natural energy space is identified along with a notion of solution to (1.1) as a function of space and time. The momentum equation is understood in the Ito sense in time and in the sense of distributions (analytically) in space. The continuity equation is understood exactly as in the deterministic setting, P almost surely in ω. Following [7], the temperature equation is replaced by an inequality in the sense of distributions. To make this precise, we introduce a class of test functions D temp . The stochastic Navier-Stokes equations 415 4. Temperature inequality: For all ϕ ∈ D temp , the following inequality holds P a.s. where K(θ ) = θ 1 κ(θ)dz. 5. The following energy estimate holds: for all p ≥ 1, (1.10)

Main result
Next we present the main result of the article. The proof of this theorem relies on a four-level approximating scheme. Three of the levels are inspired by the theory of Feireisl [7] for the treatment of the deterministic Navier-Stokes equations for compressible, non-isentropic fluids. The lowest level uses a time splitting scheme and is similar to the technique used by Berthelin/Vovelle [2] for the 1-d compressible Euler system. Each layer involves a compactness step and an identification procedure. The compactness step involves proving an appropriate tightness result and applying a recent generalization of the Skorohod theorem due to Jakubowski [9] and Vaart/Wellner [16] (Theorem 7.18). The identification procedure involves several ingredients. The first step is to use a martingale method to make a preliminary passage to the limit in the momentum equation, up to a modification in the pressure law. The second is to use this partial stability result to upgrade from strong to weak convergence of the densities. The last step is to use the strong convergence of the density to prove the convergence of the temperature away from vacuum and then use a renormalized limit in the vacuum regions.

Outline
The outline of this article is as follows. Section 1 presents the fundamental problem introducing the governing equations and the basic assumptions on the system, constitutive relations, noise, boundary, and initial data. In addition, it presents the notion of weak martingale solution to our initial boundary value problem as well as the main result on the global existence of weak solutions. Section 2 presents the formal a priori bounds. It is worth observing that, unlike the barotropic case, the total energy does not dissipate. Nonetheless, using coloring Hypothesis 1.5, for all p < ∞ we obtain L p ( ) bounds on the total energy. Using these bounds together with a suitable variant of the Poincare inequality, Lemma 2.1, the renormalized form of temperature equation gives L p ( ; H 1 ) bounds on θ and u. Section 3 presents the existence for the lowest level of our approximating scheme (the τ layer) to stochastic Navier-Stokes system 1.1. The construction is based on a splitting scheme similar to the one used in [2] for the compressible, stochastic Euler equation. Half of the time, we neglect the evolution of the noise and run the continuity and momentum equation at twice the usual speed. The other half of the time, the noise evolves at √ 2 times the usual speed. However, the splitting of the temperature equation is slightly more subtle. Two of the terms in the equation, δθ 3 and div(κ(θ )∇θ), run at speed 1 for all times, while the remaining terms run at twice the usual speed, but only when the noise is turned off. Section 4 presents the existence proof for the n layer of the approximating scheme. The compactness step involves two types of uniform bounds on the sequence of approximations {(ρ τ , u τ , θ τ )} τ >0 . The first type is analogous to the formal estimates obtained in Sect. 2. The second type of estimates is encoded in a tightness proof. These are necessary to overcome an oscillatory factor h τ det arising in the weak form, which converges (as τ → 0) only weakly in time. To combat these oscillations, we need strong convergence in time of any terms in the weak form multiplied by this factor. This difficulty was addressed in detail in the work [14]. The trick is to obtain estimates on the density and velocity which give stronger bounds on these quantities as a function of (t, x), at the cost of taking a weaker norm as a function of ω. Namely, these estimates are not in terms of L p ( ), but only in measure. The proofs of [14] carry through to this context, so our main focus is to obtain similar improvements on the bounds for the temperature.
Section 5 presents the existence result for the -layer approximation. As n → ∞, a subtlety arises in the compactness analysis of the temperature equation which does not seem to have a deterministic counterpart. Namely, in order to apply the Aubin/Lionstype Lemma from Feireisl [7] (Proposition 7.23), we require uniform L ∞ t (L 1 x ) bounds on {(ρ n + δ)θ n } ∞ n=1 which hold pointwise in ω. However, due to the presence of stochastic integrals in the total energy balance, we only have uniform L p ( ; L ∞ t (L 1 x )) bounds for p < ∞. This requires an additional argument in the Skorohod compactness step.
In Sect. 6, we send → 0 to obtain an existence proof for the δ-layer. In comparison with τ → 0 and n → ∞, the main difficulty in this section is the complexity of the proof of strong convergence of the density. This is essentially already proved in [14], but the pressure law now depends on both density and temperature, so we sketch a proof for completeness.
The proof of main result Theorem 1.8 is presented in Sect. 7. As δ → 0, two regularizations in the temperature equation degenerate. In particular, we no longer Vol. 18 (2018) The stochastic Navier-Stokes equations 417 have a free L p ( ; L 3 t,x ) bound on the sequence {θ δ } δ>0 . Instead, this has to be proved directly from the weak form at the δ layer. Additional difficulties lie in the strong convergence of the densities, due to the fact that the limiting density may not lie in L 2 t,x .

Notation
Here, and in what follows, we use the notation: Functional spaces: (i) The shorthand notation , respectively, where each space is understood to be endowed with its strong topology. Also, we use M x to denote the finite, signed radon measures on D. The abbreviation ; M x . We will often use the same notation to denote scalar functions in L q t (L p x ) and vector-valued functions (with d components) in [L q t (L p x )] d , but the meaning will always be clear from the context. To emphasize when one of the spaces above is endowed with its weak topology, we write 0,x is the closure of the smooth compactly supported functions, C ∞ c (D), with respect to the W k, p (D) norm. Moreover, we denote W 1,2 0,x as H 1 0,x . Probability space: (ii) Given a probability space ( , F, P) and a Banach space E, let L p ; E be the collection of equivalence classes of F measurable mappings X : → E such that the p th moment of the E norm is finite. Again, we write L p w ; E when emphasizing that the space is endowed with its weak topology. To define the sigma algebra generated by various random variables, we use a restriction operator r t : C [0, T ]; E → C [0, t]; E which realizes a mapping f : [0, T ] → E as a mapping r t f : [0, t] → E. The same notation is used for the restriction of an equivalence class f ∈ L p [0, T ]; E to r t f ∈ L p [0, t]; E .
Operators and operations: (iii) We denote A = ∇ −1 , understood to be well defined on compactly supported distributions in R d . Given two d × d matrices A, B, A : B denotes a Frobenius matrix product. The notation A B denotes inequality up to an insignificant constant. The notion of insignificance will be clear from the context.

Formal energy estimates
In this section, we derive the formal a priori bounds for SPDE (1.1). In Sect. 2.1, we derive the evolution of the total energy, which is the sum of internal and kinetic contributions. The evolution of the internal energy is obtained through the renormalized form of the continuity equation. Namely, given β : R + → R, multiplying the continuity equation by β (ρ) yields: On the other hand, the evolution of the kinetic energy is obtained from the momentum equation and the Ito formula. Since the stochastic forcing is non-conservative, the total energy is not conserved. In fact, it undergoes random fluctuations produced by a stochastic integral. Moreover, since the noise is understood in the Ito sense, a nonnegative correction term appears. As the stochastic integral is an unbounded stochastic process, it is not possible to obtain L ∞ ( ) bounds for the total energy. However, as in [14], coloring Hypothesis 1.5 leads to bounds in L p ( ) for every p ≥ 1. In subsections 2.2 and 2.3, we obtain further bounds on the temperature and the velocity. The basic tool is the following: for H : R + → R, multiplying the temperature equation by H (θ ) and combining with the continuity equation yields: Integrating over D, using the boundary conditions, and rearranging gives: For concave H , the main difficulty in using this identity to control temperature and velocity gradients is the last term on the RHS of (2.3). Using a trick of Lions from [12], in Sect. 2.2 we introduce an entropy to remove this term and obtain a preliminary bound on θ . In Sect. 2.3, we revisit identity (2.3) to obtain an additional temperature estimate and a bound for the velocity. Finally, to transition from gradient estimates to H 1 bounds, we will need the following variant of the Poincare inequality: The proof of Lemma 2.1 uses the method of Mellet/Vasseur [13], but tracking a bit more carefully the dependence of the estimate on |g| L γ . This is important for our purposes since this quantity will be a random process. Recall that p m was defined in Hypothesis 1.1. Set β(ρ) = ρ P m (ρ) in renormalized form (2.1) and use the identity ρ (ρ P m (ρ)) − P m (ρ) = p m (ρ). Combining with the temperature equation, we find the evolution of the internal energy: Using Ito's formula together with the momentum equation in (1.1) yields the identity: Combining (2.6) and (2.7), integrating over D × [0, t], and using the boundary conditions give: Applying the Burkholder/Davis/Gundy inequality followed by Hölder and Hypothesis 1.5 yields for all p > 1 E sup Maximizing over [0, T ] and taking L p ( ) norms of both sides of the total energy identity yields: In view of Hypothesis 1.1, ρ P m (ρ) ∼ ρ γ . By Young's inequality we deduce the following a priori bound: The constant C depends only on p, d, D, T and the norms of the initial data discussed in Hypothesis 1.3.

τ layer existence
In this section, we build the first layer of our approximating scheme, the τ layer. Each of the parameters n, , and δ is present in the notion of solution, Definition 3.3 below, but they are frozen in this section, so we only indicate dependence of the approximating sequence on τ , the time splitting parameter. We partition the time interval [0, T ] into T τ time intervals of length τ , where T τ is assumed to be an even integer. Denoting t j = jτ , we define the functions h τ det and h τ st via The main result of this section is the following: Now we give the precise definition of a τ layer approximation. There are three elements of our approximating scheme we must introduce: a finite dimensional space where the velocity evolves, a regularization of the multiplicative structure of the noise, and an artificial pressure.
To truncate the high modes of the velocity field, we introduce of collection { n } ∞ n=1 of linear operators satisfying the following: The collection { n } ∞ n=1 can be constructed using a wavelet expansion. For more details on wavelet expansions in domains, see [15]. These operators are accompanied by a collection of finite dimensional spaces {X n } ∞ n=1 defined by X n = n (L 2 (D; R d )). In view of these remarks, let C + (D) be the cone of positive functions in C(D) and η δ a standard mollifier space/time mollifier. Define the operator σ k,τ,n,δ : where (ρ, m, θ) are understood to be extended by zero outside of [0, T ] × D to give a meaning to the convolution. For the remainder of this section, we will use the abbreviation σ k,τ . Finally, the original pressure in the momentum equation will be replaced by an "artificial" one of the form p m (ρ) + δρ β + θ p θ (ρ) for a sufficiently large power β. For technical reasons which will be clear later in the article, we require that Vol. 18 (2018) The stochastic Navier-Stokes equations 423 DEFINITION 3.3. A triple (ρ τ , u τ , θ τ ) is defined to be a τ layer approximation to (1.1) provided there exists a stochastic basis ( τ , F τ , 4. For all φ ∈ X n and t ∈ [0, T ], the following equality holds P τ a.s. 5. For all ϕ ∈ C ∞ (D) with ∂ϕ ∂n | ∂ D = 0, the following equation holds P τ almost surely: In the definition above, we have replaced the initial data (ρ 0 , m 0 , θ 0 ) by a triple (ρ 0,δ , m 0,δ , θ 0,δ ) satisfying: HYPOTHESIS 3.4. For each δ > 0, ρ 0,δ and θ 0,δ belong to C ∞ (D) and obey the bounds Both are assumed to satisfy a Neumann boundary condition. The sequence {(ρ 0,δ , θ 0,δ )} δ>0 converges strongly to Finally, we define m 0,δ = m 0 1 {ρ 0,δ ≥ρ 0 } and assume |{ρ 0,δ < ρ 0 }| → 0.

Machinery from the deterministic theory
For each ρ ∈ C + (D), define a multiplication-type operator M[ρ] : X n → X * n as follows: for u, η ∈ X n , Proof. This is a very short exercise left to the reader, see Section 7 in Chapter 7 of [7] for more discussion.
Let us also introduce the mapping N : PROPOSITION 3.6. Let s < t be initial and final times and suppose initial data If u in ∈ X n , then u(s) = u in . The temperature equation is understood to hold almost everywhere in [s, t] × D. Moreover, the solution map is continuous.
Proof. The proof of this result is given in Section 7 of Chapter 7 of [7].
Vol. 18 (2018) The stochastic Navier-Stokes equations 425 We also require the following classical result on nonlinear parabolic equations.
PROPOSITION 3.7. Let ρ ∈ C ∞ (D) be nonnegative and δ > 0. For all times s < t and θ in ∈ L 2 (D), there exists a unique classical solution to Proof. Since ρ is time-independent and smooth, we may appeal to Theorem 8.1 in Chapter 5 of [11].

A classical SPDE result
Let provided that for all S ∈ [s, t] the following equality (in X n ) holds P a.s.
There exists a unique solution u ∈ L 2 ; C ([s, T ]; X n ) to (3.10) in the sense of (3.11).
Proof. The τ layer regularization of the noise coefficient ensures that the proposition can be established in the classical way using the Cauchy-Lipschitz theorem. For more details on this style of argument, see Chapter 7 of [5].

Proof of Theorem 3.1
We are now prepared to establish an existence theorem for the lowest level of our scheme.
in D (3.12) To extend the solution to the interval (t 2 j+1 , t 2 j+2 ] we appeal first to Proposition 3.7 to solve for the temperature and then to Propositions 3.8 to solve for the velocity. Observe that the evolution of the temperature does not involve the velocity, allowing us to decouple the two equations. In this manner, we find a unique triple (ρ, u, θ) satisfying (3.13) Using the Ito formula and the inductive hypothesis, one may check that (3.4)-(3.6) continue to hold for t ∈ [t 2 j , t 2 j+2 ]. The desired measurability, part 2 of Definition 3.3, follows from the continuity of the solution map to the deterministic problem (guaranteed by Proposition 3.6), together with the fact the that we obtain a stochastically strong solution during each time interval where the stochastic forcing evolves.

n layer existence
In this section, we apply Theorem 3.1 to build the next layer of the approximating scheme, the n layer. Our goal is to establish the following: THEOREM 4.1. For each n ≥ 1, there exists an n layer approximation (in the sense of Definition 4.2 below) (ρ n , u n , θ n ) of (1.1) relative to a stochastic basis ). Let us introduce the n layer regularization of the multiplicative noise structure. Define an operator σ k,n,δ : For the remainder of this section, we will use the abbreviation σ k,n . DEFINITION 4.2. A triple (ρ n , u n , θ n ) is defined to be an n layer approximation to (1.1) provided there exists a stochastic basis ( n , F n , Vol. 18 (2018) The stochastic Navier-Stokes equations 427 2. For all φ ∈ C ∞ (D) and all times t ∈ [0, T ] the following equality holds P n a.s.
3. For all φ ∈ X n and all times t ∈ [0, T ] the following equality holds P n a.s.
4. For all ϕ ∈ C ∞ (D) with ∂ϕ ∂n | ∂ D = 0, the following equation holds P n almost surely: For each n fixed we apply Theorem 3.1 to obtain a sequence of τ layer approximations {(ρ τ,n , u τ,n , θ τ,n )} τ >0 . In Sect. 4.1, we prove a compactness result for this sequence and extract a candidate n layer approximation (ρ n , u n , θ n ) built on a convenient choice of probability space ( n , F n , P n ). In Sect. 4.2, we use the compactness result to verify (ρ n , u n , θ n ) is an n layer approximation in the sense of Definition 4.2.

τ → 0 Compactness step
A key tool in this section is a τ layer analogue of renormalized temperature equation (2.1). For the convenience of the reader, we will now explain briefly how to derive this in the current context. For simplicity of notation, we drop dependence on τ for the moment. The following computations can be justified using the further regularity properties proved in Lemma 7.4 of [7]. Begin by observing that the continuity and temperature equations can be written in the compact form: Hence, we obtain the following renormalized form: The next lemma obtains estimates of the type derived in Sect. 2.
Proof. The first two lines of (4.7) may be obtained all at once. Indeed, we can apply the Ito formula to find the evolution of the total energy. For all t ∈ [0, T ]: Using the same approach as in Sect. 2, the L p ( ; L ∞ t ) norms of the RHS of (4.8) can be estimated in terms of the L p ( ; L ∞ t ) norms of the LHS of (4.8). Indeed, the only Vol. 18 (2018) The stochastic Navier-Stokes equations 429 additional fact needed is the L p x boundedness of the operators n . This follows from Hypothesis 3.2 and the Banach/Steinhaus theorem.
To obtain the remaining estimates, use (4.6) with H (θ ) = log(θ )and integrate over [0, T ] × D to find the following P a.s. inequality: Integrating by parts, we observe that: Hence, for any γ > 0, there exists a C γ such that x . Thus, we obtain: Finally, using modified Poincare inequality (2.1) as in Sect. 2 yields the last line in (4.7).
The next proposition is the main compactness step, yielding a candidate n layer approximation and a new sequence of τ layer approximations with improved compactness properties. PROPOSITION 4.4. There exists a probability space ( n , F n , P n ), a collection of independent Brownian motions {β k n } n k=1 , a limit point (ρ n , u n , θ n ), and a sequence of measurable maps { T τ } τ >0 such that: T τ constitutes a τ layer approximation to (1.1) relative to the stochastic basis T t=0 is the filtration generated by W τ . Moreover, the initial data are recovered in the sense that 3. The uniform bounds in Lemma 4.3 hold withρ τ ,θ τ ,ũ τ replacing ρ τ , θ τ , u τ and P n replacing P. 4. As τ → 0, the following convergences hold pointwise n : 14) where W = {β k } n k=1 and W n = {β k n } n k=1 .
The proof of Proposition 4.4 begins with a tightness lemma. For each τ > 0, let Observe that Y τ induces a measure on the topological space can be established using the bootstrapping arguments in [14]. In the course of these arguments, the following useful fact is proved: To prove the tightness of {P• (ρ τ +δ)θ τ , θ τ x ] w , we require some further estimates. Use renormalized form (4.6) with H (θ ) = 1 2 θ 2 and integrate over D to find: Integrating by parts, we find that for any small γ > 0, Vol. 18 (2018) The stochastic Navier-Stokes equations 431 Using that κ(θ) ∼ θ 2 , the first term above can be absorbed into the LHS of the estimate after integrating over [0, T ]. Hence, we obtain the following P a.s. inequality: Using (4.15) and the L p ( ; L 2 t (H 1 x )) bounds on θ τ , ρ τ , we find that: Using once more the L p ( ; L 2 t (H 1 x )) bounds on θ τ , we can apply (4.16) and Banach-Alaoglu to deduce the tightness of . Toward this end, recall that for any p > 1 and M > 0, the following set is compact in We will show that for p = 4 3 , the induced measures above become uniformly concentrated on such sets, up to small probability. Start by noting the inequality: x ) . Hence, (4.15) and (4.16) imply: (4.18) Next, write the temperature equation as follows: Estimating each term on the RHS gives: Applying (4.15) and Lemma 4.3, we find that Next we apply the tightness result above together with a version of Skorohod Theorem 7.18 to complete our compactness step.
Proof of Proposition 4.4. Note that (E, τ ) is a Jakubowski space. Hence, in view of Lemma 4.5, we may apply Jakubowski/Skorohod Theorem 7.18 to the sequence {Y τ } τ >0 in order to obtain a probability space ( n , F n , P n ), a sequence of measurable maps { T τ } τ >0 , and a limit point Y such that Y τ • T τ converges pointwise to Y . In components, we write Y = (ρ n , u n , T n , θ n , W n ). Noting that (ρ τ +δ)ũ τ → (ρ n +δ)u n in D t,x pointwise in n , we identify T n = (ρ n + δ)θ n . This yields parts 1 and 4 of Proposition 4.4. The uniform energy bounds in Part 3 now follow from the fact that T τ pushes forward P n to P together with the estimates obtained already in Lemma 4.3. Finally, using the explicit relationship betweenρ τ ,ũ τ ,θ τ and ρ τ , u τ , θ τ , we are able to check that that the equation is preserved, that is, Part 2 holds. Similarly, Part 3 can be deduced from the bounds in Lemma 4.3. For more details on this last point, see [1].

This implies that
Next we define a filtration (F n t ) T t=0 via F n t = σ (r t X n ) where X n = ρ n , ρ n u n , ρ n u n , W n , ρ n θ n and r t : Proof. By Part 2,ρ τ ,ũ τ satisfy(on n ) the same parabolic equation as ρ τ , u τ . Using pointwise convergences (5.10) and (4.11), together with the observation that h τ det → 1 2 weakly in L p [0, T ] for all p ≥ 1, we may pass to the limit in the weak form and deduce that ρ n , u n satisfy parabolic equation (4.2). Now let φ ∈ X n and define the continuous, Similarly, we define the process ( M φ τ (t)) T t=0 in terms ofρ τ ,ũ τ ,θ τ and an additional oscillating factor 2h τ det . Let us check that Indeed, estimating each term we find that Proof. By part 2 of Proposition 4.4, for all ϕ ∈ C ∞ (D) with ∂ϕ ∂n | ∂ D = 0, the following equality holds P n almost surely: We will pass to the limit pointwise in ω ∈ n , proceeding term by term in the equality above from left to right. For the first term, use that (ρ τ + δ)θ τ → (ρ n + δ)θ n in C t ([L 2 x ] w ). For the next two terms, use thatθ τ → θ n in L 4 t,x . This is sufficient since κ(θ) ∼ θ 2 implies K(θ ) ∼ θ 3 . The regularized data do not depend on n and can be left alone.
For the next three terms, recall that h τ det → 1 2 weakly in L p t for any p ∈ [1, ∞). For the flux term, use Lemma 4.6 together with Proposition 4.4 to deduce thatρ τũτθτ → ρ n u n θ n in L 3 t (L p x ) where 1 p = 1 β + 1 3 . Note that p > 1 since β > 4, so we find that 2h τ detρ τũτθτ → ρ n u n θ n in D t,x . For the next term, the C t (X n ) convergence of the velocity clearly implies 2 ( x . This completes the proof.

layer existence
This section is devoted to the layer existence theory; sending n → ∞ our goal is to prove: THEOREM 5.1. For every > 0, there exists an layer approximation (in the sense of Definition 5.2 below) ρ , u , θ to (1.1), relative to a stochastic basis   (ρ , u , θ ) is an layer approximation to (1.1) provided there exists a stochastic basis ( , F , (F t ) T t=0 , P , β k k≥1 ) such that and all t ∈ [0, T ], the following equality holds P a.s.
3. For all φ ∈ C ∞ c (D) d and all t ∈ [0, T ], the following equality holds P a.s.

By Hypothesis 3.2 and the Banach/Steinhaus theorem, the operators
Hence, the first part of the lemma is obtained in the same way as the formal estimates Sect. 2. The second part of the lemma can be proved with the same technique as in Lemma 4.3. The δ dependence of the constant arises from the L p ( ; L 2 t,x ) bound for div u n . Next we establish the following compactness result: There exists a probability space ( , F , P ), a collection of independent Brownian motions {β k } ∞ k=1 , limit points ρ , u , θ , √ ρ u , and a sequence of measurable maps { T n } ∞ n=1 such that 1. For each n ≥ 1, T n : ( , F , P ) → ( n , F n , P n ) and ( T n ) # P = P n .
2. The new sequence {(ρ n ,ũ n ,θ n )} ∞ n=1 defined by ρ n ,ũ n ,θ n = (ρ n , u n , θ n ) • T n constitutes an n layer approximation relative to the stochastic basis ( , F ,P , ( F t n ) T t=0 , W n ), where W n := W n • T n and F t n = T −1 n • F t n .
Vol. 18 (2018) The stochastic Navier-Stokes equations 437 3. The uniform bounds in Lemma 5.3 hold with ρ n , θ n , u n replaced byρ n ,θ n ,ũ n and P n replaced by P . Moreover, P almost surely,

The following additional convergences hold
for all q ≥ 1.
Now we proceed to a tightness lemma. Define the sequence of random variables {Y n } n≥1 , where Y n = ρ n , u n , n (ρ n u n ), θ n , (ρ n + δ)θ n , {β k n } n k=1 Our convention is that given a topological vector space G, a finite sequence {x k } n k=1 is viewed as an element of G ∞ where x j = 0 for j ≥ n. For each n ≥ 1, Y n induces a measure on the topological space E, where Proof. The tightness of {P n • (ρ n , u n , n (ρ n u n ), W n ) −1 } ∞ n=1 parallels the treatment in [14]. Indeed, the only additional fact needed is the P n a.s. inequality 438 S. A. Smith and K. Trivisa Next we note that for all M ≥ 0, the ball of radius M in L ∞ t (L 1 x ) is compact with respect to the weak-topology on in L ∞ t (M x ). Hence, by Chebyshev and the uniform bounds from Lemma 5.3, we obtain the tightness of Similarly, using Banach-Alaoglu and the uniform bounds in Lemma 5.3, we obtain the tightness of Now we can complete the proof of our compactness step.
Proof of Proposition 5.4. Since E is a Jakubowski space, Lemma 5.5 allows us to apply Theorem 7.18. The remainder of the proof follows essentially the same arguments as in [14]. The one point worthy of a remark regards the proof of (5.9) and (5.14). Indeed, Lemma 5.5 yields directly the pointwise convergence (ρ n +δ)θ n → (ρ +δ)θ in [L 1 t (C 0 (D))] * . However, due to the explicit representatioñ ρ n (θ n + δ) = ρ n (θ n + δ) •T n , we obtain also pointwise uniform bounds (5.9). These two facts together are now sufficient to deduce pointwise convergence (5.14).

n → ∞ identification step
Next we define a filtration Proof. The proof uses Proposition 5.4 with the same arguments as in [14].
Fix ω ∈ . In the language of Proposition 7.23, let f n = (ρ n (ω) + δ)θ n (ω), f = (ρ (ω) + δ)θ (ω) and The temperature equation implies ∂ t f n ≤ g n in D t,x . By (5.9), the sequence ) for some p > 1. For this purpose, we use the pointwise convergences in Proposition 5.4 to select a constant C(ω) independent of n ≥ 1 such that Next we will estimate each term in the definition of g n . For simplicity of notation, we drop dependence on ω.
Since K(θ ) ∼ θ 3 , we find that Choose a p > 1 such that ). With these observations at hand, we may apply Proposition 7.23 and deduce (ρ n + δ)θ n → (ρ + δ)θ in L 2 t (H −1 x ). Moreover, by Proposition 5.4,θ n → θ in L 2 t (H 1 x ). Hence, (ρ n + δ)θ 2 n → (ρ + δ)θ 2 in D t,x . This completes the proof of the strong convergence of the temperature. Finally, to pass the limit in the temperature equation, work pointwise in and follow exactly the arguments in [7], page 174. Proof. In view of Proposition 5.4 and Lemmas 5.6/5.7, we may proceed as in [14] to prove the lemma. Indeed, there are just two differences with the corresponding proposition in [14]: the first is the form of the pressure, and the second is the dependence of σ k,n on ρ n θ n . To treat the first point, the only additional fact required is that P(ρ n ,θ n ) + δρ β n → P(ρ , θ ) + δρ β strongly in L 1 t,x , P almost surely. This follows from Lemmas 5.6, 5.7, and the pointwise(in ω) uniform control ofθ n p θ (ρ n ) in Moreover, the fourth moments of the pressure contribution to the weak form can be estimated as in (4.21), from τ → 0 identification of the momentum martingale. To treat the second point use the P almost sure to ρ θ together with the compactness of the mollification operator.

Conclusion of the proof
Proof of Theorem 5.1. For each > 0, we obtain an layer approximation (ρ ,ũ ) using our compactness step, Proposition 5.4 together with Lemmas 5.6 and 5.8. To obtain the uniform bounds, use the weak and weak-L p ( ) convergences in Proposition 5.4 as in [14] to treat all the terms which do not involve the temperature. To treat the uniform bounds for the temperature use the lower semicontinuity of the x ) with respect to weak convergence. Finally, note that by another lower-semicontinuity argument, (5.20) Since the total variation norm and the L 1 x norm agree for absolutely continuous measures, we find that for any q > 1, we have the continuous embedding t,x for all ω ∈ , we deduce that (5.21)

δ layer existence
This section is devoted to the δ layer existence theory; sending → 0 our goal is to prove: THEOREM 6.1. For every δ > 0, there exists a δ layer approximation (in the sense of Definition 6.2 below) ρ δ , u δ , θ δ to (1.1), relative to a stochastic basis Moreover, for all p ≥ 1, there exists a constant C p > 0 independent of δ such that Vol. 18 (2018) The stochastic Navier-Stokes equations 441 Recall that R is the collection of admissible renormalizations defined in the previous section. DEFINITION 6.2. A triple (ρ δ , u δ , θ δ ) is a δ layer approximation to (1.1) provided there exists a stochastic basis ( δ , F δ , where P is the predictable σ -algebra generated by (F t δ ) T t=0 and and all t ∈ [0, T ], the following equality holds P δ a.s.
Proof. The result follows using similar line of argument as in the earlier section. Proof. In view of Proposition 6.3 and Lemmas 5.6/5.7, we may proceed following similar line of argument as in [14] to prove the lemma. Since our work treats the nonisentropic case, and in light of the form of the pressure P(ρ ,θ ) the convergence of the pressure term P(ρ ,θ ) + δρ β → P(ρ δ , θ δ ) + δρ β δ strongly in L 1 t,x , P almost surely will be of use. The proof of the lemma follows by combining Lemma 5.6, Lemma 5.7, and the pointwise(in ω) uniform control of Moreover, the fourth moments of the pressure contribution to the weak form can be estimated as in the earlier layer.

Strong convergence of the density
Now to proceed to the proof of the strong convergence of the density. The first step is the following weak continuity result: LEMMA 6.7. Let K ⊂⊂ D be arbitrary, then the weak continuity of the effective viscous pressure holds on average, that is: Proof. Consider the operator A = ∇ −1 and let η be a bump function supported in D. Define the following two random test functionsφ = ηA[ηρ ] and ϕ δ = ηA[ηρ δ ] with ϕ (0) = ϕ δ (0). Using these test functions in the weak formulation of the momentum equation and following the line of argument in [14], applying Ito rule as needed and taking expectation in the resulting relation (as a way to eliminate the stochastic integrals) we arrive after some analysis at the following relations: 2 + I C, More precisely, The terms I A, 1 , I A, 2 account for the presence of artificial viscosity, are given as follows, and tend to 0 as → 0.
Making the appropriate adaptations to the arguments in [14] and using the monotonicity of p θ as in [7], one can apply Lemma 6.7 to deduce the following strong convergence result for the density. We present here an outline of the proof for completeness. Following similar line of argument as in [14] we arrive at In particular, as p θ is a non-decreasing function of the density we get In addition, as the function z → δz β is increasing, we get Taking into consideration (6.34), (6.35), (6.36) and noting that for any Lebesgue point s ∈ [0, T ] of the function s → E P ρ δ log ρ δ − ρ δ log ρ δ ]dx (s), we may choose a sequence of test functions that approximate 1 [0,s] (t), so that their time derivatives approximate the negative of a dirac mass centered at the point s, we conclude that the following estimate holds almost everywhere in time Choosing K close enough to D and taking into consideration the convexity of the function ρ log ρ we conclude that ρ δ log ρ δ = ρ δ log ρ δ a.e. in˜ δ × D. Vol. 18 (2018) The stochastic Navier-Stokes equations 447 LEMMA 6.9. For P almost all ω ∈ δ and q < 3,θ (ω) → θ δ (ω) in L q t,x . Moreover, ρ δ , u δ , and θ δ satisfy renormalized temperature inequality, (6.4) Proof. The proof of the strong convergence claim follows the argument in Lemma 5.7. The only additional detail is to explain how to pass from the renormalized form to an inequality in D t,x directly on ∂ t (ρ + δ)θ . Toward this end, for each m ∈ N, It is straightforward to verify h m ∈ R. Using the recovery maps, we may transfer the renormalized temperature inequality to the new probability space to find for each > 0 and m ∈ N, the P δ a.s. inequality Finally, passing to the limit in the renormalized temperature equation follows the same arguments as in the proof of Lemma 5.7. The only additional detail is to check that the term arising from the vanishing viscosity regularization tends to zero as → 0.

Conclusion of the proof
Proof of Theorem 6.1. For each δ > 0 we apply Proposition 6.3 to construct a candidate δ layer approximation (ρ δ , u δ , θ δ ). Combining Lemmas 6.5 and 6.6 with the strong convergence of the density we are able to identify P(ρ δ , θ δ ) + δρ δ β = P(ρ δ , θ δ ) + δρ β δ and complete the identification procedure. The uniform bounds can be argued as in the proof of Theorem 5.1.

Some further estimates
As δ → 0, the bounds based on the total energy are stable. However, the bounds obtained from the renormalized form of the temperature equation degenerate and need to be re-derived using the δ layer temperature inequality. In particular, the following estimates can be proved.
Proof. The proof of this lemma is essentially the same as in the formal estimates section, so we will only sketch the proof. The main difference is that we must use δ layer temperature inequality (6.4), which a priori holds only in a weak, renormalized form.
Begin by using the renormalization H m defined by: Derive the analogue of (2.3) and multiply by a factor of m ∈ N to obtain Applying Hölder's inequality yields Vol. 18 (2018) The stochastic Navier-Stokes equations 449 Using the renormalized continuity equation and sending m → ∞ yields Using the modified Poincare inequality from the formal estimates section gives: The remaining estimates are obtained in an analogous way using the renormalizations H m andĤ m defined byH A limited amount of additional details is provided in [7].
Observe that the bounds above do not control the L p ( ; L 3 t,x ) norm of the {θ δ } δ>0 , which is essential for passing to the limit in the temperature inequality. Instead, this estimate must be derived directly from the weak form of the δ layer renormalized temperature inequality. For this purpose, we adapt the approach of Feirisel [7] to the stochastic case. Namely, we begin by proving estimates away from vacuum (Lemma 7.2); then, we bootstrap these bounds and use a random test function approach to get estimates in the low density regions (Proposition 7.3)).
Choosing σ sufficiently small, we can guarantee q(σ ) > 3, leading to the following chain of inequalities: By the uniform L p ( δ ; H 1 ) bounds in {θ 3−σ 2 δ } δ>0 from Lemma 7.1 and the Sobolev embedding theorem, this concludes the proof. Now we improve on our initial estimate by adapting a method from [7] to obtain estimates on the temperature near the vacuum. PROPOSITION 7.3. For all p ∈ [1, ∞) there exists a constant C p > 0 such that Proof. For each level ν > 0, we introduce a sequence of random variables {X ν δ } δ>0 , where X ν δ : δ → R is defined by: Here, λ denotes the Lebesgue measure on R d . We will begin by proving there exists a ν 0 > 0 such that for all ν ≤ ν 0 , X ν δ > 0 a.s. with respect to P δ . Moreover, for all p ≥ 1, there exists a constant C p,ν > 0 such that the following uniform bound holds: To establish the claim above, begin by applying Holder in space, then minimizing in time to deduce the following P δ a.s. inequality: inf On the other hand, conservation of mass implies that P δ almost surely, Combining (7.8) and (7.9) yields the following lower bound for X ν δ : By the strong convergence of the initial densities, Hypothesis 3.4, we can choose a small enough ν 0 to ensure X ν δ > 0 a.s. with respect to P δ . Hence, inverting the lower bound above and applying the uniform L p ( δ ; L ∞ t (L γ x )) bounds on {ρ δ } δ>0 yields (7.7). The next step of the proof is to construct a suitable random test function. For each ν < ν 0 , let B ν : R + → [−1, 0] be a smooth function such that B ν (z) = 0 for z ≤ ν and B ν (z) = −1 for z ≥ 2ν. For each δ > 0, construct η δ ν : [0, T ]× D × δ → R such that for each (t, ω) ∈ [0, T ] × δ , the function x → η δ ν (t, x, ω) solves the following Neumann problem: The proof follows a similar line of argument as in [14] and the previous layers of this article. The only key difference is to obtain a tightness result for the renormalizations of the temperature equation. Toward this end, we define for each δ > 0 the [ Using the L p ( ; L 3 t,x ) estimates from Proposition 7.3, we will establish the following: Proof. By Proposition 7.3, there exists a constant C such that Since every sequence in E m is uniformly bounded and uniformly integrable in L 1 t,x , we may conclude that E m is a compact set in (L 1 t,x ) w . In addition, define By Tychnoff's theorem, E is a compact set in (L 1 t,x ) w ∞ . Applying Chebyshev, we find that Hence, applying inequalities (7.32) and (7.33) yields Since > 0 was arbitrary and C, D were fixed in advance, this completes the proof.
7.3. δ → 0 preliminary limit passage Next we define a filtration (F t ) T t=0 via F t = σ (r t X ) where X = ρ, ρu, W, u, ρθ and r t : Proof. In view of the strong convergence of the initial density (the initial data are assumed to be deterministic) this immediately follows from the pointwise convergences in Proposition 7.5.
At this point, we cannot make the same preliminary passage to the limit in the momentum equation as in the → 0 step. Recall that this preliminary passage to the limit was crucial for establishing the averaged weak continuity of the effective viscous pressure. Instead, one proves a partial result in this direction (Lemma 7.8), which essentially amounts to a preliminary passage to the limit in each of the terms besides the stochastic integrals. Indeed, as δ → 0, passing to the limit in the stochastic Vol. 18 (2018) The stochastic Navier-Stokes equations 455 integrals is even more difficult than passing to the limit in the pressure. This lemma turns out to be enough to prove the strong convergence of the density and temperature. We are then able to return to the task of passing to the limit in the stochastic integrals at the very end of the proof.
is a continuous, Proof. The proof uses our compactness result Proposition 7.5 following the arguments in [14].
LEMMA 7.9. Let K ⊂⊂ D be arbitrary, then the weak continuity of the effective viscous pressure holds on average, that is: Proof. The proof is established by proving an averaged Ito product rule based on the two random test functions. For more details, see the argument in [14]. The key trick is that: The other key point is that as a consequence of the integrability gains on the density.

Strong convergence of the density
Since γ may be close to d 2 , the limiting density ρ may not belong to L ∞ t (L 2 x ) for all ω ∈ . Hence, it is not a priori clear that the continuity equation for ρ may be renormalized (which is crucial for the proof of strong convergence). In [7], this issue is addressed via a thorough analysis of the so-called oscillations defect measure: In our framework, the quantity above is random. If we could show that for some p > 2 and all O ⊂ [0, T ] × D, osc p [ρ δ → ρ](O) < ∞ P almost surely, we could appeal directly to the results in [7] and deduce that ρ is a renormalized solution of the transport equation P almost surely. This would be the case, for instance, if we could show that However, it seems that this would require proving the weak continuity of the effective viscous pressure, Lemma 7.9, in the P almost sure sense, rather than in expectation only. It is our point of view that this is most likely not even true based on the information at this stage in the proof. The issue is the contribution of the stochastic integrals, which seems to lead only to a convergence in probability law, not P almost sure convergence. Effectively, the way to cure this problem is to reverse the order of operations. Namely, in the notion of oscillations defect measure, one should take an expectation prior to passing the limit supremum in δ. This remark is made precise through the following lemma. The proof follows closely the method in [7]. Combining this with observation with Lemma 7.9 and the monotonicity of p θ yields: The stochastic Navier-Stokes equations 457 Finally, using the uniform estimates for {ũ δ } δ>0 in L 2 ( ; L 2 t (H 1 x )) guaranteed by Proposition 7.5, it follows that: where˜ may be chosen arbitrarily small. Combining these inequalities and bringing the last term back to the LHS of the estimate gives the claim.
(7.38) Send δ → 0 and apply P a.s. convergences in Proposition 7.5 to pass the limit in each term above. Noting that the equation satisfied by T k (ρ) can be renormalized (due to the unlimited integrability), we find that P almost surely, (7.39) By assumption, we may choose M > 0 such that β is supported on [0, M]. Passing k → ∞ on both sides of the identity above and using the P almost sure L 1 t,x strong convergence of T k (ρ) toward ρ, we see that the proof of the Lemma will be complete as soon as we establish: Using the fact that T k (z)z ≤ T k (z), adding and subtracting T k (ρ), then appealing to lower-semicontinuity arguments once more yield: The proof is now complete in view of Lemma 7.10.
Using Lemma 7.10 together with our renormalization Lemma 7.11, we may now obtain the following strong convergence of the density. Proof. Since ρ is a renormalized solution, the proof now follows along the lines of the arguments in [7,14]. The role of the higher integrability ρ in the case γ > 2 is now replaced by the use of Lemma 7.10. The treatment of the temperature part of the pressure does not differ substantially from the arguments given at the → 0 construction of the δ layer.
To summarize, we have the following Corollary of Lemma 7.13. COROLLARY 7.14. For all q < 3, ω ∈ , we have the following strong convergence away from vacuum: Proof. Use the δ layer renormalized temperature inequality satisfied byρ δ ,θ δ ,ũ δ with the renormalization H m defined above. Sending m → ∞ and using Corollary 7.14 together with the arguments from Lemma 6.9 yields: By the definition of θ , it follows that K(θ )(ω) = K(ω), pointwise in . This completes the lemma.
Raising to the power β we find: Using our lower bound for |G|, we find: Combining this estimate with the lemma gives the claim.