# Measure-valued solutions to the stochastic compressible Euler equations and
incompressible limits

Martina Hofmanová 11footnotemark: 1 Ujjwal Koley 22footnotemark: 2 Utsab
Sarkar 33footnotemark: 3

###### Abstract

We introduce a new concept of dissipative measure-valued martingale solutions
to the stochastic compressible Euler equations. These solutions are weak in
the probabilistic sense i.e., the probability space and the driving Wiener
process are an integral part of the solution. We derive the relative energy
inequality for the stochastic compressible Euler equations and, as a
corollary, we exhibit pathwise weak-strong uniqueness principle. Moreover,
making use of the relative energy inequality, we investigate the low Mach
limit (incompressible limit) of underlying system of equations. As a main
novelty with respect to the related literature, our results apply to general
nonlinear multiplicative stochastic perturbations of Nemytskij type.

∗ Fakultät für Mathematik, Universität Bielefeld

D-33501 Bielefeld, Germany

<EMAIL_ADDRESS>
† Centre for Applicable Mathematics, Tata Institute of Fundamental Research

P.O. Box 6503, GKVK Post Office, Bangalore 560065, India

<EMAIL_ADDRESS>
‡ Centre for Applicable Mathematics, Tata Institute of Fundamental Research

P.O. Box 6503, GKVK Post Office, Bangalore 560065, India

<EMAIL_ADDRESS>
Keywords: Euler system; Navier–Stokes system; Compressible fluids; Stochastic
forcing; Measure-valued solution; Dissipative solution; Weak-strong
uniqueness; Low Mach limit.

###### Contents

  1. 1 Introduction
  2. 2 Mathematical Framework and Main Results
    1. 2.1 Analytic framework
      1. 2.1.1 Young measures, concentration defect measures
    2. 2.2 Stochastic framework
    3. 2.3 Stochastic compressible Euler equations
    4. 2.4 Stochastic compressible Navier–Stokes equations
    5. 2.5 Measure-valued solutions
      1. 2.5.1 Dissipative measure-valued martingale solutions
    6. 2.6 Statements of main results
  3. 3 Proof of Theorem 2.14
    1. 3.1 Compactness and almost sure representations
      1. 3.1.1 Passage to the limit
  4. 4 Weak-Strong Uniqueness Principle
    1. 4.1 Relative energy inequality
    2. 4.2 Proof of Theorem 2.16
  5. 5 Singular Limits
    1. 5.1 Solutions of the incompressible Euler system
    2. 5.2 Main result
    3. 5.3 Proof of Theorem 5.2
      1. 5.3.1 Relative energy inequality

##  1 Introduction

The concept of measure-valued solutions to deterministic partial differential
equations (PDEs) was first introduced by DiPerna ([24]) in the context of
conservation laws. The measure-valued solutions for compressible fluid
dynamics equations has been introduced by Neustupa in [39], and subsequently
revisited by Feireisl et. al. in [27], where the authors have introduced the
concept of so-called _dissipative_ measure-valued solutions. In this paper, we
introduce a notion of dissipative measure-valued solution for the
stochastically forced system of the _compressible barotropic Euler system_
describing the time evolution of the mass density $\varrho$ and the bulk
velocity ${\bf u}$ of a fluid. The system of equations reads

$\displaystyle{\rm d}\varrho+{\rm div}_{x}(\varrho{\bf u})\,{\rm d}t$
$\displaystyle=0,$ (1.1) $\displaystyle{\rm d}(\varrho{\bf u})+\left[{\rm
div}_{x}(\varrho{\bf u}\otimes{\bf u})+\nabla_{x}p(\varrho)\right]\,{\rm d}t$
$\displaystyle=\mathbb{G}(\varrho,\varrho{\bf u})\,{\rm d}W.$ (1.2)

Here $p(\varrho):=\varrho^{\gamma}$ is the pressure; $\gamma>1$ denotes the
adiabatic exponent, and to avoid the well-known difficulties related to the
behaviour of fluid flows near the boundary, we study (1.1)–(1.2) on the
$3$-dimensional torus $\mathbb{T}^{3}$. The driving stochastic process $W$ is
a cylindrical Wiener process defined on some probability space
$(\Omega,\mathfrak{F},\mathbb{P})$ and the coefficient $\mathbb{G}$ is
generally nonlinear and satisfies suitable growth assumptions (see Section 2
for the complete list of assumptions). The problem is closed by prescribing
the random _initial_ data

$\varrho(0,\cdot)=\varrho_{0},\ \varrho{\bf u}(0,\cdot)=(\varrho{\bf u})_{0},$
(1.3)

with sufficient spatial regularity specified later.

It is well-known that smooth solutions to deterministic counterpart of
(1.1)–(1.2) exist only for a finite lap of time after which singularities may
develop for a generic class of initial data, and weak solutions must be
sought. Moreover, weak solutions may not be uniquely determined by their
initial data and an admissibility condition must be imposed to single out the
physically correct solution. However, in light of the seminal work by De
Lellis $\&$ Székelyhidi [22, 23], and further confirmed by Chiodaroli et. al.
[18], it turns out that the deterministic compressible Euler system is
desperately ill-posed. Even if the initial data is smooth, the global
existence and uniqueness of solutions can fail. Moreover, the question of
existence of global-in-time weak solutions to (1.1)–(1.2) for _general_
initial data remains open. In the stochastic set-up, existence of a martingale
solution for (1.1)–(1.2) was established in [7] in _one_ space-dimension. A
stochastic variant of deterministic results, as depicted in [22, 26], for the
stochastic compressible Euler system have been achieved recently by Breit et.
al. in [12]. In fact, it was shown in [12] that the same difficulty (ill-
posedness) persists even in the presense of a random forcing. Incidentally,
the existence of global strong solutions for (1.1)–(1.2) (i.e. the stopping
time $\mathfrak{t}$ in Definition 2.5 reaches $T$) is not expected.

In view of the above discussion, our goal is to present a suitable concept of
_dissipative_ measure-valued solutions to (1.1)–(1.2) and discuss their basic
properties. Broadly speaking, they are measure-valued solutions of the
stochastic compressible Euler system (1.1)–(1.2) satisfying an appropriate
form of energy inequality and exist globally in time for any finite energy
initial data. The reason we consider measure-valued solutions is twofold:
First, we see measure-valued solutions as possibly the largest class in which
family of smooth (classical) solutions is stable. This is an undemanding
consequence of weak (measure-valued)-strong uniqueness principle; second,
solutions to an expansive class of approximate problems - emanating from
numerical schemes - can be shown to converge to a measure-valued solution
whereas convergence to a weak solution is either not known or requires
strenuous efforts to prove. Measure-valued solution for fluid flows driven by
_multiplicative_ noise is a relatively new area of pursuit, and to the best of
our knowledge, this is the first attempt to characterize the concept of
measure-valued solution for such equations. Having said this, we mention that
for incompressible Euler equations driven by _additive_ noise, the existence
of measure valued solution was first established by Kim in [32]. He has shown
that the existence can be obtained from a sequence of solutions of stochastic
Navier Stokes equations using tools from martingale selection. Recently, a
concept of dissipative solutions to incompressible Euler equations driven by
_additive_ noise has been introduced by Breit $\&$ Moyo in [11].

To motivate our definition of measure-valued solutions, we need a sequence of
approximate solutions to (1.1)–(1.2) together with an appropriate energy
inequality. To give a concrete example, we consider an approximation by
viscous compressible fluids subject to stochastic forcing described by the
Navier–Stokes system

$\displaystyle{\rm d}\varrho_{\varepsilon}+{\rm
div}_{x}(\varrho_{\varepsilon}{\bf u}_{\varepsilon})\,{\rm d}t$
$\displaystyle=0,$ (1.4) $\displaystyle{\rm d}(\varrho_{\varepsilon}{\bf
u}_{\varepsilon})+\left[{\rm div}_{x}(\varrho_{\varepsilon}{\bf
u}_{\varepsilon}\otimes{\bf
u}_{\varepsilon})+\nabla_{x}p(\varrho_{\varepsilon})\right]\,{\rm d}t$
$\displaystyle={\varepsilon}\,{\rm div}_{x}\mathbb{S}(\nabla_{x}{\bf
u}_{\varepsilon})\,dt+\mathbb{G}(\varrho_{\varepsilon},\varrho_{\varepsilon}{\bf
u}_{\varepsilon})\,{\rm d}W.$ (1.5)

Here $\mathbb{S}(\nabla_{x}{\bf u}_{\varepsilon})$ is the viscous stress
tensor for which we assume Newton’s rheological law

$\mathbb{S}(\nabla_{x}{\bf u}_{\varepsilon})=\delta\left(\nabla_{x}{\bf
u}_{\varepsilon}+\nabla_{x}^{t}{\bf u}_{\varepsilon}-\frac{2}{3}{\rm
div}_{x}{\bf u}_{\varepsilon}\mathbb{I}\right)+\lambda\,{\rm div}_{x}{\bf
u}_{\varepsilon}\mathbb{I},\qquad\delta>0,\ \lambda\geq 0.$

It is also well-known that smooth solutions to (1.4)–(1.5) satisfy the
following energy inequality

$\displaystyle\begin{aligned}
&\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\varrho_{\varepsilon}|{\bf
u}_{\varepsilon}|^{2}+\frac{\rho_{\varepsilon}^{\gamma}}{\gamma-1}\bigg{]}\,{\rm
d}{x}+{\varepsilon}\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathbb{S}(\nabla_{x}{\bf
u}_{\varepsilon}):\nabla_{x}{\bf u}_{\varepsilon}\,{\rm d}{x}\,{\rm
d}s\leq\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}(0)|^{2}}{\varrho_{\varepsilon}(0)}+\frac{\varrho_{\varepsilon}^{\gamma}(0)}{\gamma-1}\bigg{]}\,{\rm
d}{x}\\\
&\qquad+\sum_{k=1}^{\infty}\int_{0}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\mathbf{G}_{k}(\varrho_{\varepsilon},\varrho_{\varepsilon}{\bf
u}_{\varepsilon})\cdot{\bf u}_{\varepsilon}\,{\rm d}{x}\bigg{)}{\rm
d}W_{k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{t}\int_{\mathbb{T}^{3}}\varrho_{\varepsilon}^{-1}|\mathbf{G}_{k}(\varrho_{\varepsilon},\varrho_{\varepsilon}{\bf
u}_{\varepsilon})|^{2}\,{\rm d}s\end{aligned}$ (1.6)

The study of the system (1.4)–(1.5) was first initiated by Breit $\&$
Hofmanova in [9], where the global-in-time existence of _finite energy weak
martingale solutions_ is shown. These solutions are weak in the analytical
sense (derivatives only exists in the sense of distributions) and weak in the
probabilistic sense (the probability space is an integral part of the
solution) as well. Moreover, a relative energy inequality, based on the
pioneering work of Dafermos ([19]), has been recently established in Breit et.
al. [8]. As a corollary, they also demonstrated pathwise (and in law) weak-
strong uniqueness property for the system (1.4)–(1.5). The main deterrence
against plausible extension of the deterministic argument, in deriving
relative energy inequality [8, Section 3], essentially lies in successful
seizure of “noise-noise” interaction term (a perfect square term) between a
smooth given strong solution and a martingale solution of (1.4)–(1.5). Note
that, in the realm of stochastic scalar conservation laws (refer to [3, 4, 5,
34, 35]), “noise-noise” interaction term plays a pivotal role in the “doubling
of variables” argument, à la Kružkov. Furthermore, we mention that weak-strong
property was used in [8] to show convergence of weak solutions in the
inviscid-incompressible limit.

In this paper, our interest is directed to the weak (measure-valued)–strong
uniqueness principle for dissipative measure-valued solution. In what follows,
we first introduce a notion of dissipative measure-valued solution for
stochastic compressible Euler equations, keeping in mind that this framework
will enable us to exhibit weak-strong uniquness property. Roughly speaking,
the major stumbling block to emulate the ideas from [8] lies in the successful
identification (in the limit) of the martingale terms present in momentum
equations (1.5) and energy inequality (1.6). In general, due to lack of
regular estimates for approximate solutions, it seems not possible to
identify, in the limit, the martingale term in the momentum equation. In
particular, it is neither possible to borrow results from Debussche et. al.
[20] nor to use strong/weak continuity property of linear operators (like
Itô’s integral) between two Banach spaces, to pass to the limit in the
martingale terms. We can only conclude that the limit object is a martinagle.
On the other hand, a closer look at the proof of weak-strong uniqueness in [8]
reveals that it is enough to have knowledge about the cross variation of a
martingale solution with a strong solution. We are able to successfully
leverage this observation further to establish the weak (measure-
valued)–strong principle for (1.1)–(1.2). Indeed, as a conceivable remedy,
this is done by stipulating the correct cross variation with a smooth process
in the definition of measure-valued solution (refer to Definition 2.11).

Finally, to display the effectiveness of the relative energy inequality, we
apply the same to find out the incompressible limit of stochastic compressible
flows. Note that it is arguably rather difficult to show directly that the
solutions of compressible Euler equations converge to the solutions of
incompressible Euler equations in the low Mach number regime. But it is much
easier, as it shown in Section 5, to show that the compressible Euler
solutions converge to some strong kind of solutions to the incompressible
Euler equations, thanks to weak-strong uniqueness property. In the realm of
low Mach limit, there is a large literature related to incompressible limit
for the deterministic counterpart of (1.1)–(1.2), being the first important
contribution by Kleinermann and Majda [33]. For the so called _well-prepared_
data, justification of incompressible limit has been made rigorous by many
authors ([1, 40, 38, 42]). All these authors assume that the solutions of the
compressible flow are smooth. However, as is well known, solutions of the
compressible Euler system develop singularities in a finite time no matter how
smooth and/or small the initial data are. The hypothesis of smoothness of
solutions is therefore quite restrictive and even not appropriate in the
context of compressible inviscid fluids. In light of the above discussions, to
find out the incompressible limit, we follow the strategy of Feireisl et. al.
[28] based on measure-valued solution. We also mention that this concept of
dissipative measure valued solution and weak (measure-valued)–strong
uniqueness principle are used recently in the analysis of convergence of
certain numerical schemes [16]. Furthermore, a suitable concept of dissipative
measure valued solution has been introduced for stochastic equations of
incompressible fluid flow in [17].

The rest of the paper is organized as follows: we describe all
mathematical/technical framework and state the main results in Section 2. In
Section 3, we present a proof of existence of dissipative measure-valued
martingale solutions using stochastic compactness. Section 4 is devoted on
deriving the weak (measure-valued) – strong uniqueness principle by making use
of a suitable relative energy inequality. Finally, in Section 5, we explore
another application of weak (measure-valued)–strong uniqueness property by
justifying rigorously low Mach limit of the stochastic compressible Euler
system (1.1)–(1.2).

##  2 Mathematical Framework and Main Results

In this section we present relevant mathematical tools to be used in the
subsequent analysis and state main results of this paper. To begin, we fix an
arbitrary large time horizon $T>0$. Throughout this paper, we use the letter
$C$ to denote various generic constants. There are situations where constant
may change from line to line, but the notation is kept unchanged so long as it
does not impact the central idea. For a generic set $E$, let
$\mathcal{M}_{b}(E)$ denotes the space of bounded Borel measures on $E$ whose
norm is given by the total variation of measures. It is the dual space to the
space of continuous functions vanishing at infinity $C_{0}(E)$ equipped with
the supremum norm. Moreover, let $\mathcal{P}(E)$ be the space of probability
measures on $E$.

###  2.1 Analytic framework

Let $H$ be a separable Hilbert space. Given $\alpha\in(0,1)$, let
$W^{\alpha,2}(0,T;H)$ denotes a $H$-valued Sobolev space which is
characterized by its norm

$\|u\|^{2}_{W^{\alpha,2}(0,T;H)}:=\int_{0}^{T}\|u(t)\|^{2}_{H}\,dt+\int_{0}^{T}\int_{0}^{T}\frac{\|u(t)-u(s)\|^{2}_{H}}{|t-s|^{1+2\alpha}}\,dt\,ds.$

We recall the following compact embedding result from Flandoli $\&$ Gatarek
[29, Theorem 2.2]

######  Lemma 2.1.

If $A\subset B$ are two Banach spaces with compact embedding, and real numbers
$\alpha\in(0,1)$ satisfy $\alpha>1/2$, then the embedding

$W^{\alpha,2}(0,T;A)\subset C([0,T];B)$

is compact.

####  2.1.1 Young measures, concentration defect measures

We shall briefly recapitulate the notion of Young measures and related results
which have been used frequently in the text. For a nice overview of
applications of the Young measure theory to hyperbolic conservation laws, we
refer to Balder [2]. In what follows, let us first assume that
$(X,\mathcal{M},\mu)$ is a sigma finite measure space. A Young measure from
$X$ into $\mathbb{R}^{M}$ is a weakly measurable function
$\mathcal{V}:X\rightarrow\mathcal{P}(\mathbb{R}^{M})$ in the sense that
$x\mapsto\mathcal{V}_{x}(B)$ is $\mathcal{M}$-measurable for every Borel set
$B$ in $\mathbb{R}^{M}$. In what follows, we make use of the following
generalization of the classical result on Young measures, see [13, Section
2.8].

######  Lemma 2.2.

Let $N,M\in\mathbb{N}$, $\mathcal{O}\subset\mathbb{R}^{N}\times(0,T)$ and let
$({\bf U}_{n})_{n\in\mathbb{N}}$, ${\bf
U}_{n}:\Omega\times\mathcal{O}\to\mathbb{R}^{M}$, be a sequence of random
variables such that

$\mathbb{E}\big{[}\|{\bf U}_{n}\|^{p}_{L^{p}(\mathcal{O})}\big{]}\leq
C,\,\,\text{for a certain}\,\,p\in(1,\infty).$

Then there exists a new subsequence $(\tilde{\bf U}_{n})_{n\in\mathbb{N}}$
(not relabeled), defined on the standard probability space
$\big{(}[0,1],\overline{\mathcal{B}[0,1]},\mathcal{L}\big{)}$, and a
parametrized family
${\\{\mathcal{\tilde{V}}^{\omega}_{x}\\}}_{x\in\mathcal{O}}$ (superscript
$\omega$ emphasises the dependence on $\omega$) of random probability measures
on $\mathbb{R}^{M}$, regarded as a random variable taking values in
$\big{(}L_{w^{*}}^{\infty}(\mathcal{O};\mathcal{P}(\mathbb{R}^{M})),w^{*}\big{)}$,
such that ${\bf U}_{n}$ has the same law as $\tilde{\bf U}_{n}$, i.e. ${\bf
U}_{n}\sim_{d}\tilde{\bf U}_{n},$ and the following property holds: for any
Carathéodory function $G=G(x,Z),x\in\mathcal{O},Z\in\mathbb{R}^{M}$, such that

$|G(x,Z)|\leq C(1+|Z|^{q}),\quad 1\leq q<p,\,\,\text{uniformly in}\,\,x,$

implies $\mathcal{L}$-a.s.,

$G(\cdot,\tilde{\bf
U}_{n})\rightharpoonup\overline{G}\,\,\text{in}\,\,L^{p/q}(\mathcal{O}),\,\,\text{where}\,\,\overline{G}(x)=\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot)};G(x,\cdot)\rangle:=\int_{\mathbb{R}^{M}}G(x,z)\,{\rm
d}\mathcal{\tilde{V}}^{\omega}_{x}(z),\,\,\text{for a.a.}\,\,x\in\mathcal{O}.$

It is well-known that Young measure theory works well when we try to extract
limits of bounded continuous functions. Next, we examine what happens for
those functions $H$ for which we only know that

$\mathbb{E}\big{[}\|H({\bf U}_{n})\|^{p}_{L^{1}(\mathcal{O})}\big{]}\leq
C,\,\,\text{for a certain}\,\,p\in(1,\infty),\,\mbox{uniformly in }n.$

In fact, using the fact that $L^{1}(\mathcal{O})$ is embedded in the space of
bounded Radon measures $\mathcal{M}_{b}(\mathcal{O})$, we can infer that
$\mathbb{P}$-a.s.

$\displaystyle\mbox{weak-* limit
in}\,\mathcal{M}_{b}(\mathcal{O})\,\,\mbox{of}\,\,H({\bf
U}_{n})=\langle\mathcal{\tilde{V}}^{\omega}_{x};H\rangle\,dx+H_{\infty},$

where $H_{\infty}\in\mathcal{M}_{b}(\mathcal{O})$, and $H_{\infty}$ is called
_concentration defect measure (or concentration Young measure)_. Note that, a
simple truncation analysis and Fatou’s lemma reveal that $\mathbb{P}$-a.s.
$\|\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot)};H\rangle\|_{L^{1}(\mathcal{O})}\leq
C$ and thus $\mathbb{P}$-a.s.
$\langle\mathcal{\tilde{V}}^{\omega}_{x};H\rangle$ is finite for a.e.
$x\in\mathcal{O}$. In what follows, regarding the concentration defect
measure, we shall make use of the following crucial lemma. For a proof of this
lemma modulo cosmetic changes, we refer to Feireisl et. al. [27, Lemma 2.1].

######  Lemma 2.3.

Let $\\{{\bf U}_{n}\\}_{n>0}$, ${\bf
U}_{n}:\Omega\times\mathcal{O}\rightarrow\mathbb{R}^{M}$ be a sequence
generating a Young measure $\\{\mathcal{V}^{\omega}_{y}\\}_{y\in\mathcal{O}}$,
where $\mathcal{O}$ is a measurable set in $\mathbb{R}^{N}\times(0,T)$. Let
$G:\mathbb{R}^{M}\rightarrow[0,\infty)$ be a continuous function such that

$\sup_{n>0}\mathbb{E}\big{[}\|G({\bf
U}_{n})\|^{p}_{L^{1}(\mathcal{O})}\big{]}<\infty,\,\text{for a
certain}\,\,p\in(1,\infty),$

and let $F$ be continuous such that

$F:\mathbb{R}^{M}\rightarrow\mathbb{R},\quad|F(\bm{z})|\leq G(\bm{z}),\mbox{
for all }\bm{z}\in\mathbb{R}^{M}.$

Let us denote $\mathbb{P}$-a.s.

${F_{\infty}}:={\tilde{F}}-\langle\mathcal{\tilde{V}}^{\omega}_{y},F(\textbf{v})\rangle\,dy,\quad{G_{\infty}}:={\tilde{G}}-\langle\mathcal{\tilde{V}}^{\omega}_{y},G(\textbf{v})\rangle\,dy.$

Here ${\tilde{F}},{\tilde{G}}\in\mathcal{M}_{b}(\mathcal{O})$ are weak-$*$
limits of $\\{F({\bf U}^{n})\\}_{n>0}$, $\\{G({\bf U}^{n})\\}_{n>0}$
respectively in $\mathcal{M}_{b}(\mathcal{O})$. Then $\mathbb{P}$-a.s.
$|F_{\infty}|\leq G_{\infty}$.

###  2.2 Stochastic framework

Here we specify the definition of the stochastic forcing term. Let
$(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P})$ be a
stochastic basis with a complete, right-continuous filtration. The stochastic
process $W$ is a cylindrical $(\mathfrak{F}_{t})$-Wiener process in a
separable Hilbert space $\mathfrak{U}$. It is formally given by the expansion

$W(t)=\sum_{k\geq 1}e_{k}W_{k}(t),$

where $\\{W_{k}\\}_{k\geq 1}$ is a sequence of mutually independent real-
valued Brownian motions relative to $(\mathfrak{F}_{t})_{t\geq 0}$ and
$\\{e_{k}\\}_{k\geq 1}$ is an orthonormal basis of $\mathfrak{U}$. To give the
precise definition of the diffusion coefficient $\mathbb{G}$, consider
$\varrho\in L^{\gamma}(\mathbb{T}^{3})$, $\varrho\geq 0$, and $\mathbf{u}\in
L^{2}(\mathbb{T}^{3})$ such that $\sqrt{\varrho}\mathbf{u}\in
L^{2}(\mathbb{T}^{3})$. Denote $\bf m=\varrho\bf u$ and let
$\,\mathbb{G}(\varrho,{\bf m}):\mathfrak{U}\rightarrow L^{1}(\mathbb{T}^{3})$
be defined as follows

$\mathbb{G}(\varrho,{\bf m})e_{k}=\mathbf{G}_{k}(\cdot,\varrho(\cdot),{\bf
m}(\cdot)).$

The coefficients
$\mathbf{G}_{k}:\mathbb{T}^{3}\times\mathbb{R}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$
are $C^{1}$-functions that satisfy uniformly in $x\in\mathbb{T}^{3}$

$\displaystyle|{\bf G}_{k}(\cdot,0,0)|+|\partial_{\varrho}{\bf
G}_{k}|+|\nabla_{{\bf m}}{\bf G}_{k}|$
$\displaystyle\leq\beta_{k},\quad\sum_{k\geq 1}\beta_{k}<\infty.$ (2.1)

As discussed in [9], we understand the stochastic integral as a process in the
Hilbert space $W^{-m,2}(\mathbb{T}^{3})$, $m>3/2$. Indeed, it is easy to check
that under the above assumptions on $\varrho$ and $\bf m$, the mapping
$\mathbb{G}(\varrho,\varrho\bf m)$ belongs to
$L_{2}(\mathfrak{U};W^{-m,2}(\mathbb{T}^{3}))$, the space of Hilbert–Schmidt
operators from $\mathfrak{U}$ to $W^{-m,2}(\mathbb{T}^{3})$. Consequently,
if111Here $\mathcal{P}$ denotes the predictable $\sigma$-algebra associated to
$(\mathfrak{F}_{t})$.

$\displaystyle\varrho$ $\displaystyle\in
L^{\gamma}(\Omega\times(0,T),\mathcal{P},\mathrm{d}\mathbb{P}\otimes\mathrm{d}t;L^{\gamma}(\mathbb{T}^{3})),$
$\displaystyle\sqrt{\varrho}\mathbf{u}$ $\displaystyle\in
L^{2}(\Omega\times(0,T),\mathcal{P},\mathrm{d}\mathbb{P}\otimes\mathrm{d}t;L^{2}(\mathbb{T}^{3})),$

and the mean value $(\varrho(t))_{\mathbb{T}^{3}}$ is essentially bounded then
the stochastic integral

$\int_{0}^{t}\mathbb{G}(\varrho,\varrho{\bf u})\ {\rm d}W=\sum_{k\geq
1}\int_{0}^{t}{\bf G}_{k}(\cdot,\varrho,\varrho{\bf u})\ {\rm d}W_{k}$

is a well-defined $(\mathfrak{F}_{t})$-martingale taking values in
$W^{-m,2}(\mathbb{T}^{3})$. Note that the continuity equation (1.1) implies
that the mean value $(\varrho(t))_{\mathbb{T}^{3}}$ of the density $\varrho$
is constant in time (but in general depends on $\omega$). Finally, we define
the auxiliary space $\mathfrak{U}_{0}\supset\mathfrak{U}$ via

$\mathfrak{U}_{0}:=\bigg{\\{}u=\sum_{k\geq 1}\beta_{k}e_{k};\;\sum_{k\geq
1}\frac{\beta_{k}^{2}}{k^{2}}<\infty\bigg{\\}},$

endowed with the norm

$\|u\|^{2}_{\mathfrak{U}_{0}}=\sum_{k\geq 1}\frac{\beta_{k}^{2}}{k^{2}},\quad
v=\sum_{k\geq 1}\beta_{k}e_{k}.$

Note that the embedding $\mathfrak{U}\hookrightarrow\mathfrak{U}_{0}$ is
Hilbert–Schmidt. Moreover, trajectories of $W$ are $\mathbb{P}$-a.s. in
$C([0,T];\mathfrak{U}_{0})$.

Finally, we recall the “Kolmogorov test” for the existence of continuous
modifications of real-valued stochastic processes.

######  Lemma 2.4.

Let $Y={\\{Y(t)\\}}_{t\in[0,T]}$ be a real-valued stochastic process defined
on a probability space $(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq
0},\mathbb{P})$. Suppose that there are constants $\alpha>1,\beta>0$, and
$C>0$ such that for all $s,t\in[0,T]$,

$\displaystyle\mathbb{E}[|Y(t)-Y(s)|^{\alpha}]\leq C|t-s|^{1+\beta}.$

Then there exists a continuous modification of $Y$ and the paths of $Y$ are
$\gamma$-Hölder continuous for every $\gamma\in[0,\frac{\beta}{\alpha})$.

###  2.3 Stochastic compressible Euler equations

Since the weak (measure-valued)–strong uniqueness principle for dissipative
measure-valued solutions require the existence of strong solution, we first
recall the notion of local strong pathwise solution for stochastic
compressible Euler equations, being first introduced in [10]. Such a solution
is strong in both the probabilistic and PDE sense, at least locally in time.
To be more precise, system (1.1)–(1.2) will be satisfied pointwise (not only
in the sense of distributions) on the given stochastic basis associated to the
cylindrical Wiener process $W$.

######  Definition 2.5 (Local strong pathwise solution).

Let $\left(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P}\right)$
be a stochastic basis with a complete right-continuous filtration. Let ${W}$
be an $(\mathfrak{F}_{t})$-cylindrical Wiener process and
$(\varrho_{0},\mathbf{u}_{0})$ be a $W^{s,2}(\mathbb{T}^{3})\times
W^{s,2}(\mathbb{T}^{3})$-valued $\mathfrak{F}_{0}$-measurable random variable,
for some $s>7/2$. A triplet $(\varrho,{\bf u},\mathfrak{t})$ is called a local
strong pathwise solution to the system (1.1)–(1.3) provided

  1. (a)

$\mathfrak{t}$ is an a.s. strictly positive $(\mathfrak{F}_{t})$-stopping
time;

  2. (b)

the density $\varrho$ is a $W^{s,2}(\mathbb{T}^{3})$-valued
$(\mathfrak{F}_{t})$-progressively measurable process satisfying

$\varrho(\cdot\wedge\mathfrak{t})>0,\ \varrho(\cdot\wedge\mathfrak{t})\in
C([0,T];W^{s,2}(\mathbb{T}^{3}))\quad\mathbb{P}\text{-a.s.};$

  3. (c)

the velocity ${\bf u}$ is a $W^{s,2}(\mathbb{T}^{3})$-valued
$(\mathfrak{F}_{t})$-progressively measurable process satisfying

${\bf u}(\cdot\wedge\mathfrak{t})\in
C([0,T];W^{s,2}(\mathbb{T}^{3}))\quad\mathbb{P}\text{-a.s.};$

  4. (d)

there holds $\mathbb{P}$-a.s.

$\begin{split}\varrho(t\wedge\mathfrak{t})&=\varrho_{0}-\int_{0}^{t\wedge\mathfrak{t}}{\rm
div}_{x}(\varrho{\bf u})\ \mathrm{d}s,\\\ (\varrho{\bf
u})(t\wedge\mathfrak{t})&=\varrho_{0}{\bf
u}_{0}-\int_{0}^{t\wedge\mathfrak{t}}{\rm div}_{x}(\varrho{\bf u}\otimes{\bf
u})\ \mathrm{d}s-\int_{0}^{t\wedge\mathfrak{t}}\nabla_{x}\varrho^{\gamma}\
\mathrm{d}s+\int_{0}^{t\wedge\mathfrak{t}}{\mathbb{G}}(\varrho,\varrho{\bf
u})\ {\rm d}W,\end{split}$

for all $t\in[0,T]$.

Note that classical solutions require spatial derivatives of $\varrho$ and
${\bf u}$ to be continuous $\mathbb{P}$-a.s. This motivates the following
definition.

######  Definition 2.6 (Maximal strong pathwise solution).

Fix a stochastic basis with a cylindrical Wiener process and an initial
condition as in Definition 2.5. A quadruplet

$(\varrho,{\bf u},(\mathfrak{t}_{R})_{R\in\mathbb{N}},\mathfrak{t})$

is a maximal strong pathwise solution to system (1.1)–(1.3) provided

  1. (a)

$\mathfrak{t}$ is an a.s. strictly positive $(\mathfrak{F}_{t})$-stopping
time;

  2. (b)

$(\mathfrak{t}_{R})_{R\in\mathbb{N}}$ is an increasing sequence of
$(\mathfrak{F}_{t})$-stopping times such that $\mathfrak{t}_{R}<\mathfrak{t}$
on the set $[\mathfrak{t}<T]$,
$\lim_{R\to\infty}\mathfrak{t}_{R}=\mathfrak{t}$ a.s. and

$\sup_{t\in[0,\mathfrak{t}_{R}]}\|{\bf u}(t)\|_{1,\infty}\geq
R\quad\text{on}\quad[\mathfrak{t}<T];$ (2.2)

  3. (c)

each triplet $(\varrho,{\bf u},\mathfrak{t}_{R})$, $R\in\mathbb{N}$, is a
local strong pathwise solution in the sense of Definition 2.5.

Regarding existence of a maximal pathwise solution, there are number of works
available in literature in the context of various SPDE or SDE models, see for
instance [14, 25]. For the specific result on compressible Euler equations, we
refer to the work by Breit $\&$ Mensah in [10, Theorem 2.4].

######  Theorem 2.7.

Let $s>7/2$ and $(\varrho_{0},\mathbf{u}_{0})$ be an
$\mathfrak{F}_{0}$-measurable, $W^{s,2}(\mathbb{T}^{3})\times
W^{s,2}(\mathbb{T}^{3})$-valued random variable such that $\varrho_{0}>0$
$\mathbb{P}$-a.s. Then, under certain assumptions imposed on the coefficients
$\mathbf{G}_{k}$, there exists a unique maximal strong pathwise solution, in
the sense of Definition 2.6, $(\varrho,{\bf
u},(\mathfrak{t}_{R})_{R\in\mathbb{N}},\mathfrak{t})$ to problem (1.1)–(1.3)
with the initial condition $(\varrho_{0},{\bf u}_{0})$.

###  2.4 Stochastic compressible Navier–Stokes equations

There is a relatively small literature related to stochastic compressible
Navier–Stokes equations, with the first contribution by Breit $\&$ Hofmanova
in [9]. In this connection we also mention [8] and [13], which improves the
results in [9]. All the papers complement (1.4)–(1.5) with periodic boundary
conditions. Here we recall the concept of finite energy weak martingale
solutions to (1.4)–(1.5). These solutions are weak in the analytical sense
(derivatives only exists in the sense of distributions) and weak in the
probabilistic sense (the probability space is an integral part of the
solution) as well. Moreover, the time-evolution of the energy can be
controlled in terms of its initial state. They exists globally in time.

######  Definition 2.8 (Finite energy weak martingale solution).

Let $\Lambda_{\varepsilon}$ be a Borel probability measure on
$L^{\gamma}(\mathbb{T}^{3})\times L^{2\gamma/\gamma+1}(\mathbb{T}^{3})$. Then
$\big{[}\big{(}\Omega_{\varepsilon},\mathfrak{F}_{\varepsilon},(\mathfrak{F}_{{\varepsilon},t})_{t\geq
0},\mathbb{P}_{\varepsilon}\big{)};\varrho_{\varepsilon},\mathbf{u}_{\varepsilon},W_{\varepsilon}\big{]}$
is a finite energy weak martingale solution of (1.4)–(1.5) if

  1. (a)

$\big{(}\Omega_{\varepsilon},\mathfrak{F}_{\varepsilon},(\mathfrak{F}_{{\varepsilon},t})_{t\geq
0},\mathbb{P}_{\varepsilon}\big{)}$ is a stochastic basis with a complete
right-continuous filtration,

  2. (b)

$W_{\varepsilon}$ is a $(\mathfrak{F}_{{\varepsilon},t})$-cylindrical Wiener
process,

  3. (c)

the density $\varrho_{\varepsilon}$ satisfies $\varrho_{\varepsilon}\geq 0$,
$t\mapsto\langle\varrho_{\varepsilon}(t,\cdot),\phi\rangle\in C[0,T]$ for any
$\phi\in C^{\infty}(\mathbb{T}^{3})$ $\mathbb{P}_{\varepsilon}$-a.s., the
function $t\mapsto\langle\varrho_{\varepsilon}(t,\cdot),\phi\rangle$ is
progressively measurable and

$\displaystyle\mathbb{E}_{\varepsilon}\,\bigg{[}\sup_{t\in(0,T)}\|\varrho_{\varepsilon}(t,\cdot)\|_{L^{\gamma}(\mathbb{T}^{3})}^{p}\bigg{]}<\infty$

for all $1\leq p<\infty$,

  4. (d)

the velocity field $\mathbf{u}_{\varepsilon}$ is
$(\mathfrak{F}_{{\varepsilon},t})$-adapted and

$\displaystyle\mathbb{E}_{\varepsilon}\,\bigg{[}\int_{0}^{T}\|\mathbf{u}_{\varepsilon}\|^{2}_{W^{1,2}(\mathbb{T}^{3})}\,\mathrm{d}t\bigg{]}^{p}<\infty$

for all $1\leq p<\infty$,

  5. (e)

the momentum $\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}$ satisfies
$t\mapsto\langle\varrho_{\varepsilon}\mathbf{u}_{\varepsilon},\bm{\varphi}\rangle\in
C[0,T]$ for any $\bm{\varphi}\in C^{\infty}(\mathbb{T}^{3})$
$\mathbb{P}$-a.s., the function
$t\mapsto\langle\varrho_{\varepsilon}\mathbf{u}_{\varepsilon},\bm{\phi}\rangle$
is progressively measurable and

$\displaystyle\mathbb{E}_{\varepsilon}\,\bigg{[}\sup_{t\in(0,T)}\|\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}\|_{L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})}^{p}\bigg{]}<\infty$

for all $1\leq p<\infty$,

  6. (f)

$\Lambda_{\varepsilon}=\mathbb{P}_{\varepsilon}\circ\big{[}\varrho_{\varepsilon}(0),\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}(0)\big{]}^{-1}$,

  7. (g)

for all $\phi\in C^{\infty}(\mathbb{T}^{3})$ and $\bm{\varphi}\in
C^{\infty}(\mathbb{T}^{3})$ we have

$\displaystyle\langle\varrho_{\varepsilon}(t),\phi\rangle$
$\displaystyle=\langle\varrho_{\varepsilon}(0),\phi\rangle-\int_{0}^{t}\langle\varrho_{\varepsilon}\mathbf{u}_{\varepsilon},\nabla_{x}\phi\rangle\mathrm{d}s$
(2.3)
$\displaystyle\langle\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}(t),\bm{\varphi}\rangle$
$\displaystyle=\langle\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}(0),\bm{\varphi}\rangle-\int_{0}^{t}\langle\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}\otimes\mathbf{u}_{\varepsilon},\nabla_{x}\bm{\varphi}\rangle\,\mathrm{d}s+{\varepsilon}\,\int_{0}^{t}\langle\mathbb{S}(\nabla_{x}\mathbf{u}_{\varepsilon})\,,\nabla_{x}\bm{\varphi}\rangle\mathrm{d}s$
$\displaystyle-\int_{0}^{t}\langle
p(\varrho_{\varepsilon}),\mathrm{div}_{x}\,\bm{\varphi}\rangle\,\mathrm{d}s+\int_{0}^{t}\langle\mathbb{G}(\varrho_{\varepsilon},\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}),\bm{\varphi}\rangle\,\mathrm{d}W$

$\mathbb{P}$-a.s. for all $t\in[0,T]$,

  8. (h)

the energy inequality

$\displaystyle\begin{aligned}
&-\int_{0}^{T}\partial_{t}\psi\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\varrho_{\varepsilon}|{\bf
u}_{\varepsilon}|^{2}+\frac{\varrho_{\varepsilon}^{\gamma}}{\gamma-1}\bigg{]}\,dx\,dt+{\varepsilon}\,\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\mathbb{S}(\nabla_{x}{\bf
u}_{\varepsilon}):\nabla_{x}{\bf u}_{\varepsilon}\,{\rm d}{x}\,\,{\rm d}t\\\
&\qquad\leq\psi(0)\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}(0)|^{2}}{\varrho_{\varepsilon}(0)}+\frac{\varrho_{\varepsilon}^{\gamma}(0)}{\gamma-1}\bigg{]}\,{\rm
d}{x}+\sum_{k=1}^{\infty}\int_{0}^{T}\psi\bigg{(}\int_{\mathbb{T}^{3}}\mathbf{G}_{k}(\varrho_{\varepsilon},\varrho_{\varepsilon}{\bf
u}_{\varepsilon})\cdot{\bf u}_{\varepsilon}\,{\rm d}{x}\bigg{)}{\rm d}W_{k}\\\
&\qquad\qquad+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\varrho_{\varepsilon}^{-1}|\mathbf{G}_{k}(\varrho_{\varepsilon},\varrho_{\varepsilon}{\bf
u}_{\varepsilon})|^{2}\,dx\,dt\end{aligned}$ (2.4)

holds $\mathbb{P}$-a.s., for all $\psi\in C_{c}^{\infty}([0,T))$, $\psi\geq
0$.

######  Remark 2.9.

Note that we may take, for every $\varepsilon$, $\Omega_{\varepsilon}=[0,1]$,
with $\mathfrak{F}_{\varepsilon}$ the $\sigma$-algebra of the Borelians on
$[0,1]$, and $\mathbb{P}_{\varepsilon}$ the Lebesgue measure on $[0,1]$, see
Skorohod [41]. Moreover, we can assume without loss of generality that there
exists one common Wiener space $W$ for all $\varepsilon$. Indeed, this can be
achieved by performing the compactness argument from any chosen subsequence
${\\{\varepsilon_{n}\\}}_{n\in\mathbb{N}}$ at once. However, we stress the
fact that it may not be possible to preclude the dependence of $\varepsilon$
in the filtration $(\mathfrak{F}_{{\varepsilon},t})_{t\geq 0}$, due to lack of
pathwise uniqueness for the underlying system.

The following existence theorem is shown in [9, Theorem 1] (see also [13,
Chapter 4] for a more general version).

######  Theorem 2.10.

Let $\gamma>\frac{3}{2}$ and assume that $\Lambda_{\varepsilon}$ is a Borel
probability measure on $L^{\gamma}(\mathbb{T}^{3})\times
L^{2\gamma/\gamma+1}(\mathbb{T}^{3})$ such that

$\displaystyle\Lambda_{\varepsilon}\Bigg{\\{}(\varrho,\mathbf{m})\in
L^{\gamma}(\mathbb{T}^{3})\times
L^{2\gamma/\gamma+1}(\mathbb{T}^{3})\,:\,0<M_{1}\leq\varrho\leq
M_{2},\,\mathbf{m}|_{\\{\varrho=0\\}}=0\Bigg{\\}}=1$

with constants $0<M_{1}<M_{2}$. Furthermore, assume that the following moment
estimate

$\displaystyle\int_{L^{\gamma}_{x}\times
L^{2\gamma/\gamma+1}_{x}}\bigg{\|}\frac{1}{2}\frac{|\mathbf{m}|^{2}}{\varrho}+\frac{\varrho^{\gamma}}{\gamma-1}\bigg{\|}^{p}_{L^{1}(\mathbb{T}^{3})}\,\mathrm{d}\Lambda_{\varepsilon}(\varrho,\mathbf{m})<\infty,$

holds for all $1\leq p<\infty$. Finally, assume that (2.1) holds. Then there
exists a finite energy weak martingale solution of (1.4)–(1.5) in the sense of
Definition 2.8 with initial law $\Lambda_{\varepsilon}$.

###  2.5 Measure-valued solutions

To introduce the notion of measure-valued solution, it is more convenient to
reformulate the problem (1.1)–(1.2) in the _conservative_ variables $\varrho$
and ${\bf m}=\varrho{\bf u}$:

$\displaystyle{\rm d}\varrho+{\rm div}_{x}{\bf m}\,{\rm d}t$
$\displaystyle=0,$ (2.5) $\displaystyle{\rm d}{\bf m}+\left[{\rm
div}_{x}\bigg{(}\frac{{\bf m}\otimes{\bf
m}}{\varrho}\bigg{)}+\nabla_{x}p(\varrho)\right]\,{\rm d}t$
$\displaystyle=\mathbb{G}(\varrho,{\bf m})\,{\rm d}W.$ (2.6)

Note that, in general, the energy inequality (2.4) seems to be the only source
of a priori bounds. However, it is well known that uniformly bounded sequence
in $L^{1}(\mathbb{T}^{3})$, in general, does not imply weak convergence of it
due to the presence of oscillations and concentration effects. In fact, to
accommodate the above mentioned singularities two kinds of tools are used:

  * •

the Young measures, which are probability measures on the phase space and
account for the persistence of oscillations in the solution;

  * •

concentration defect measures, which are measures on physical space-time,
account for blow up type collapse due to possible concentration points.

####  2.5.1 Dissipative measure-valued martingale solutions

Motivated by the previous discussion, we are ready to introduce the concept of
dissipative measure–valued martingale solution to the stochastic compressible
Euler system. In what follows, let

$\mathcal{S}=\left\\{[\varrho,{\bf m}]\ \Big{|}\ \varrho\geq 0,\ {\bf
m}\in\mathbb{R}^{3}\right\\}$

be the phase space associated to the Euler system.

######  Definition 2.11 (Dissipative measure-valued martingale solution).

Let $\Lambda$ be a Borel probability measure on
$L^{\gamma}(\mathbb{T}^{3})\times
L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})$. Then
$\big{[}\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq
0},\mathbb{P}\big{)};\mathcal{V}^{\omega}_{t,x},W\big{]}$ is a dissipative
measure-valued martingale solution of (2.5)–(2.6), with initial condition
$\mathcal{V}^{\omega}_{0,x}$; if

  1. (a)

$\mathcal{V}^{\omega}$ is a random variable taking values in the space of
Young measures on
$L^{\infty}_{w^{*}}\big{(}[0,T]\times\mathbb{T}^{3};\mathcal{P}\big{(}\mathcal{S})\big{)}$.
In other words, $\mathbb{P}$-a.s.
$\mathcal{V}^{\omega}_{t,x}:(t,x)\in[0,T]\times\mathbb{T}^{3}\rightarrow\mathcal{P}(\mathcal{S})$
is a parametrized family of probability measures on $\mathcal{S}$,

  2. (b)

$\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P}\big{)}$ is
a stochastic basis with a complete right-continuous filtration,

  3. (c)

$W$ is a $(\mathfrak{F}_{t})$-cylindrical Wiener process,

  4. (d)

the average density $\langle\mathcal{V}^{\omega}_{t,x};\varrho\rangle$
satisfies
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};\varrho\rangle(t,\cdot),\phi\rangle\in
C[0,T]$ for any $\phi\in C^{\infty}(\mathbb{T}^{3})$ $\mathbb{P}$-a.s., the
function
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};\varrho\rangle(t,\cdot),\phi\rangle$
is progressively measurable and

$\displaystyle\mathbb{E}\,\bigg{[}\sup_{t\in(0,T)}\|\langle\mathcal{V}^{\omega}_{t,x};\varrho\rangle(t,\cdot)\|_{L^{\gamma}(\mathbb{T}^{3})}^{p}\bigg{]}<\infty$

for all $1\leq p<\infty$,

  5. (e)

the average momentum $\langle\mathcal{V}^{\omega}_{t,x};\textbf{m}\rangle$
satisfies
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};\textbf{m}\rangle(t,\cdot),\bm{\varphi}\rangle\in
C[0,T]$ for any $\bm{\varphi}\in C^{\infty}(\mathbb{T}^{3})$
$\mathbb{P}$-a.s., the function
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};\textbf{m}\rangle(t,\cdot),\bm{\phi}\rangle$
is progressively measurable and

$\displaystyle\mathbb{E}\,\bigg{[}\sup_{t\in(0,T)}\|\langle\mathcal{V}^{\omega}_{t,x};\textbf{m}\rangle(t,\cdot)\|_{L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})}^{p}\bigg{]}<\infty$

for all $1\leq p<\infty$,

  6. (f)

$\Lambda=\mathcal{L}[\mathcal{V}^{\omega}_{0,x}]$,

  7. (g)

the integral identity

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{\tau,x};\varrho\rangle\,\varphi(\tau,\cdot)\,dx-\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{0,x};\varrho\rangle\,\varphi(0,\cdot)\,dx=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{t,x};\textbf{m}\rangle\cdot\nabla_{x}\varphi\,dx\,dt,$
(2.7)

holds $\mathbb{P}$-a.s., for all $\tau\in[0,T)$, and for all $\varphi\in
C^{\infty}(\mathbb{T}^{3})$,

  8. (h)

there exists a $W^{-m,2}(\mathbb{T}^{3})$-valued square integrable continuous
martingale $M^{1}_{E}$, such that the integral identity

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{\tau,x};\textbf{m}\rangle\cdot\bm{\varphi}(\tau,\cdot)dx-\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{0,x};\textbf{m}\rangle\cdot\bm{\varphi}(0,\cdot)dx$
(2.8)
$\displaystyle\qquad=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\left[\left\langle\mathcal{V}^{\omega}_{t,x};\frac{\textbf{m}\otimes\textbf{m}}{\varrho}\right\rangle:\nabla_{x}\bm{\varphi}+\langle\mathcal{V}^{\omega}_{t,x};p(\varrho)\rangle\operatorname{div}_{x}\bm{\varphi}\right]dxdt$
$\displaystyle\qquad\qquad+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\bm{\varphi}\,dx\,dM^{1}_{E}(t)+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\nabla_{x}\bm{\varphi}:d\mu_{m},$

holds $\mathbb{P}$-a.s., for all $\tau\in[0,T)$, and for all $\bm{\varphi}\in
C^{\infty}(\mathbb{T}^{3};\mathbb{R}^{3})$, where $\mu_{m}\in
L^{\infty}_{w^{*}}\big{(}[0,T];\mathcal{M}_{b}({\mathbb{T}^{3}})\big{)}$,
$\mathbb{P}$-a.s., is a tensor–valued measure,

  9. (i)

there exists a real-valued square integrable continuous martingale
$M^{2}_{E}$, such that the following energy inequality

$\displaystyle\begin{aligned} \mathcal{E}(t+)&\leq\mathcal{E}(s-)\\\
&\quad+\frac{1}{2}\int_{s}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\left\langle\mathcal{V}^{\omega}_{\tau,x};\varrho^{-1}|\mathbf{G}_{k}(\varrho,\textbf{m})|^{2}\right\rangle\bigg{)}\,d\tau+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}d\mu_{e}+\int_{s}^{t}dM^{2}_{E}\end{aligned}$
(2.9)

holds $\mathbb{P}$-a.s., for all $0\leq s<t$ in $(0,T)$ with

$\displaystyle\mathcal{E}(t-):=\lim_{\tau\rightarrow
0+}\frac{1}{\tau}\int_{t-\tau}^{t}\Bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{D}(s)\Bigg{)}\,ds$
$\displaystyle\mathcal{E}(t+):=\lim_{\tau\rightarrow
0+}\frac{1}{\tau}\int_{t}^{t+\tau}\Bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{D}(s)\Bigg{)}\,ds$

Here $\mu_{e}\in
L^{\infty}_{w^{*}}\big{(}[0,T];\mathcal{M}_{b}({\mathbb{T}^{3}})\big{)}$,
$\mathbb{P}$-almost surely, $\mathcal{D}\geq 0$, $\mathbb{P}$-almost surely,
and
$\mathbb{E}\big{[}\operatorname*{ess\,sup}_{t\in(0,T)}\mathcal{D}(t)\big{]}<\infty$,
with initial energy

$\mathcal{E}(0-)=\int_{\mathbb{T}^{3}}\bigg{(}\frac{1}{2}\frac{|\textbf{m}_{0}|^{2}}{\varrho_{0}}+P(\varrho_{0})\bigg{)}\,\,{\rm
d}{x}.$

  10. (j)

there exists a constant $C>0$ such that

$\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\mu_{m}|+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\mu_{e}|\leq
C\int_{0}^{\tau}\mathcal{D}(t)\,\,{\rm d}t,$ (2.10)

holds $\mathbb{P}$-a.s., for every $\tau\in(0,T)$.

  11. (k)

Given any random stochastic process $f$ adapted to $(\mathfrak{F}_{t})_{t\geq
0}$ given by

$\mathrm{d}f=D^{d}f\,\mathrm{d}t+\mathbb{D}^{s}f\,\mathrm{d}W,$

satisfying

$f\in C([0,T];W^{1,q}\cap
C(\mathbb{T}^{3})),\quad\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|f\|_{W^{1,q}}^{2}\bigg{]}^{q}<\infty,\quad\text{$\mathbb{P}$-a.s.
for all }\ 1\leq q<\infty,$

and

$\displaystyle D^{d}f\in
L^{q}(\mathbb{T}^{3};L^{q}(0,T;W^{1,q}(\mathbb{T}^{3}))),\quad\mathbb{D}^{s}f\in
L^{2}(\mathbb{T}^{3};L^{2}(0,T;L_{2}(\mathfrak{U};L^{2}(\mathbb{T}^{3})))),$
$\displaystyle\bigg{(}\sum_{k\geq
1}|\mathbb{D}^{s}f(e_{k})|^{q}\bigg{)}^{\frac{1}{q}}\in
L^{q}(\mathbb{T}^{3};L^{q}(0,T;L^{q}(\mathbb{T}^{3}))),$

the cross variation between $f$ and the square integrable continuous
martingale $M^{1}_{E}$ is given by

$\displaystyle\Big{<}\Big{<}f(t),M^{1}_{E}(t)\Big{>}\Big{>}=\sum_{i,j}\Bigg{(}\sum_{k=1}^{\infty}\int_{0}^{t}\left\langle\left\langle\mathcal{V}^{\omega}_{s,x};\mathbf{G}_{k}(\varrho,{\bf
m})\right\rangle,g_{i}\right\rangle\,\left\langle\mathbb{D}^{s}f(e_{k}),g_{j}\right\rangle\,ds\Bigg{)}g_{i}\otimes
g_{j},$

where $g_{j}$’s are orthonormal basis for the space
$W^{-r,2}(\mathbb{T}^{N})$, for some $r>0$, and the bracket
$\big{<}\cdot,\cdot\big{>}$ denotes inner product in the same space.

######  Remark 2.12.

Since our interest is directed to the weak (measure-valued)-strong uniqueness
principle for dissipative measure-valued solution, the regularity assumptions
in the item (k) on the process $f$ are indeed taylor made to successfully
demonstrate the same principle for stochastic Euler equations, by making use
of the relative energy inequality.

######  Remark 2.13.

Note that, in view of a standard Lebesgue point argument applied to (2.9),
energy inequality holds for a.e. $0\leq s<t$ in $(0,T)$:

$\displaystyle\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{t,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{D}(t)$
(2.11)
$\displaystyle\qquad\leq\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{D}(s)+\frac{1}{2}\int_{s}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\left\langle\mathcal{V}^{\omega}_{\tau,x};\varrho^{-1}|\mathbf{G}_{k}(\varrho,\textbf{m})|^{2}\right\rangle\bigg{)}\,d\tau$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}d\mu_{e}+\int_{s}^{t}dM^{2}_{E},\,\,\mathbb{P}-a.s.$

However, for technical reasons, as pointed out in Section 4, we require energy
inequality to hold for _all_ $s,t\in(0,T)$. This can be achieved following the
argument depicted in Section 3.

###  2.6 Statements of main results

We now state main results of this paper. To begin with, regarding the
existence of dissipative measure-valued martingale solutions, we have the
following theorem.

######  Theorem 2.14.

Suppose that $\gamma>\frac{3}{2}$ and ${\bf G}_{k}$ be continuously
differentiable satisfying (2.1). Let $(\varrho_{\varepsilon},{\bf
m}_{\varepsilon}=\varrho_{\varepsilon}\,{\bf u}_{\varepsilon})$ be a family of
finite energy weak martingale solutions to the stochastic compressible
Navier–Stokes system (1.4)–(1.5). Let the corresponding initial data
$\varrho_{0},{\bf m}_{0}$, the initial law $\Lambda$, given on the space
$L^{\gamma}(\mathbb{T}^{3})\times
L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3};R^{3})$, be independent of
$\varepsilon$ satisfying

$\displaystyle\Lambda\Bigg{\\{}(\varrho,\mathbf{q})\in
L^{\gamma}(\mathbb{T}^{3})\times
L^{2\gamma/\gamma+1}(\mathbb{T}^{3})\,:\,0<M_{1}\leq\varrho\leq
M_{2},\,\mathbf{m}|_{\\{\varrho=0\\}}=0\Bigg{\\}}=1$

with constants $0<M_{1}<M_{2}$. Furthermore, assume that the following moment
estimate

$\displaystyle\int_{L^{\gamma}_{x}\times
L^{2\gamma/\gamma+1}_{x}}\bigg{\|}\frac{1}{2}\frac{|\mathbf{m}|^{2}}{\varrho}+\frac{\varrho^{\gamma}}{\gamma-1}\bigg{\|}^{p}_{L^{1}(\mathbb{T}^{3})}\,\mathrm{d}\Lambda(\varrho,\mathbf{m})<\infty$
(2.12)

holds for all $1\leq p<\infty$. Then the family
${\\{\varrho_{\varepsilon},{\bf m}_{\varepsilon}\\}}_{\varepsilon>0}$
generates, as $\varepsilon\mapsto 0$, a Young measure
${\\{\mathcal{V}^{\omega}_{t,x}\\}}_{t\in[0,T];x\in\mathbb{T}^{3}}$ which is a
dissipative measure-valued martingale solution to the stochastic compressible
Euler system (1.1)–(1.2), in the sense of Definition 2.11, with initial data
$\mathcal{V}^{\omega}_{0,x}=\delta_{\varrho_{0}(x),{\bf m}_{0}(x)}$
$\mathbb{P}$-a.s.

######  Remark 2.15.

We remark that the assumption $\gamma>\frac{3}{2}$ in the above Theorem 2.14
is merely technical and a manifestation of our particular choice of
approximate equations, i.e., Navier–Stokes equations, for which existence
result is available only for $\gamma>\frac{3}{2}$. However, for our purpose,
we only need an approximate solution (which works for all $\gamma>1$ ) with
appropriate energy bounds, e.g, one may consider “multipolar fluid” type
approximation [36]. In this way, we can exhibit existence of mesure-valued
solution to (1.1)–(1.2) for all $\gamma>1$.

We then establish the following weak (measure-valued)-strong uniqueness
principle:

######  Theorem 2.16 (Weak-Strong Uniqueness).

Let $\big{[}\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq
0},\mathbb{P}\big{)};\mathcal{V}^{\omega}_{t,x},W\big{]}$ be a dissipative
measure-valued martingale solution to the system (1.1)–(1.2). On the same
stochastic basis $\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq
0},\mathbb{P}\big{)}$, let us consider the unique maximal strong pathwise
solution to the Euler system (1.1)–(1.2) given by $(\bar{\varrho},\bar{{\bf
u}},(\mathfrak{t}_{R})_{R\in\mathbb{N}},\mathfrak{t})$ driven by the same
cylindrical Wiener process $W$ with the initial data
$\bar{\varrho}(0),\bar{\varrho}\bar{\bf u}(0)$ satisfies

$\mathcal{V}^{\omega}_{0,x}=\delta_{\bar{\varrho}(0,x),(\bar{\varrho}\bar{\bf
u})(0,x)},\,\mathbb{P}-\mbox{a.s.,}\,\mbox{for a.e. }x\in\mathbb{T}^{3}.$

Then for a.e. $t\in[0,T]$, $\mathcal{D}(t\wedge\mathfrak{t})=0$,
$\mathbb{P}$-a.s., and for a.e. $(t,x)\in[0,T]\times\mathbb{T}^{3}$

$\mathcal{V}^{\omega}_{t\wedge\mathfrak{t},x}=\delta_{\bar{\varrho}(t\wedge\mathfrak{t},x),(\bar{\varrho}\bar{\bf
u})(t\wedge\mathfrak{t},x)},\,\mathbb{P}-\mbox{a.s.}$

##  3 Proof of Theorem 2.14

The crucial argument used in this proof is based on a compactness method where
one needs uniform estimates to demonstrate tightness results and this yields
the convergence of the approximate sequence on another probability space and
the existence of dissipative measure-valued martingale solution follows,
thanks to Young measure theory. Observe that the existence of a pathwise
solution (typically obtained by Gyöngy-Krylov’s characterization of
convergence in probability) seems not possible due to the lack of pathwise
uniqueness for the approximate problem (1.4)–(1.5).

Let us first note that, thanks to the Theorem 2.10, the existence of finite
energy weak martingale solution of stochastic Navier–Stokes system (1.4)–(1.5)

$\big{[}\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{{\varepsilon},t})_{t\geq
0},\mathbb{P}\big{)};\varrho_{\varepsilon},\mathbf{u}_{\varepsilon},W\big{]}$

is well established. As alluded to before, lack of pathwise uniqueness for
(1.4)–(1.5) necessitates the filtration to depend on $\varepsilon$, while the
probability space and the Brownian motion can be chosen independent of
$\varepsilon$. The functions $\varrho_{\varepsilon}$ and
$\mathbf{u}_{\varepsilon}$ satisfy the energy inequality, i.e. for any $1\leq
p<\infty$ we have

$\displaystyle\mathbb{E}\bigg{[}\sup_{0\leq t\leq
T}\int_{\mathbb{T}^{3}}\Big{(}\frac{1}{2}\frac{|{\bf
m}_{\varepsilon}|^{2}}{\varrho_{\varepsilon}}+\frac{1}{\gamma-1}\varrho^{\gamma}_{\varepsilon}\Big{)}\,\mathrm{d}x$
(3.1)
$\displaystyle\qquad+\varepsilon\int_{0}^{T}\int_{\mathbb{T}^{3}}\delta|\nabla_{x}\mathbf{u}_{\varepsilon}|^{2}+(\lambda+\delta)|\operatorname{div}_{x}\mathbf{u}_{\varepsilon}|^{2}\,\mathrm{d}x\,\mathrm{d}s\bigg{]}^{p}$
$\displaystyle\qquad\qquad\leq\,C_{p}\bigg{(}1+\int_{L^{\gamma}\times
L^{\frac{2\gamma}{\gamma+1}}}\bigg{\|}\frac{1}{2}\frac{|{\bf
m}|^{2}}{\varrho}+\frac{1}{\gamma-1}\varrho^{\gamma}\bigg{\|}_{L^{1}}^{p}\,\mathrm{d}\Lambda(\varrho,{\bf
m})\bigg{)}\leq C(p,T).$

For a proof of (3.1), we refer to the paper by Breit $\&$ Hofmanova [9]. The
above relation leads to the following uniform bounds

$\displaystyle\mathbf{u}_{\varepsilon}$ $\displaystyle\in
L^{p}(\Omega;L^{2}(0,T;W^{1,2}(\mathbb{T}^{3}))),$ (3.2)
$\displaystyle\frac{{\bf m}_{\varepsilon}}{\sqrt{\varrho_{\varepsilon}}}$
$\displaystyle\in L^{p}(\Omega;L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))),$ (3.3)
$\displaystyle\varrho_{\varepsilon}$ $\displaystyle\in
L^{p}(\Omega;L^{\infty}(0,T;L^{\gamma}(\mathbb{T}^{3}))),$ (3.4)
$\displaystyle{\bf m}_{\varepsilon}$ $\displaystyle\in
L^{p}(\Omega;L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3}))),$
(3.5) $\displaystyle\frac{{\bf m}_{\varepsilon}\otimes{\bf
m}_{\varepsilon}}{\varrho_{\varepsilon}}$ $\displaystyle\in
L^{p}(\Omega;L^{2}(0,T;L^{\frac{6\gamma}{4\gamma+3}}(\mathbb{T}^{3}))).$ (3.6)

###  3.1 Compactness and almost sure representations

It is well known that, without assuming any topological structure on $\Omega$,
establishing a result of compactness in the probability variable
($\omega$-variable) is a non-trivial task. In what follows, to obtain strong
(a.s.) convergence in the $\omega$-variable, we make use of Skorokhod-
Jakubowski’s representation theorem (cf. [31]), linked to tightness of
probability measures and a.s. representations of random variables in quasi-
Polish spaces adapted to this situation. Note that since our path spaces are
not Polish, the classical Skorokhod theorem is not suitable; instead we use
the Jakubowski-Skorokhod representation theorem [31].

To establish the tightness of the laws generated by the approximations, we
first denote the path space $\mathcal{X}$ to be the product of the following
spaces:

$\displaystyle\mathcal{X}_{\varrho}$
$\displaystyle=C_{w}([0,T];L^{\gamma}(\mathbb{T}^{3})),$
$\displaystyle\mathcal{X}_{\mathbf{u}}$
$\displaystyle=\big{(}L^{2}(0,T;W^{1,2}(\mathbb{T}^{3})),w\big{)},$
$\displaystyle\mathcal{X}_{\varrho\mathbf{u}}$
$\displaystyle=C_{w}([0,T];L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})),$
$\displaystyle\mathcal{X}_{W}$ $\displaystyle=C([0,T];\mathfrak{U}_{0}),$
$\displaystyle\mathcal{X}_{C}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)},$
$\displaystyle\mathcal{X}_{P}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)},$
$\displaystyle\mathcal{X}_{E}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)},$
$\displaystyle\mathcal{X}_{\mathcal{V}}$
$\displaystyle=\big{(}L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4})),w^{*}\big{)},$
$\displaystyle\mathcal{X}_{M}$
$\displaystyle=\big{(}W^{\alpha,2}(0,T;W^{-m,2}(\mathbb{T}^{3})),w\big{)},$
$\displaystyle\mathcal{X}_{N}$ $\displaystyle=C([0,T];\mathbb{R}),$
$\displaystyle\mathcal{X}_{D}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)}.$

Let us denote by $\mu_{\varrho_{\varepsilon}}$,
$\mu_{\mathbf{u}_{\varepsilon}}$,
$\mu_{\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}}$, and
$\mu_{W_{\varepsilon}}$ respectively, the law of $\varrho_{\varepsilon}$,
$\mathbf{u}_{\varepsilon}$, $\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}$,
and $W_{\varepsilon}$ on the corresponding path space. Moreover, let
$\mu_{M_{\varepsilon}}$, and $\mu_{N_{\varepsilon}}$ denote the law of two
martingales
$M_{\varepsilon}:=\int_{0}^{t}{\mathbb{G}}(\varrho_{\varepsilon},\textbf{m}_{\varepsilon})\,{\rm
d}W$, and $N_{\varepsilon}:=\sum_{k\geq
1}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf u}_{\varepsilon}\cdot{{\bf
G}_{k}}(\varrho_{\varepsilon},\textbf{m}_{\varepsilon})\,dx\,{\rm d}W$ on the
corresponding path spaces. Furthermore, let $\mu_{C_{\varepsilon}}$,
$\mu_{D_{\varepsilon}}$, $\mu_{E_{\varepsilon}}$, $\mu_{P_{\varepsilon}}$, and
$\mu_{{\mathcal{V}}_{\varepsilon}}$ denote the law of

$\displaystyle
C_{\varepsilon}:=\frac{\textbf{m}_{\varepsilon}\otimes\textbf{m}_{\varepsilon}}{\varrho_{\varepsilon}},\quad
D_{\varepsilon}:=\sum_{k\geq
1}\rho_{\varepsilon}^{-1}|\mathbf{G}_{k}(\rho_{\varepsilon},\rho_{\varepsilon}{\bf
v}_{\varepsilon})|^{2},$ $\displaystyle
E_{\varepsilon}:=\frac{1}{2}\frac{|{\bf
m}_{\varepsilon}|^{2}}{\varrho_{\varepsilon}}+\frac{1}{\gamma-1}\varrho^{\gamma}_{\varepsilon},\quad
P_{\varepsilon}=\varrho^{\gamma}_{\varepsilon},\quad{\mathcal{V}}_{\varepsilon}:=\delta_{[\varrho_{\varepsilon},\textbf{m}_{\varepsilon}]},$

respectively, on the corresponding path spaces. Finally, let
$\mu^{\varepsilon}$ denotes joint law of all the variables on $\mathcal{X}$.
To proceed further, it is necessary to establish tightness of
$\\{\mu^{\varepsilon};\,\varepsilon\in(0,1)\\}$. To this end, we observe that
tightness of $\mu_{W_{\varepsilon}}$ is immediate. So we show tightness of
other variables.

######  Proposition 3.1.

The sets $\\{\mu_{\varrho_{\varepsilon}};\,\varepsilon\in(0,1)\\}$,
$\\{\mu_{\mathbf{u}_{\varepsilon}};\,\varepsilon\in(0,1)\\}$, and
$\\{\mu_{\varrho_{\varepsilon}\mathbf{u}_{\varepsilon}};\,\varepsilon\in(0,1)\\}$
are tight on $\mathcal{X}_{\varrho}$, $\mathcal{X}_{\mathbf{u}}$, and
$\mathcal{X}_{\varrho\mathbf{u}}$ respectively.

###### Proof.

For the details of this proof, we refer to [9]. ∎

######  Proposition 3.2.

The set
$\\{\mu_{C_{\varepsilon}},\mu_{D_{\varepsilon}},\mu_{E_{\varepsilon}},\mu_{P_{\varepsilon}};\,\varepsilon\in(0,1),\,k\geq
1\\}$ is tight on
$\mathcal{X}_{C}\times\mathcal{X}_{D}\times\mathcal{X}_{E}\times\mathcal{X}_{P}$.

###### Proof.

These follow immediately from the _a priori_ bounds using the fact that all
bounded sets in $L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3}))$ are
relatively compact with respect to the weak-$*$ topology. ∎

######  Proposition 3.3.

The set $\\{\mu_{{\mathcal{V}}_{\varepsilon}};\,\varepsilon\in(0,1)\\}$ is
tight on $\mathcal{X}_{\mathcal{V}}$.

###### Proof.

The aim is to apply the compactness criterion in
$\big{(}L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4})),w^{*}\big{)}$.
Define the set

$\displaystyle
B_{R}:=\Big{\\{}{\mathcal{V}}\in\big{(}L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4})),w^{*}\big{)};\int_{0}^{T}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{4}}\Big{(}|\xi_{1}|^{\gamma}+|\xi_{2}|^{\frac{2\gamma}{\gamma+1}}\Big{)}\,d{\mathcal{V}}_{t,x}(\xi)\,dx\,dt\leq
R\Big{\\}},$

which is relatively compact in
$\big{(}L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4})),w^{*}\big{)}$.
Note that

$\displaystyle\mathcal{L}[{\mathcal{V}}_{\varepsilon}](B^{c}_{R})$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{T}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{4}}\Big{(}|\xi_{1}|^{\gamma}+|\xi_{2}|^{\frac{2\gamma}{\gamma+1}}\Big{)}\,d{\mathcal{V}}_{t,x}(\xi)\,dx\,dt>R\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{(}|\varrho_{\varepsilon}|^{\gamma}+|\textbf{m}_{\varepsilon}|^{\frac{2\gamma}{\gamma+1}}\Big{)}\,dx\,dt>R\Bigg{)}\leq\frac{1}{R}\mathbb{E}\Big{[}\|\varrho_{\varepsilon}\|_{L^{\gamma}}^{\gamma}+\|\textbf{m}_{\varepsilon}\|_{L^{\frac{2\gamma}{\gamma+1}}}^{\frac{2\gamma}{\gamma+1}}\Big{]}\leq\frac{C}{R}.$

The proof is complete. ∎

######  Proposition 3.4.

The set $\\{\mu_{M_{\varepsilon}};\,\varepsilon\in(0,1)\\}$ is tight on
$\mathcal{X}_{M}$.

###### Proof.

First note that
$M_{\varepsilon}=\int_{0}^{t}\,\mathbb{G}(\varrho_{\varepsilon},\varrho_{\varepsilon}\mathbf{u}_{\varepsilon})\,\mathrm{d}W(s)\in
L^{p}\big{(}\Omega;W^{\alpha,2}(0,T;W^{-m,2}(\mathbb{T}^{3}))\big{)}$, and
hence tightness follows from weak compactness. ∎

######  Proposition 3.5.

The set $\\{\mu_{N_{\varepsilon}};\,\varepsilon\in(0,1)\\}$ is tight on
$\mathcal{X}_{N}$.

###### Proof.

First observe that, for each $\varepsilon$, $N_{\varepsilon}(t)=\sum_{k\geq
1}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf u}_{\varepsilon}\cdot{{\bf
G}_{k}}(\varrho_{\varepsilon},\textbf{m}_{\varepsilon})\,dx\,{\rm d}W$ is a
square integrable martingale. Note that for $r>2$

$\displaystyle\mathbb{E}\Big{[}\Big{|}\sum_{k\geq
1}\int_{s}^{t}\int_{\mathbb{T}^{3}}{\bf u}_{\varepsilon}\cdot{{\bf
G}_{k}}(\varrho_{\varepsilon},\textbf{m}_{\varepsilon})\Big{|}^{r}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\int_{s}^{t}\sum_{k=1}^{\infty}\Big{|}\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}\cdot{{\bf
G}_{k}}(\varrho_{\varepsilon},\textbf{m}_{\varepsilon})\Big{|}^{2}\Big{]}^{r/2}$
$\displaystyle\leq|t-s|^{r/2}\,\Big{(}1+\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\|\sqrt{\varrho_{\varepsilon}}u_{\varepsilon}\|^{r}_{L^{2}}\Big{]}\Big{)}\leq
C|t-s|^{r/2},$

and the Kolmogorov continuity criterion (cf. Lemma 2.4) applies. This in
particular implies that, for some $\alpha>1$

$\sum_{k\geq 1}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}\cdot{{\bf
G}_{k}}(\varrho_{\varepsilon},\textbf{m}_{\varepsilon})\,dx\,{\rm d}W\in
L^{r}(\Omega;C^{\alpha}(0,T;\mathbb{R})).$

Therefore, tightness of law follows from the compact embedding of $C^{\alpha}$
into $C^{0}$. ∎

Combining all the informations obtained from Proposition 3.1, Proposition 3.2,
Proposition 3.3, Proposition 3.4, and Proposition 3.5, we conclude that

######  Corollary 3.6.

The set $\\{\mu^{\varepsilon};\,\varepsilon\in(0,1)\\}$ is tight on
$\mathcal{X}$.

At this point, we are ready to apply Jakubowski-Skorokhod representation
theorem (see also Brzezniak et. al. [15]) to extract a.s convergence on a new
probability space. In what follows, passing to a weakly convergent subsequence
$\mu^{\varepsilon}$ (and denoting by $\mu$ the limit law) we infer the
following result:

######  Proposition 3.7.

There exists a subsequence $\mu^{\varepsilon}$ (not relabelled), a probability
space $(\tilde{\Omega},\tilde{\mathfrak{F}},\tilde{\mathbb{P}})$ with
$\mathcal{X}$-valued Borel measurable random variables
$(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon},\tilde{\textbf{u}}_{\varepsilon},\tilde{W}_{\varepsilon},\tilde{C}_{\varepsilon},\tilde{D}_{\varepsilon},\tilde{E}_{\varepsilon},\tilde{P}_{\varepsilon},\tilde{M}_{\varepsilon},\tilde{N}_{\varepsilon},\tilde{\nu}_{\varepsilon})$,
$\varepsilon\in(0,1)$, and $(\tilde{\varrho},\tilde{\bf
m},\tilde{\textbf{u}},\tilde{W},\tilde{C},\tilde{D},\tilde{E},\tilde{P},\tilde{M},\tilde{N},\tilde{\nu})$
such that

  1. (1)

the law of $(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon},\tilde{\textbf{u}}_{\varepsilon},\tilde{W}_{\varepsilon},\tilde{C}_{\varepsilon},\tilde{D}_{\varepsilon},\tilde{E}_{\varepsilon},\tilde{P}_{\varepsilon},\tilde{M}_{\varepsilon},\tilde{N}_{\varepsilon},\tilde{\nu}_{\varepsilon})$
is given by $\mu^{\varepsilon}$, $\varepsilon\in(0,1)$,

  2. (2)

the law of $(\tilde{\varrho},\tilde{\bf
m},\tilde{\textbf{u}},\tilde{W},\tilde{C},\tilde{D},\tilde{E},\tilde{P},\tilde{M},\tilde{N},\tilde{\nu})$,
denoted by $\mu$, is a Radon measure,

  3. (3)

$(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon},\tilde{\textbf{u}}_{\varepsilon},\tilde{W}_{\varepsilon},\tilde{C}_{\varepsilon},\tilde{D}_{\varepsilon},\tilde{E}_{\varepsilon},\tilde{P}_{\varepsilon},\tilde{M}_{\varepsilon},\tilde{N}_{\varepsilon},\tilde{\nu}_{\varepsilon})$
converges $\,\tilde{\mathbb{P}}$-almost surely to
$(\tilde{\varrho},\tilde{\bf
m},\tilde{\textbf{u}},\tilde{W},\tilde{C},\tilde{D},\tilde{E},\tilde{P},\tilde{M},\tilde{N},\tilde{\nu})$
in the topology of $\mathcal{X}$, i.e.,

$\displaystyle\tilde{\varrho}_{\varepsilon}\rightarrow\tilde{\varrho}\,\,\text{in}\,\,C_{w}([0,T];L^{\gamma}(\mathbb{T}^{3})),\quad$
$\displaystyle\tilde{\textbf{m}}_{\varepsilon}$
$\displaystyle\rightarrow\tilde{\textbf{m}}\,\,\text{in}\,\,C_{w}([0,T];L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})),$
$\displaystyle\tilde{\textbf{u}}_{\varepsilon}\rightarrow\tilde{\textbf{u}}\,\,\text{weak
in}\,\,L^{2}(0,T;W^{1,2}(\mathbb{T}^{3})),\quad$
$\displaystyle\tilde{W}_{\varepsilon}$
$\displaystyle\rightarrow\tilde{W}\,\,\text{in}\,\,C([0,T];\mathfrak{U}_{0})),$
$\displaystyle\tilde{C}_{\varepsilon}\rightarrow\tilde{C}\,\,\text{weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),\quad$
$\displaystyle\tilde{N}_{\varepsilon}$
$\displaystyle\rightarrow\tilde{N}\,\,\text{in}\,\,C([0,T];\mathbb{R}),$
$\displaystyle\tilde{E}_{\varepsilon}\rightarrow\tilde{E}\,\,\text{weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),\quad$
$\displaystyle\tilde{P}_{\varepsilon}$
$\displaystyle\rightarrow\tilde{P}\,\,\text{weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{\nu}_{\varepsilon}\rightarrow\tilde{\nu}\,\,\text{weak-$*$
in}\,\,L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4})),\quad$
$\displaystyle\tilde{M}_{\varepsilon}$
$\displaystyle\rightarrow\tilde{M}\,\,\text{weak
in}\,\,W^{\alpha,2}(0,T;W^{-m,2}(\mathbb{T}^{3})),$
$\displaystyle\tilde{D}_{\varepsilon}\rightarrow\tilde{D}\,\,\text{weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})).$

  4. (4)

For any Carathéodory function $H=H(t,x,\varrho,\textbf{m})$, where
$(t,x)\in(0,T)\times\mathbb{T}^{3}$ and
$(\varrho,\textbf{m})\in\mathbb{R}^{4}$, satisfying for some $p,q$ the growth
condition

$\displaystyle|H(t,x,\varrho,\textbf{m})|\leq
1+|\varrho|^{p}+|\textbf{m}|^{q},$

uniformly in $(t,x)$. Then we have $\tilde{\mathbb{P}}$-a.s.

$H(\tilde{\varrho}_{\varepsilon},\tilde{\textbf{m}}_{\varepsilon})\rightarrow\overline{H(\tilde{\varrho},\tilde{\textbf{m}})}\,\,\text{in}\,\,L^{r}((0,T)\times\mathbb{T}^{3}),\,\,\text{for
all}\,\,1<r\leq\frac{\gamma}{p}\wedge\frac{2\gamma}{q(\gamma+1)}.$

###### Proof.

Proof of the items $(1)$, $(2)$, and $(3)$ directly follow from Jakubowski-
Skorokhod representation theorem. For the proof of the item $(4)$, we refer to
the Lemma 2.3. ∎

We remark that the energy inequality (3.1) continues to hold on the new
probability space, thanks to the equality of joint laws. In other words, all
the _a priori_ estimates (3.2)–(3.6) also hold for the new random variables.

####  3.1.1 Passage to the limit

Given the above convergences, our aim is to pass to the limit in approximate
equations (1.4)–(1.5), and energy inequality (1.6). We first show that the
approximations $\tilde{\varrho}_{\varepsilon},\tilde{u}_{\varepsilon}$ solve
equations (1.4)–(1.5) on the new probability space
$(\tilde{\Omega},\tilde{\mathfrak{F}},\tilde{\mathbb{P}})$. To that context,
let $(\tilde{\mathfrak{F}}_{t}^{\varepsilon})$ and
$(\tilde{\mathfrak{F}}_{t})$, respectively, be the
$\tilde{\mathbb{P}}$-augmented canonical filtration of the processes
$(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon},\tilde{W}_{\varepsilon},\tilde{M}_{\varepsilon},\tilde{N}_{\varepsilon})$
and
$(\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\varrho}\rangle,\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\textbf{m}}\rangle,\tilde{W},\tilde{M},\tilde{N})$,
respectively, that is

$\begin{split}\tilde{\mathfrak{F}}_{t}^{\varepsilon}&=\sigma\big{(}\sigma\big{(}\mathbf{r}_{t}\tilde{\varrho}_{\varepsilon},\,\mathbf{r}_{t}\tilde{\bf
m}_{\varepsilon},\,\mathbf{r}_{t}\tilde{W}_{\varepsilon},\mathbf{r}_{t}\tilde{M}_{\varepsilon},\mathbf{r}_{t}\tilde{N}_{\varepsilon}\big{)}\cup\big{\\{}N\in\tilde{\mathfrak{F}};\;\tilde{\mathbb{P}}(N)=0\big{\\}}\big{)},\quad
t\in[0,T],\\\
\tilde{\mathfrak{F}}_{t}&=\sigma\big{(}\sigma\big{(}\mathbf{r}_{t}\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\varrho}\rangle,\,\mathbf{r}_{t}\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\textbf{m}}\rangle,\,\mathbf{r}_{t}\tilde{W},\mathbf{r}_{t}\tilde{M},\mathbf{r}_{t}\tilde{N}\big{)}\cup\big{\\{}N\in\tilde{\mathfrak{F}};\;\tilde{\mathbb{P}}(N)=0\big{\\}}\big{)},\quad
t\in[0,T],\end{split}$

where we denote by $\mathbf{r}_{t}$ the operator of restriction to the
interval $[0,t]$ acting on various path spaces.

######  Proposition 3.8.

For every $\varepsilon\in(0,1)$,
$\big{(}(\tilde{\Omega},\tilde{\mathfrak{F}},(\tilde{\mathfrak{F}}_{\varepsilon,t}),\tilde{\mathbb{P}}),\tilde{\varrho}_{\varepsilon},\tilde{\mathbf{u}}_{\varepsilon},\tilde{W}\big{)}$
is a finite energy weak martingale solution to (1.4)–(1.6) with the initial
law $\Lambda_{\varepsilon}$.

###### Proof.

Proof of the above proposition directly follows form the Theorem 2.9.1 of the
monograph by Breit et. al. [13]. ∎

We note that the above proposition implies that the new random variables
satisfy the following equations and the energy inequality on the new
probability space

  * •

for all $\phi\in C^{\infty}(\mathbb{T}^{3})$ and $\bm{\varphi}\in
C^{\infty}(\mathbb{T}^{3})$ we have

$\displaystyle\langle\tilde{\varrho}_{\varepsilon}(t),\phi\rangle$
$\displaystyle=\langle\tilde{\varrho}_{\varepsilon}(0),\phi\rangle-\int_{0}^{t}\langle\tilde{\bf
m}_{\varepsilon},\nabla_{x}\phi\rangle\mathrm{d}s$ (3.7)
$\displaystyle\langle\tilde{\bf m}_{\varepsilon}(t),\bm{\varphi}\rangle$
$\displaystyle=\langle\tilde{\bf
m}_{\varepsilon}(0),\bm{\varphi}\rangle-\int_{0}^{t}\bigg{\langle}\frac{\tilde{\bf
m}_{\varepsilon}\otimes\tilde{\bf
m}_{\varepsilon}}{\tilde{\varrho}_{\varepsilon}},\nabla_{x}\bm{\varphi}\bigg{\rangle}\,\mathrm{d}s+{\varepsilon}\,\int_{0}^{t}\langle\mathbb{S}(\nabla_{x}\tilde{\mathbf{u}}_{\varepsilon})\,\nabla_{x}\bm{\varphi}\rangle\mathrm{d}s$
$\displaystyle-\int_{0}^{t}\langle\tilde{\varrho}_{\varepsilon}^{\gamma},\mathrm{div}_{x}\,\bm{\varphi}\rangle\,\mathrm{d}s+\int_{0}^{t}\langle\mathbb{G}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon}),\bm{\varphi}\rangle\,\mathrm{d}\tilde{W}_{\varepsilon}$

$\mathbb{P}$-a.s. for all $t\in[0,T]$,

  * •

the energy inequality

$\displaystyle\begin{aligned}
&-\int_{0}^{T}\partial_{t}\psi\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|\tilde{\bf
m}_{\varepsilon}|^{2}}{\tilde{\varrho}_{\varepsilon}}+\frac{\tilde{\varrho}_{\varepsilon}^{\gamma}}{\gamma-1}\bigg{]}\,dx\,dt+{\varepsilon}\,\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\mathbb{S}(\nabla_{x}\tilde{\bf
u}_{\varepsilon}):\nabla_{x}\tilde{\bf u}_{\varepsilon}\,dx\,dt\\\
&\qquad\leq\psi(0)\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|\tilde{\bf
m}_{\varepsilon}(0)|^{2}}{\tilde{\varrho}_{\varepsilon}(0)}+\frac{\tilde{\varrho}_{\varepsilon}^{\gamma}(0)}{\gamma-1}\bigg{]}\,{\rm
d}{x}+\sum_{k=1}^{\infty}\int_{0}^{T}\psi\bigg{(}\int_{\mathbb{T}^{3}}\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon})\cdot\tilde{\bf u}_{\varepsilon}\,{\rm d}{x}\bigg{)}{\rm
d}\tilde{W}_{\varepsilon,k}\\\
&\qquad\qquad+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\tilde{\varrho}_{\varepsilon}^{-1}|\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon})|^{2}\,dx\,dt\end{aligned}$ (3.8)

holds for all $\psi\in C_{c}^{\infty}([0,T))$, $\psi\geq 0$, $\mathbb{P}$-a.s.

Next we would like to pass to the limit in $\varepsilon$ in (3.7) and (3.8).
To do this, we first recall that a-priori estimates (3.2)–(3.6) continue to
hold for the new random variables. Thus, making use of the item $(5)$ of
Proposition 3.7, we conclude that $\tilde{\mathbb{P}}$-a.s.,

$\displaystyle\tilde{\varrho}_{\varepsilon}\rightharpoonup\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\varrho}\rangle,\,\,\text{weakly
in}\,\,L^{\gamma}((0,T)\times\mathbb{T}^{3}),$
$\displaystyle\tilde{\textbf{m}}_{\varepsilon}\rightharpoonup\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\textbf{m}}\rangle,\,\,\text{weakly
in}\,\,L^{\frac{2\gamma}{\gamma+1}}((0,T)\times\mathbb{T}^{3}).$

In order to pass to the limit in the nonlinear terms present in the equations,
we first introduce the corresponding concentration defect measures

$\displaystyle\tilde{\mu}_{C}$
$\displaystyle=\tilde{C}-\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\frac{\tilde{\bf
m}\otimes\tilde{\bf m}}{\tilde{\varrho}}\right\rangle dxdt,\,\,$
$\displaystyle\tilde{\mu}_{P}=\tilde{P}-\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};p(\tilde{\varrho})\rangle
dxdt,$ $\displaystyle\tilde{\mu}_{E}$
$\displaystyle=\tilde{E}-\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\frac{1}{2}\frac{|\tilde{\bf
m}|^{2}}{\tilde{\varrho}}+P(\tilde{\varrho})\right\rangle dx,\,\,$
$\displaystyle\tilde{\mu}_{D}=\tilde{D}-\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\sum_{k\geq
1}\frac{|{\bf G}_{k}(\tilde{\varrho},\tilde{\bf
m})|^{2}}{\tilde{\varrho}}\right\rangle dxdt.$

With the help of these concentration defect measures, thanks to the discussion
in Subsection 2.1.1, we can conclude that $\mathbb{\tilde{P}}$-a.s.

$\displaystyle\tilde{C}_{\varepsilon}\rightharpoonup\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\frac{\tilde{\bf
m}\otimes\tilde{\bf m}}{\tilde{\varrho}}\right\rangle
dxdt+\tilde{\mu}_{C},\,\,\text{ weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{D}_{\varepsilon}\rightharpoonup\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\sum_{k\geq
1}\frac{|{\bf G}_{k}(\tilde{\varrho},\tilde{\bf
m})|^{2}}{\tilde{\varrho}}\right\rangle
dxdt+\tilde{\mu}_{D},\,\,\text{weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{E}_{\varepsilon}\rightharpoonup\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\frac{1}{2}\frac{|\tilde{\bf
m}|^{2}}{\tilde{\varrho}}+P(\tilde{\varrho})\right\rangle
dx+\tilde{\mu}_{E},\,\,\text{weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}^{+}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{P}_{\varepsilon}\rightharpoonup\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};p(\tilde{\varrho})\rangle
dxdt+\tilde{\mu}_{P},\,\,\text{weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})).$

Regarding the convergence of martingale terms $\tilde{M}_{\varepsilon}$,
appearing in the momentum equation, and $\tilde{N}_{\varepsilon}$, appearing
in the energy inequality, we have following propositions.

######  Proposition 3.9.

For each $t$, $\tilde{N}_{\varepsilon}(t)\rightarrow\tilde{N}(t)$ in
$\mathbb{R}$, $\mathbb{P}$-a.s., and $\tilde{N}(t)$ is a real valued square-
integrable continuous martingale with respect to the filtration
$(\tilde{\mathfrak{F}}_{t})$.

###### Proof.

Note that, thanks to Proposition 3.7, we have the information
$\tilde{N}_{\varepsilon}\rightarrow\tilde{N}$ $\mathbb{P}$-a.s. in
$C([0,T];\mathbb{R})$. To conclude that $\tilde{N}(t)$ is a martingale, it is
enough to show that

$\tilde{\mathbb{E}}[\tilde{N}(t)|\mathcal{\tilde{F}}_{s}]=\tilde{N}(s),$

for all $t,s\in[0,T]$ with $s\leq t$. To prove this, it is sufficient to show
that

$\tilde{\mathbb{E}}\Big{[}L_{s}(\tilde{\Phi})\big{(}\tilde{N}(t)-\tilde{N}(s)\big{)}\Big{]}=0,$

where we defined $\tilde{\Phi}=(\tilde{\varrho},\tilde{\bf
m},\tilde{W},\tilde{M},\tilde{N})$, and $L_{s}$ is any bounded continuous
functional, depending only on the values of $\tilde{\Phi}$ restricted to
$[0,s]$, on the path space
$\underline{\mathcal{X}}:=\mathcal{X}_{\varrho}\times\mathcal{X}_{{\bf
m}}\times\mathcal{X}_{W}\times\mathcal{X}_{M}\times\mathcal{X}_{N}$. Now using
the information that $\tilde{N}_{\varepsilon}(t)$ is a martingale, we know
that

$\tilde{\mathbb{E}}\Big{[}L_{s}(\tilde{\Phi}_{\varepsilon})\big{(}\tilde{N}_{\varepsilon}(t)-\tilde{N}_{\varepsilon}(s)\big{)}\Big{]}=0,$

for all bounded continuous functional $L_{s}$ on the same path space, and
$\tilde{\Phi}_{\varepsilon}=(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon},\tilde{W}_{\varepsilon},\tilde{M}_{\varepsilon},\tilde{N}_{\varepsilon})$.
Note that Proposition 3.7 reveals that
$\tilde{\Phi}_{\varepsilon}\rightarrow\tilde{\Phi}$, $\mathbb{P}$-a.s. in the
(weak) topology of $\underline{\mathcal{X}}$. This, in particular, implies
that $L_{s}(\tilde{\Phi}_{\varepsilon})\rightarrow L_{s}(\tilde{\Phi})$
$\mathbb{P}$-a.s. This information along with higher order moment estimate
e.g. $\tilde{N}_{\varepsilon}(t)\in L^{2}(\tilde{\Omega})$, with the help of
Vitali’s convergence theorem, we can pass to the limit in $\varepsilon$ to
conclude that $\tilde{N}(t)$ is a martingale. In this way, we loose the
structure of the martingale $\tilde{N}(t)$, which is expected due of lack of
sufficient informations. ∎

######  Proposition 3.10.

For each $t$ and $r=m+1$, $\tilde{M}_{\varepsilon}(t)\rightarrow\tilde{M}(t)$,
$\mathbb{P}$-a.s. in the (weak) topology of $W^{-r,2}(\mathbb{T}^{3})$.
Moreover, $\tilde{M}(t)$ is also a $W^{-r,2}(\mathbb{T}^{3})$-valued
martingale with respect to the filtration $(\tilde{\mathfrak{F}}_{t})$.

###### Proof.

As before, thanks to Proposition 3.7 and the compact embedding given in Lemma
2.1, we conclude that for each $t$,
$\tilde{M}_{\varepsilon}(t)\rightarrow\tilde{M}(t)$, $\mathbb{P}$-a.s. in the
topology of $W^{-r,2}(\mathbb{T}^{N})$. To show that $\tilde{M}(t)$ is a
martingale, it is sufficient to prove that for all $j\geq 1$

$\tilde{\mathbb{E}}\Big{[}L_{s}(\tilde{\Phi})\big{<}\tilde{M}(t)-\tilde{M}(s),g_{j}\big{>}\Big{]}=0,$

where $g_{j}$’s are orthonormal basis for the space
$W^{-r,2}(\mathbb{T}^{N})$, and the bracket $\big{<}\cdot,\cdot\big{>}$
denotes inner product in the same space. Again, we use the fact that

$\tilde{\mathbb{E}}\Big{[}L_{s}(\tilde{\Phi}_{\varepsilon})\big{<}\tilde{M}_{\varepsilon}(t)-\tilde{M}_{\varepsilon}(s),g_{j}\big{>}\Big{]}=0,$

for all $j\geq 1$. Then we can pass to the limit in $\varepsilon$, as before,
to conclude that $\tilde{M}(t)$ is a martingale. Indeed, we can show uniform
integrabilty using BDG inequality:

$\displaystyle\tilde{\mathbb{E}}\Big{[}\big{|}\big{<}\tilde{M}_{\varepsilon}(t),g_{j}\big{>}\big{|}^{p}\Big{]}$
$\displaystyle=\tilde{\mathbb{E}}\bigg{[}\bigg{|}\Big{<}\int_{0}^{t}{\mathbb{G}}(\tilde{\varrho}_{\varepsilon},\tilde{\textbf{m}}_{\varepsilon})\,{\rm
d}\tilde{W},g_{j}\Big{>}\bigg{|}^{p}\bigg{]}\leq
C\tilde{\mathbb{E}}\Bigg{[}\sup_{0\leq t\leq
T}\bigg{\|}\int_{0}^{t}{\mathbb{G}}(\tilde{\varrho}_{\varepsilon},\tilde{\textbf{m}}_{\varepsilon})\,{\rm
d}\tilde{W}\bigg{\|}^{p}_{W^{-r,2}(\mathbb{T}^{3})}\Bigg{]}$
$\displaystyle\leq
C\tilde{\mathbb{E}}\Bigg{[}\bigg{(}\int_{0}^{T}\|{\mathbb{G}}(\tilde{\varrho}_{\varepsilon},\tilde{\textbf{m}}_{\varepsilon})\|^{2}_{L_{2}(\mathfrak{U},W^{-r,2}(\mathbb{T}^{3}))}\,ds\bigg{)}^{p/2}\Bigg{]}\leq
C.$

∎

Note that collating all the above informations, we can pass to the limit in
both equations of (3.7) to infer that

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{\tilde{V}}^{\omega}_{t,x};\varrho\rangle\,\varphi(\tau,\cdot)\,dx-\int_{\mathbb{T}^{3}}\langle\mathcal{\tilde{V}}^{\omega}_{0,x};\varrho\rangle\,\varphi(0,\cdot)\,dx=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\langle\mathcal{\tilde{V}}^{\omega}_{t,x};\textbf{m}\rangle\cdot\nabla_{x}\varphi\,dx\,dt,$

holds $\tilde{\mathbb{P}}$-a.s., for all $\tau\in[0,T)$, and for all
$\varphi\in C^{\infty}(\mathbb{T}^{3})$. Moreover,

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\textbf{m}\rangle\cdot\bm{\varphi}(\tau,\cdot)dx-\int_{\mathbb{T}^{3}}\langle\mathcal{\tilde{V}}^{\omega}_{0,x};\textbf{m}\rangle\cdot\bm{\varphi}(0,\cdot)dx$
$\displaystyle\qquad=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\left[\left\langle\mathcal{\tilde{V}}^{\omega}_{t,x};\frac{\textbf{m}\otimes\textbf{m}}{\varrho}\right\rangle:\nabla_{x}\bm{\varphi}+\langle\mathcal{\tilde{V}}^{\omega}_{t,x};p(\varrho)\rangle\operatorname{div}_{x}\bm{\varphi}\right]dxdt$
$\displaystyle\qquad\qquad+\int_{\mathbb{T}^{3}}\bm{\varphi}\,\int_{0}^{\tau}d\tilde{M}(t)\,dx+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\nabla_{x}\bm{\varphi}:d\Big{(}\tilde{\mu}_{C}+\tilde{\mu}_{P}\mathbb{I}\Big{)},$

holds $\tilde{\mathbb{P}}$-a.s., for all $\tau\in[0,T)$, and for all
$\bm{\varphi}\in C^{\infty}(\mathbb{T}^{3};\mathbb{R}^{3})$, where
$\Big{(}\tilde{\mu}_{C}+\tilde{\mu}_{P}\mathbb{I}\Big{)}\in
L^{\infty}_{w^{*}}\big{(}[0,T];\mathcal{M}_{b}({\mathbb{T}^{3}})\big{)}$,
$\mathbb{P}$-a.s., is a tensor–valued measure, Therefore, we conclude that
(LABEL:first_condition_measure-valued_solution), and (2.8) hold. To conclude
(2.9), we proceed as follows. First, note that we can pass to the limit in
$\varepsilon$ in (3.8) to obtain the following energy inequality in the new
probability space:

$\displaystyle-\int_{0}^{T}\partial_{t}\psi\Bigg{[}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{t,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{\tilde{D}}(t)\Bigg{]}\,dt\leq\psi(0)\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{0,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx$
(3.9)
$\displaystyle\quad+\frac{1}{2}\int_{0}^{T}\psi\bigg{(}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\varrho^{-1}|\mathbf{G}_{k}(\varrho,{\textbf{m}})|^{2}\right\rangle\bigg{)}\,dt\,dx+\frac{1}{2}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}d\tilde{\mu}_{D}+\int_{0}^{T}\psi\,d{\tilde{N}}.$

Here $0\leq\mathcal{\tilde{D}}(t):=\tilde{\mu}_{E}(t)(\mathbb{T}^{3})$. Next,
we choose a specific test function $\psi_{\tau}:[0,T]\rightarrow\mathbb{R}$ as
follows: Fix any $t_{0}$ and $t$ such that $0<t_{0}<t<T$. For any $\tau>0$
with $0<t_{0}-\tau<t+\tau<T$, let $\psi_{\tau}$ be a Lipschitz function that
is linear on $[t_{0}-\tau,t_{0}]\cup[t,t+\tau]$ and satisfies

$\displaystyle\psi_{\tau}(t)=\begin{cases}0,\,&\text{if
}t\in[0,t_{0}-\tau]\,\,\text{or}\,\,t\in[t+\tau,T]\\\ 1,\,&\text{if
}t\in[t_{0},t].\end{cases}$

Then, via a standard regularization argument, we see that $\psi_{\tau}$ is an
admissible test function in (3.9). Inserting this $\psi_{\tau}$ into (3.9)
gives

$\displaystyle\frac{1}{\tau}\int_{t}^{t+\tau}\Bigg{[}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{t,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{\tilde{D}}(t)\Bigg{]}\,dt$
$\displaystyle\qquad\leq\frac{1}{\tau}\int_{t_{0}-\tau}^{t_{0}}\Bigg{[}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{t,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{\tilde{D}}(t)\Bigg{]}\,dt+\frac{1}{2}\int_{t_{0}-\tau}^{t+\tau}\int_{\mathbb{T}^{3}}d\tilde{\mu}_{D}+\int_{t_{0}-\tau}^{t+\tau}d{\tilde{N}}(s)$
$\displaystyle\qquad\quad+\frac{1}{2}\int_{t_{0}-\tau}^{t+\tau}\bigg{(}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\varrho^{-1}|\mathbf{G}_{k}(\varrho,{\textbf{m}})|^{2}\right\rangle\bigg{)}\,dt\,dx.$

Now taking limit as $\tau\rightarrow 0^{+}$, we conclude that (2.9) holds.
Note that for $t_{0}=0$, we need to slightly modify the test function. In this
case we take $\psi_{\tau}$ which takes the value $1$ in $[0,t]$, linear on
$[t,t+\tau]$, and zero otherwise and apply the same argument as before.

Therefore, we are only left with the verifications of (2.10), and item (k) of
Definition 2.11. To proceed, we start with the following lemma.

######  Lemma 3.11.

Given a stochastic process $f$, as in item $(k)$ of Definition 2.11

$\mathrm{d}f=D^{d}_{t}f\,\mathrm{d}t+\mathbb{D}^{s}_{t}f\,\mathrm{d}\tilde{W},$

the cross variation with $\tilde{M}$ is given by

$\displaystyle\Big{<}\Big{<}f(t),\tilde{M}(t)\Big{>}\Big{>}=\sum_{i,j}\Bigg{(}\sum_{k=1}^{\infty}\int_{0}^{t}\left\langle\left\langle\mathcal{\tilde{V}}^{\omega}_{s,x};\mathbf{G}_{k}(\tilde{\varrho},\tilde{\bf
m})\right\rangle,g_{i}\right\rangle\,\left\langle\mathbb{D}_{t}^{s}f(e_{k}),g_{j}\right\rangle\,ds\Bigg{)}\,g_{i}\otimes
g_{j}.$

###### Proof.

First note that, according to Da Prato $\&$ Zabczyk [21], and the definition
of $\tilde{M}_{\varepsilon}$, we have

$\displaystyle\Big{<}\Big{<}f(t),\tilde{M}_{\varepsilon}(t)\Big{>}\Big{>}=\sum_{i,j}\Big{<}\Big{<}\left\langle\tilde{M}_{\varepsilon}(t),g_{i}\right\rangle,\,\left\langle
f(t),g_{j}\right\rangle\Big{>}\Big{>}\,g_{i}\otimes g_{j}.$

Moreover,

$\displaystyle\Big{<}\Big{<}\left\langle\tilde{M}_{\varepsilon}(t),g_{i}\right\rangle,\,\left\langle
f(t),g_{j}\right\rangle\Big{>}\Big{>}$
$\displaystyle=\Big{<}\Big{<}\sum_{k=1}^{\infty}\int_{0}^{t}\left\langle\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon}),g_{i}\right\rangle\mathrm{d}\tilde{W}_{k}\,,\,\sum_{k=1}^{\infty}\int_{0}^{t}\left\langle\mathbb{D}^{s}_{t}f(e_{k}),g_{j}\right\rangle\mathrm{d}\tilde{W}_{k}\Big{>}\Big{>}$
(3.10)
$\displaystyle=\sum_{k=1}^{\infty}\int_{0}^{t}\left\langle\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon}),g_{i}\right\rangle\,\left\langle\mathbb{D}^{s}_{t}f(e_{k}),g_{j}\right\rangle\,ds.$

Therefore

$\displaystyle\Big{<}\Big{<}f(t),\tilde{M}_{\varepsilon}(t)\Big{>}\Big{>}=\sum_{i,j}\Bigg{(}\sum_{k=1}^{\infty}\int_{0}^{t}\left\langle\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon}),g_{i}\right\rangle\,\left\langle\mathbb{D}^{s}_{t}f(e_{k}),g_{j}\right\rangle\,ds\Bigg{)}\,g_{i}\otimes
g_{j}.$

Observe that (3.10) equivalently implies that

$\displaystyle\tilde{\mathbb{E}}\Big{[}L_{s}(\tilde{\Phi}_{\varepsilon})\Big{(}\big{<}\tilde{M}_{\varepsilon}(t),g_{i}\big{>}\big{<}f(t),g_{j}\big{>}-\big{<}\tilde{M}_{\varepsilon}(s),g_{i}\big{>}\big{<}f(s),g_{j}\big{>}$
$\displaystyle\hskip
170.71652pt-\sum_{k=1}^{\infty}\int_{s}^{t}\big{<}\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon}),g_{i}\big{>}\big{<}\mathbb{D}^{s}_{t}f(e_{k}),g_{j}\big{>}\,ds\Big{)}\Big{]}=0.$

We may apply item (4) of Proposition 3.7, to the composition
$\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf m}_{\varepsilon})$,
$k\in\mathbb{N}$. This gives

$\mathbf{G}_{k}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon})\rightharpoonup\Big{\langle}\mathcal{\tilde{V}}^{\omega}_{t,x};\mathbf{G}_{k}(\tilde{\varrho},\tilde{\bf
m})\Big{\rangle}\,\,\mbox{weakly in}\,\,L^{q}((0,T)\times\mathbb{T}^{3}),$

$\mathbb{P}$-a.s., for some $q>1$. Moreover, for $m>3/2$, we have by Sobolev
embedding

$\displaystyle\tilde{\mathbb{E}}\Big{[}\int_{0}^{T}\|\mathbb{G}(\tilde{\varrho}_{\varepsilon},\tilde{\bf
m}_{\varepsilon})\|^{2}_{L_{2}(\mathfrak{U};W^{-m,2})}\,dt\Big{]}\leq\tilde{\mathbb{E}}\Big{[}\int_{0}^{T}(\tilde{\varrho}_{\varepsilon})_{\mathbb{T}^{3}}\int_{\mathbb{T}^{3}}(\tilde{\varrho}_{\varepsilon}+\tilde{\varrho}_{\varepsilon}|\tilde{\bf
u}_{\varepsilon}|^{2})\,dx\,dt\Big{]}\leq c(r).$

This implies that $\mathbb{P}$-a.s.

$\mathbf{G}_{k}(\tilde{\varrho},\tilde{\bf
m})\rightharpoonup\Big{\langle}\mathcal{\tilde{V}}^{\omega}_{t,x};\mathbf{G}_{k}(\tilde{\varrho},\tilde{\bf
m})\Big{\rangle}\,\,\mbox{weakly
in}\,\,L^{2}((0,T);W^{-m,2}(\mathbb{T}^{3})).$

This will imply that, thanks to uniform integrabilty, we can pass to the limit
in $\varepsilon\rightarrow 0$ to conclude

$\displaystyle\tilde{\mathbb{E}}\Big{[}L_{s}(\tilde{\Phi})$
$\displaystyle\Big{(}\big{<}\tilde{M}(t),g_{i}\big{>}\big{<}f(t),g_{j}\big{>}-\big{<}\tilde{M}(s),g_{i}\big{>}\big{<}f(s),g_{j}\big{>}$
$\displaystyle\hskip
113.81102pt-\sum_{k=1}^{\infty}\int_{s}^{t}\big{<}\left\langle\mathcal{\tilde{V}}^{\omega}_{s,x};\mathbf{G}_{k}(\tilde{\varrho},\tilde{\bf
m})\right\rangle,g_{i}\big{>}\big{<}\mathbb{D}^{s}_{t}f(e_{k}),g_{j}\big{>}\,ds\Big{)}\Big{]}=0.$

This finishes the proof the lemma.

∎

######  Lemma 3.12.

The concentration defect
$0\leq\mathcal{\tilde{D}}(\tau):=\tilde{\mu}_{E}(\tau)(\mathbb{T}^{3})$
dominates defect measures $\tilde{\mu}_{C},\tilde{\mu}_{D}$, and
$\tilde{\mu}_{P}$ in the sense of Lemma 2.3. More precisely, there exists a
constant $C>0$ such that

$\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\tilde{\mu}_{C}|+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\tilde{\mu}_{D}|+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\tilde{\mu}_{P}|\leq
C\int_{0}^{\tau}\mathcal{\tilde{D}}(\tau)\,dt,$

for every $\tau\in(0,T)$, $\mathbb{P}$-a.s.

###### Proof.

Following deterministic argument, we can conclude that $\tilde{\mu}_{E}$
dominates defect measures $\tilde{\mu}_{C},\tilde{\mu}_{P}$. To show the
dominance of $\tilde{\mu}_{E}$ over $\tilde{\mu}_{D}$, observe that by virtue
of hypotheses (2.1), the function

$[\varrho,{\bf m}]\mapsto\sum_{k\geq 1}\frac{|{\bf G}_{k}(\varrho,{\bf
m})|^{2}}{\varrho}\ \mbox{is continuous},$

and as such dominated by the total energy

$\sum_{k\geq 1}\frac{|{\bf G}_{k}(\varrho,{\bf m})|^{2}}{\varrho}\leq
c\left(\varrho+\frac{|{\bf m}|^{2}}{\varrho}\right)\leq
c\left(\frac{1}{2}\frac{|{\bf m}|^{2}}{\varrho}+P(\varrho)\right)+1.$

Hence, a simple application of the Lemma 2.3 finishes the proof of the lemma.
∎

##  4 Weak-Strong Uniqueness Principle

In this section we prove Theorem 2.16 through couple of auxiliary results. We
start with the following technical lemma, a variant of [8, Lemma 3.1], to
which we refer for the proof. Note that this lemma plays a pivotal role in the
proof of the Proposition 4.2.

######  Lemma 4.1.

Let $q$ be a stochastic process on
$\left(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P}\right)$
such that for some $b\in\mathbb{R}$,

$q\in C_{\rm weak}([0,T];W^{-b,2}(\mathbb{T}^{3}))\cap
L^{\infty}(0,T;L^{1}(\mathbb{T}^{3}))\quad\text{$\mathbb{P}$-a.s.},$ ${\rm
d}q=D^{d}_{t}q\,\,{\rm d}t+{\rm d}M.$

Here $M$ is a continuous square integrable $W^{-b,2}(\mathbb{T}^{3})$ valued
martingale, and $D^{d}_{t}q$ is progressively measurable with

$\displaystyle\begin{aligned}
D^{d}_{t}q\in&L^{p}(\Omega;L^{1}(0,T;W^{-b,a}(\mathbb{T}^{3})),\end{aligned}$
(4.1)

for some $a>1$ and some $m\in\mathbb{N}$.

Let $w$ be a stochastic process on
$\left(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P}\right)$
satisfying

$w\in C([0,T];W^{b,a^{\prime}}\cap
C(\mathbb{T}^{3}))\quad\text{$\mathbb{P}$-a.s.},$
$\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|w\|_{W^{b,a^{\prime}}\cap
C(\mathbb{T}^{3})}^{p}\bigg{]}<\infty,\ 1\leq p<\infty,$ ${\rm
d}w=D^{d}_{t}w+\mathbb{D}^{s}_{t}w\,{\rm d}W.$

Here $D^{d}_{t}w,\mathbb{D}^{s}_{t}w$ are progressively measurable with

$\displaystyle\begin{aligned} D^{d}_{t}w\in
L^{p}&(\Omega;L^{1}(0,T;W^{b,a^{\prime}}\cap
C(\mathbb{T}^{3})),\quad\mathbb{D}^{s}_{t}w\in
L^{2}(\Omega;L^{2}(0,T;L_{2}(\mathfrak{U};W^{-m,2}(\mathbb{T}^{3})))),\\\
&\sum_{k\geq
1}\int_{0}^{T}\|\mathbb{D}^{s}_{t}w(e_{k})\|^{2}_{W^{b,a^{\prime}}\cap
C(\mathbb{T}^{3})}\,{\rm d}t\in L^{p}(\Omega),\quad 1\leq
p<\infty.\end{aligned}$

Let $Q$ be $[b+2]$-continuously differentiable function satisfying

$\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|Q^{(j)}(w)\|_{W^{b,a^{\prime}}\cap
C(\mathbb{T}^{3})}^{p}\bigg{]}<\infty,\quad j=0,1,2,\quad 1\leq p<\infty.$

Then

$\begin{split}{\rm
d}\left(\int_{\mathbb{T}^{3}}qQ(w)\,dx\right)&=\bigg{(}\int_{\mathbb{T}^{3}}\Big{[}q\Big{(}Q^{\prime}(w)D^{d}_{t}w+\frac{1}{2}\sum_{k\geq
1}Q^{\prime\prime}(w)\left|\mathbb{D}^{s}_{t}w(e_{k})\right|^{2}\Big{)}\Big{]}\,dx+\left<Q(w),D^{d}_{t}q\right>\bigg{)}{\rm
d}t\\\
&+\int_{\mathbb{T}^{3}}\Big{<}\Big{<}f(t),\tilde{M}(t)\Big{>}\Big{>}\,{\rm
d}t+{\rm d}\mathbb{M},\end{split}$

where

$\mathbb{M}=\int_{0}^{t}\int_{\mathbb{T}^{3}}\Big{[}q\,Q^{\prime}(w)\,{\rm
d}M+Q(w)\,{\rm d}M\Big{]}\,dx\,{\rm d}s.$

###  4.1 Relative energy inequality

We proceed further and introduce the relative energy (entropy) functional. The
commonly used form of the relative energy functional in the context of
measure-valued solutions to the compressible Euler system reads

$\displaystyle\mathcal{E}^{1}_{\mathrm{mv}}\left(\varrho,\textbf{m}\ \Big{|}\
r,{\bf
U}\right)(t):=\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega}_{t,x}};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\Bigg{\rangle}\,dx+\mathcal{D}(t)-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\,\cdot{\bf
U}\,dx$ (4.2) $\displaystyle\hskip
56.9055pt+\frac{1}{2}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}\,|{\bf
U}|^{2}\,dx-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}\,P^{\prime}(r)\,dx-\int_{\mathbb{T}^{3}}\big{[}P^{\prime}(r)r-P(r)\big{]}\,dx.$

We remark that the above relative energy functional is defined for all
$t\in[0,T]\setminus N$, where the set $N$, may depends on $\omega$, has
Lebesgue measure zero. To define relative energy functional for all $t\in N$,
we consider

$\displaystyle\mathcal{E}^{2}_{\mathrm{mv}}\left(\varrho,\textbf{m}\ \Big{|}\
r,{\bf U}\right)(t):=\lim_{\tau\rightarrow
0+}\frac{1}{\tau}\int_{t}^{t+\tau}\Bigg{[}\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega}_{s,x}};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\Bigg{\rangle}\,dx+\mathcal{D}(s)\Bigg{]}\,ds$
(4.3)
$\displaystyle-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\,\cdot{\bf
U}\,dx+\frac{1}{2}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}\,|{\bf
U}|^{2}\,dx-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}\,P^{\prime}(r)\,dx-\int_{\mathbb{T}^{3}}\big{[}P^{\prime}(r)r-P(r)\big{]}\,dx.$

Using above, we define relative energy functional as follows

$\displaystyle\mathcal{E}_{\mathrm{mv}}\left(\varrho,\textbf{m}\ \Big{|}\
r,{\bf
U}\right)(t):=\begin{cases}\mathcal{E}^{1}_{\mathrm{mv}}\left(\varrho,\textbf{m}\
\Big{|}\ r,{\bf U}\right)(t),\,&\text{if}\,\,t\in[0,T]\setminus N\\\
\mathcal{E}^{2}_{\mathrm{mv}}\left(\varrho,\textbf{m}\ \Big{|}\ r,{\bf
U}\right)(t),\,&\text{if}\,\,t\in N.\end{cases}$ (4.4)

Note that the above relative energy functional is defined for all $t\in[0,T]$.
In what follows, with the help of the above defined relative energy, we derive
the relative energy inequality (LABEL:relativeEntropy). Note that, the
relative energy inequality is a tool which enables us to compare measure
valued solutions with some smooth comparison functions.

######  Proposition 4.2 (Relative Energy).

Let $\big{[}\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq
0},\mathbb{P}\big{)};{\mathcal{V}^{\omega}_{t,x}},W\big{]}$ be a dissipative
measure-valued martingale solution to the system (1.1)–(1.2). Suppose
$\big{(}r\,,\,\mathbf{U}\big{)}$ be a pair of stochastic processes which are
adapted to the filtration $(\mathfrak{F}_{t})_{t\geq 0}$ and which satisfies

$\displaystyle\mathrm{d}r$
$\displaystyle=D^{d}_{t}r\,\mathrm{d}t+\mathbb{D}^{s}_{t}r\,\mathrm{d}W,$
$\displaystyle\mathrm{d}\mathbf{U}$
$\displaystyle=D^{d}_{t}\mathbf{U}\,\mathrm{d}t+\mathbb{D}^{s}_{t}\mathbf{U}\,\mathrm{d}W,$

with

$\displaystyle\begin{aligned} r\in C([0,T];W^{1,q}(\mathbb{T}^{3})),\ {\bf
U}\in C([0,T];W^{1,q}(\mathbb{T}^{3})),\ \quad\text{$\mathbb{P}$-a.s.},\\\
\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|r\|_{W^{1,q}(\mathbb{T}^{3})}^{2}\bigg{]}^{q}+\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|{\bf
U}\|_{W^{1,q}(\mathbb{T}^{3})}^{2}\bigg{]}^{q}\leq c(q)\quad\forall\,\,2\leq
q<\infty,\end{aligned}$ $0<\underline{r}\leq
r(t,x)\leq\overline{r}\quad\text{$\mathbb{P}$-a.s.}$ (4.5)

Moreover, $r$, ${\bf U}$ satisfy

$\displaystyle D^{d}r,D^{d}{\bf U}\in
L^{q}(\mathbb{T}^{3};L^{q}(0,T;W^{1,q}(\mathbb{T}^{3}))),\quad\mathbb{D}^{s}r,\mathbb{D}^{s}{\bf
U}\in
L^{2}(\mathbb{T}^{3};L^{2}(0,T;L_{2}(\mathfrak{U};L^{2}(\mathbb{T}^{3})))),$
$\displaystyle\bigg{(}\sum_{k\geq
1}|\mathbb{D}^{s}r(e_{k})|^{q}\bigg{)}^{\frac{1}{q}},\bigg{(}\sum_{k\geq
1}|\mathbb{D}^{s}{\bf U}(e_{k})|^{q}\bigg{)}^{\frac{1}{q}}\in
L^{q}(\mathbb{T}^{3};L^{q}(0,T;L^{q}(\mathbb{T}^{3}))).$ (4.6)

Then the following _relative energy inequality_ holds $\mathbb{P}$-a.s., for
all $t\in(0,T)$:

$\displaystyle\mathcal{E}_{\mathrm{mv}}\left(\varrho,\textbf{m}\ \Big{|}\
r,{\bf U}\right)(t)\leq\mathcal{E}_{\mathrm{mv}}\left(\varrho,\textbf{m}\
\Big{|}\ r,{\bf
U}\right)(0)+M_{RE}(t)+\int_{0}^{t}\mathcal{R}_{\mathrm{mv}}\big{(}\varrho,\textbf{m}\left|\right.r,\mathbf{U}\big{)}(s)\,\mathrm{d}s$
(4.7)

where

$\displaystyle\mathcal{R}_{\mathrm{mv}}\big{(}\varrho,{\bf
m}\left|\right.r,\mathbf{U}\big{)}=\int_{\mathbb{T}^{3}}\langle{\mathcal{V}^{\omega}_{t,x}};\varrho\mathbf{U}-{\bf
m}\rangle\cdot[D^{d}_{t}\mathbf{U}+\nabla_{x}\mathbf{U}\cdot\mathbf{U}]\,dx$
(4.8)
$\displaystyle\quad+\int_{\mathbb{T}^{3}}\left\langle{\mathcal{V}^{\omega}_{t,x}};\frac{({\bf
m}-\varrho\mathbf{U})\otimes(\varrho\mathbf{U}-{\bf
m})}{\varrho}\right\rangle:\nabla_{x}\textbf{U}\,\mathrm{d}x-\int_{\mathbb{T}^{3}}\nabla_{x}\mathbf{U}:d\mu_{m}+\frac{1}{2}\int_{\mathbb{T}^{3}}d\mu_{e}$
$\displaystyle\quad+\int_{\mathbb{T}^{3}}\big{[}(r-\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle})P^{\prime\prime}(r)D^{d}_{t}r+\nabla_{x}P^{\prime}(r)\cdot(r\mathbf{U}-\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
m}\big{\rangle})\big{]}\,\mathrm{d}x$
$\displaystyle\quad+\int_{\mathbb{T}^{3}}\big{[}p(r)-\big{\langle}{\mathcal{V}^{\omega}_{t,x}};p(\varrho)\big{\rangle}\big{]}\mathrm{div}_{x}(\mathbf{U})\,\mathrm{d}x+\frac{1}{2}\sum_{k\in\mathbb{N}}\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\bigg{|}\frac{\mathbf{G}_{k}(\varrho,{\bf
m})}{\varrho}-\mathbb{D}^{s}_{t}\mathbf{U}(e_{k})\bigg{|}^{2}\Bigg{\rangle}\,\mathrm{d}x$
$\displaystyle\quad+\frac{1}{2}\sum_{k\geq
1}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}P^{\prime\prime\prime}(r)|\mathbb{D}^{s}_{t}r(e_{k})|^{2}\,dx\,{\rm
d}t+\frac{1}{2}\sum_{k\geq
1}\int_{\mathbb{T}^{3}}p^{\prime\prime}(r)|\mathbb{D}^{s}_{t}r(e_{k})|^{2}\,dx\,{\rm
d}t.$

Here $M_{RE}$ is a real valued square integrable martingale, and the norm of
this martingale depends only on the norms of $r$ and $\mathbf{U}$ in the
aforementioned spaces. Moreover, the pressure potential $P$ is defined as the
solution of the equation $rP^{\prime}(r)-P(r)=p(r)$.

###### Proof.

Note that all the integrals on the right hand side of (4.2) can be explicitly
expressed by means of either the energy inequality (2.9) or the field
equations (LABEL:first_condition_measure-valued_solution) and (2.8).
Therefore, to compute the right hand side of (4.2), we make use of Lemma 4.1
and the energy inequality (2.9).

Step 1:

To compute ${\rm
d}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\,\cdot{\bf
U}\,dx$ we recall that
$q=\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}$
satisfies hypotheses (4.1) with some $a<\infty$. Applying Lemma 4.1 we obtain

$\begin{split}&{\rm
d}\left(\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\cdot{\bf
U}\,dx\right)=\int_{\mathbb{T}^{3}}\left[\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\cdot
D^{d}_{t}{\bf
U}+\left\langle{\mathcal{V}^{\omega}_{t,x}};\frac{\textbf{m}\otimes\textbf{m}}{\varrho}\right\rangle:\nabla_{x}\textbf{U}+\big{\langle}{\mathcal{V}^{\omega}_{t,x}};p(\varrho)\big{\rangle}\,{\rm
div}_{x}{\bf U}\right]\,dx{\rm d}t\\\ &\qquad+\sum_{k\geq
1}\int_{\mathbb{T}^{3}}\mathbb{D}^{s}_{t}{\bf
U}(e_{k})\cdot\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}(\varrho,\textbf{m})\big{\rangle}\,dx\,{\rm
d}t+\int_{\mathbb{T}^{3}}\nabla_{x}{\bf U}:d\mu_{m}\,dt+{\rm
d}M_{1},\end{split}$ (4.9)

where

$M_{1}(t)=\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf
U}\,dx\,dM^{1}_{E}+\int_{0}^{t}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\cdot\mathbb{D}^{s}_{t}{\bf
U}\,dx\,{\rm d}W$

is a square integrable martingale. Note that to identify the cross variation
in (4.9), we have used item (k) of the Definition 2.11, and noticing that

$\displaystyle\Big{<}\Big{<}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle},{\bf
U}(t)\Big{>}\Big{>}$ $\displaystyle=\sum_{i,j}\Bigg{(}\sum_{k\geq
1}\int_{0}^{t}\left\langle\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}(\varrho,\textbf{m})\big{\rangle},g_{i}\right\rangle\,\left\langle\mathbb{D}_{t}^{s}{\bf
U}(e_{k}),g_{j}\right\rangle\,ds\Bigg{)}\,g_{i}\otimes g_{j}$
$\displaystyle=\sum_{k\geq
1}\int_{0}^{t}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}(\varrho,\textbf{m})\big{\rangle}\cdot\mathbb{D}^{s}_{t}{\bf
U}(e_{k})\,{\rm d}s.$

Step 2:

Similarly, we compute

$\begin{split}{\rm
d}\left(\int_{\mathbb{T}^{3}}\frac{1}{2}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}|{\bf
U}|^{2}\,dx\right)&=\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\cdot\nabla_{x}{\bf
U}\cdot{\bf U}\,dx{\rm
d}t+\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}{\bf
U}\cdot D^{d}_{t}{\bf U}\,dx{\rm d}t\\\ &\qquad+\frac{1}{2}\sum_{k\geq
1}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}|\mathbb{D}^{s}_{t}{\bf
U}(e_{k})|^{2}\,dx\ {\rm d}t+{\rm d}M_{2},\end{split}$ (4.10)

where

$M_{2}(t)=\int_{0}^{t}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}{\bf
U}\cdot\mathbb{D}^{s}_{t}{\bf U}\,dx\,{\rm d}W,$ ${\rm
d}\left(\int_{\mathbb{T}^{3}}\left[P^{\prime}(r)r-P(r)\right]\,dx\right)=\int_{\mathbb{T}^{3}}p^{\prime}(r)D^{d}_{t}r\,dx\
{\rm d}t+\frac{1}{2}\sum_{k\geq
1}\int_{\mathbb{T}^{3}}p^{\prime\prime}(r)|\mathbb{D}^{s}_{t}r(e_{k})|^{2}\,dx\,{\rm
d}t+{\rm d}M_{3},$ (4.11)

where

$M_{3}(t)=\int_{0}^{t}\int_{\mathbb{T}^{3}}p^{\prime}(r)\mathbb{D}^{s}_{t}r\,dx\,{\rm
d}W,$

and, finally,

$\begin{split}{\rm
d}\left(\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}P^{\prime}(r)\,dx\right)&=\int_{\mathbb{T}^{3}}\nabla_{x}P^{\prime}(r)\cdot\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\textbf{m}\big{\rangle}\,dx\
{\rm
d}t+\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}P^{\prime\prime}(r)D^{d}_{t}r\,dx\
{\rm d}t\\\ &\qquad+\frac{1}{2}\sum_{k\geq
1}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}P^{\prime\prime\prime}(r)|\mathbb{D}^{s}_{t}r(e_{k})|^{2}\,dx\,{\rm
d}t+{\rm d}M_{4},\end{split}$ (4.12)

where

$M_{4}(t)=\int_{0}^{t}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}P^{\prime\prime}(r)\mathbb{D}^{s}_{t}r\,dx\,{\rm
d}W.$

Step 3:

Now we can collect (4.9–4.12), define the square integrable real valued
martingale $M_{RE}(t):=M_{1}(t)+M_{2}(t)+M_{3}(t)+M_{4}(t)+M^{2}_{E}(t)$, and
summing up the resulting expressions and adding the sum with (2.9) to obtain
(LABEL:relativeEntropy). ∎

###  4.2 Proof of Theorem 2.16

As we have seen before, the _relative energy inequality_
(LABEL:relativeEntropy) is a consequence of the _energy inequality_ (2.9). Now
we wish to use the Proposition 4.2 to prove the following weak (measure-
valued)–strong uniqueness principle given by Theorem 2.16.

In what follows, our aim is to apply the relative energy inequality (2.9) to
the pair
$(r,\mathbf{U})=(\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R}),\bar{\mathbf{u}}(\cdot\wedge\mathfrak{t}_{R}))$,
where
$(\bar{\varrho},\bar{\mathbf{u}},(\mathfrak{t}_{R})_{R\in\mathbb{N}},\mathfrak{t})$
is the unique maximal strong pathwise solution to (1.1)–(1.2) which exists by
Theorem 2.7. Recall that the stopping time $\mathfrak{t}_{R}$ announces the
blow-up and satisfies

$\sup_{t\in[0,\mathfrak{t}_{R}]}\|\bar{{\bf u}}(t)\|_{1,\infty}\geq
R\quad\text{on}\quad[\mathfrak{t}<T];$

Moreover, $(r,\mathbf{U})=(\bar{\varrho},\bar{\mathbf{u}})$ satisfies an
equation of the form (4.2), where

$\displaystyle
D^{d}_{t}r=-\mathrm{div}_{x}(\bar{\varrho}\bar{\mathbf{u}}),\quad\mathbb{D}^{s}_{t}r=0,\quad
D^{d}_{t}\mathbf{U}=-\bar{\mathbf{u}}\cdot\nabla_{x}\bar{\mathbf{u}}-\frac{1}{\bar{\varrho}}\nabla_{x}p(\bar{\varrho}),\quad\mathbb{D}^{s}_{t}\mathbf{U}=\frac{1}{\bar{\varrho}}\mathbb{G}(\bar{\varrho},\bar{{\bf
m}}).$

It is easy to see that (4.5) and (4.6) are satisfied for
$t\leq\mathfrak{t}_{R}$, thanks to Theorem 2.7 and (2.1). Moreover, the lower
bound for $\bar{\varrho}$ is a consequence of standard maximum principle. So,
(LABEL:relativeEntropy) holds and we can now deduce that for every $t\in[0,T]$
and $R\in\mathbb{N}$,

$\displaystyle\mathcal{E}_{\mathrm{mv}}\left(\varrho,\textbf{m}\ \Big{|}\
\bar{\varrho},\bar{\mathbf{u}}\right)(t\wedge\mathfrak{t}_{R})$ (4.13)
$\displaystyle\qquad\qquad\leq\mathcal{E}_{\mathrm{mv}}\left(\varrho,\textbf{m}\
\Big{|}\
\bar{\varrho},\bar{\mathbf{u}}\right)(0)+M_{RE}(t\wedge\mathfrak{t}_{R})+\int_{0}^{t\wedge\mathfrak{t}_{R}}\mathcal{R}_{\mathrm{mv}}\big{(}\varrho,{\bf
m}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(s)\,\mathrm{d}s,$

where after a standard manipulation of terms in (LABEL:remainderRE), as in
[27], we obtain

$\displaystyle\mathcal{R}_{\mathrm{mv}}\big{(}\varrho,{\bf m}$
$\displaystyle\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}=\,\int_{\mathbb{T}^{3}}\left\langle{\mathcal{V}^{\omega}_{t,x}};\left|\frac{(\textbf{m}-\varrho\bar{\mathbf{u}})\otimes(\varrho\bar{\mathbf{u}}-\textbf{m})}{\varrho}\right|\right\rangle|\nabla_{x}\bar{\mathbf{u}}|\,dx$
(4.14)
$\displaystyle-\int_{\mathbb{T}^{3}}\langle{\mathcal{V}^{\omega}_{t,x}};|p(\varrho)-p^{\prime}(\bar{\varrho})(\varrho-\bar{\varrho})-p(\bar{\varrho})|\rangle|\operatorname{div}_{x}\bar{\mathbf{u}}|\,dx$
$\displaystyle+\frac{1}{2}\sum_{k\in\mathbb{N}}\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\bigg{|}\frac{\mathbf{G}_{k}(\varrho,\varrho\mathbf{u})}{\varrho}-\frac{\mathbf{G}_{k}(\bar{\varrho},\bar{\varrho}\bar{\mathbf{u}})}{\bar{\varrho}}\bigg{|}^{2}\Bigg{\rangle}\,\mathrm{d}x+\int_{\mathbb{T}^{3}}|\nabla_{x}\bar{\mathbf{u}}|\cdot
d|\mu_{m}|+\frac{1}{2}\int_{\mathbb{T}^{3}}d|\mu_{c}|.$

Since $\bar{\textbf{u}}$ has compact support we can control the terms
$|\nabla_{x}\bar{\mathbf{u}}|$ by some constants. It is also clear that there
exist a constant $c_{1}$ such that

$\left|\frac{(\textbf{m}-\varrho\bar{\mathbf{u}})\otimes(\varrho\bar{\mathbf{u}}-\textbf{m})}{\varrho}\right|\leq\frac{c_{1}}{2\varrho}|\textbf{m}-\varrho\bar{\mathbf{u}}|^{2},$

and a constant $c_{2}$ such that

$|p(\varrho)-p^{\prime}(\bar{\varrho})(\varrho-\bar{\varrho})-p(\bar{\varrho})|\leq
c_{2}(P(\varrho)-P^{\prime}(\bar{\varrho})(\varrho-\bar{\varrho})-P(\bar{\varrho})).$

To deal with the term coming from Itö correction terms, we follow [8] and
rewrite

$\displaystyle\varrho\bigg{|}\frac{\mathbf{G}_{k}(\varrho,\varrho\mathbf{u})}{\varrho}-\frac{\mathbf{G}_{k}(\bar{\varrho},\bar{\varrho}\bar{\mathbf{u}})}{\bar{\varrho}}\bigg{|}^{2}=\chi_{\\{\varrho\leq\bar{\varrho}/2\\}}\,\varrho\bigg{|}\frac{\mathbf{G}_{k}(\varrho,\varrho\mathbf{u})}{\varrho}-\frac{\mathbf{G}_{k}(\bar{\varrho},\bar{\varrho}\bar{\mathbf{u}})}{\bar{\varrho}}\bigg{|}^{2}$
(4.15)
$\displaystyle\qquad+\chi_{\\{\bar{\varrho}/2<\varrho<2\bar{\varrho}\\}}\,\varrho\bigg{|}\frac{\mathbf{G}_{k}(\varrho,\varrho\mathbf{u})}{\varrho}-\frac{\mathbf{G}_{k}(\bar{\varrho},\bar{\varrho}\bar{\mathbf{u}})}{\bar{\varrho}}\bigg{|}^{2}+\chi_{\\{\varrho\geq
2\bar{\varrho}\\}}\,\varrho\bigg{|}\frac{\mathbf{G}_{k}(\varrho,\varrho\mathbf{u})}{\varrho}-\frac{\mathbf{G}_{k}(\bar{\varrho},\bar{\varrho}\bar{\mathbf{u}})}{\bar{\varrho}}\bigg{|}^{2}$
$\displaystyle=:I_{1}+I_{2}+I_{3}$

We can now use the inequality $\varrho\leq 1+\varrho^{\gamma}$ and elementary
inequalities to conclude that

$\displaystyle I_{1}$ $\displaystyle\leq
c\,\chi_{\\{\varrho\leq\bar{\varrho}/2\\}}\,\bigg{(}\frac{1}{\varrho}\,|\mathbf{G}_{k}(\varrho,\varrho{\mathbf{u}})|^{2}+\frac{\varrho}{\bar{\varrho}^{2}}\,|\mathbf{G}_{k}(\bar{\varrho},\bar{\varrho}\bar{\mathbf{u}})|^{2}\bigg{)}$
$\displaystyle\leq
c\,\chi_{\\{\varrho\leq\bar{\varrho}/2\\}}\big{(}\varrho+\varrho|{\mathbf{u}}|^{2}+\varrho|\bar{\mathbf{u}}|^{2}\big{)}\leq
c(R)\,\chi_{\\{\varrho\leq\bar{\varrho}/2\\}}\big{(}1+\varrho^{\gamma}+\varrho|{\mathbf{u}}-\bar{\mathbf{u}}|^{2}\big{)}$

Therefore,

$\displaystyle\sum_{k\in\mathbb{N}}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};I_{1}\big{\rangle}\,\mathrm{d}x\leq
c(R)\,\mathcal{E}^{1}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)},\,\,\text{for
a.e.}\,\,t\in[0,T].$

We can estimate $I_{2}$ and $I_{3}$ in a similar fashion. Finally, we can
conclude from (LABEL:nio0) that

$\displaystyle\frac{1}{2}\sum_{k\in\mathbb{N}}\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\bigg{|}\frac{\mathbf{G}_{k}(\varrho,\varrho\mathbf{u})}{\varrho}-\frac{\mathbf{G}_{k}(\bar{\varrho},\bar{\varrho}\bar{\mathbf{u}})}{\bar{\varrho}}\bigg{|}^{2}\Bigg{\rangle}\,\mathrm{d}x\leq
c(R)\,\mathcal{E}^{1}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)},\,\,\text{for
a.e.}\,\,t\in[0,T].$ (4.16)

Collecting all the above estimates, we have shown that

$\displaystyle\int_{0}^{t\wedge\mathfrak{t}_{R}}\mathcal{R}_{\mathrm{mv}}\big{(}\varrho,{\bf
m}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}\,\mathrm{d}s\leq
c(R)\,\int_{0}^{t\wedge\mathfrak{t}_{R}}\mathcal{E}_{\mathrm{mv}}\big{(}\varrho,{\bf
m}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(s)\,\mathrm{d}s.$ (4.17)

Combining (4.17) and (LABEL:relativeEntropy1) and applying Gronwall’s lemma
yields

$\displaystyle\mathbb{E}\,\Big{[}\mathcal{E}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(t\wedge\mathfrak{t}_{R})\Big{]}\leq
c(R)\,\mathbb{E}\,\Big{[}\mathcal{E}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(0)\Big{]}.$

Note that we have

$\displaystyle\mathcal{E}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(0)$
$\displaystyle=\int_{\mathbb{T}^{3}}\Big{\langle}{\mathcal{V}^{\omega}_{0,x}};\frac{1}{2}\varrho_{0}\big{|}\mathbf{u}_{0}-\bar{\mathbf{u}}_{0}\big{|}^{2}+P\big{(}\varrho_{0},\bar{\varrho}_{0}\big{)}\Big{\rangle}\,\mathrm{d}x$

which is zero in expectation by assumptions. Therefore, we conclude that

$\displaystyle\mathbb{E}\,\Big{[}\mathcal{E}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(t\wedge\mathfrak{t}_{R})\Big{]}=0,\quad\text{for
all}\,\,t\in[0,T].$

This, in particular, implies that

$\displaystyle\lim_{\tau\rightarrow
0+}\frac{1}{\tau}\int_{t}^{t+\tau}\mathbb{E}\,\Big{[}\mathcal{E}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(s\wedge\mathfrak{t}_{R})\Big{]}\,ds=0$

Thanks to _a priori_ estimates (3.2)–(3.6) (which are preserved in the limit),
an application of Fubini’s theroem, and a Lebesgue point argument reveals that
for a.e. $t\in[0,T]$

$\displaystyle\mathbb{E}\,\Big{[}\mathcal{E}^{1}_{\mathrm{mv}}\big{(}\varrho,{\mathbf{m}}\left|\right.\bar{\varrho},\bar{\mathbf{u}}\big{)}(t\wedge\mathfrak{t}_{R})\Big{]}=0.$

In other words, for a.e. $(t,x)\in[0,T]\times\mathbb{T}^{3}$

$\mathcal{D}(t\wedge\mathfrak{t})=0,\quad\mathcal{V}^{\omega}_{t\wedge\mathfrak{t},x}=\delta_{\bar{\varrho}(t\wedge\mathfrak{t},x),(\bar{\varrho}\bar{\bf
u})(t\wedge\mathfrak{t},x)},\,\mathbb{P}-\mbox{a.s.}$

##  5 Singular Limits

In this section, we discuss another application of relative energy - a
rigorous justification of _low Mach number limit_ (also called _incompressible
limit_) for the system (1.1)–(1.2). For this purpose, let us first rescale the
deterministic counterpart of the stochastic compressible Euler system
(1.1)–(1.2) by non-dimensionalization. After combining terms appropriately
(setting the so–called Strouhal number equal to one), and adding a stochastic
force term, one reaches the following system

$\displaystyle{\rm d}\varrho_{\varepsilon}+{\rm
div}_{x}(\varrho_{\varepsilon}{\bf u}_{\varepsilon})\,dt$ $\displaystyle=0,$
(5.1) $\displaystyle{\rm d}(\varrho_{\varepsilon}{\bf
u}_{\varepsilon})+\big{[}{\rm div}_{x}(\varrho_{\varepsilon}{\bf
u}_{\varepsilon}\otimes{\bf
u}_{\varepsilon})+\frac{1}{\varepsilon^{2}}\nabla_{x}p(\varrho_{\varepsilon})\big{]}\,{\rm
d}t$ $\displaystyle=\mathbb{G}(\varrho_{\varepsilon},\varrho_{\varepsilon}{\bf
u}_{\varepsilon})\,{\rm d}W,$ (5.2)

where $\varepsilon$ is called the Mach number. It represents the norm of the
velocity divided by the sound speed. It is well known that acoustic waves are
responsible for the weak convergence of gradient part of the velocity. Hence,
the limit of stochastic forcing term $\mathbb{G}(\varrho,\varrho{\bf u})\,{\rm
d}W$ can be performed only if $\mathbb{G}$ is linear in the second variable.
Therefore, for the remainder of the paper, we only consider $\mathbb{G}$ of
the following form:

$\displaystyle\mathbb{G}(\varrho,\varrho{\bf u})=\varrho\mathbb{K}+\varrho{\bf
u}\mathbb{L},\quad\sum_{k\geq 1}\big{(}|K_{k}|+|L_{k}|\big{)}<\infty,$ (5.3)

where $K_{k},L_{k}$ are real numbers, and
$\mathbb{K}=(K_{k})_{k\in\mathbb{N}}$, $\mathbb{L}=(L_{k})_{k\in\mathbb{N}}$
are suitable Hilbert-Schmidt operators.

We consider the asymptotic limit of solutions $(\varrho_{\varepsilon},{\bf
u}_{\varepsilon})$ for $\varepsilon\rightarrow 0$. Accordingly, as
$\varepsilon\rightarrow 0$, the speed of the acoustic wave becomes infinite
and the fluid density approaches to a constant and the velocity becomes
solenoidal. The resulting limiting equations are

$\displaystyle{\rm div}_{x}{\bf v}$ $\displaystyle=0$ (5.4) $\displaystyle{\rm
d}{\bf v}+\big{[}{\bf v}\cdot\nabla_{x}{\bf v}+\nabla_{x}\Pi\big{]}\,dt$
$\displaystyle=\mathbb{G}(1,{\bf v})\,{\rm d}W$ (5.5)

There are couple of approaches available in literature to deal with the
singular limit problem. The first approach deals with the classical (strong)
solution of (1.1)–(1.2), while the second approach is based on the concept of
weak (dissipative measure-valued) solutions for the system (1.1–1.2). For the
deterministic counterpart of (1.1)–(1.2), Kleinermann and Majda [33], Schochet
[40], Masmoudi [37] and many others have successfully implemented the first
approach. On the other hand, Feireisl et.al. [28], and Bruell and Feireisl [6]
have explored the second approcah for deterministic compressible fluid
equations. The main advantage of the second approach is that measure-valued
solutions exist globally in time, while classical solutions may not exist
globally in time.

Our aim is to extend the result of Feireisl et. al. [28] to the stochastically
driven compressible fluids. To fix the ideas, let
$\big{[}\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{{\varepsilon},t})_{t\geq
0},\mathbb{P}\big{)};{\mathcal{V}^{\omega,\varepsilon}_{t,x}},W\big{]}$ be
dissipative measure-valued martingale solutions to the system (5.1)–(5.2), in
the sense of Definition 2.11. A rigorous justification of passing to the limit
as $\varepsilon\rightarrow 0$ in (5.1)–(5.2) makes use of the relative energy
inequality (LABEL:relativeEntropy). However, since the filtration
corresponding to measure-valued martingale solution depends explicitly on
$\varepsilon$, we need to justify that we can apply relative energy inequality
to the pair ${\mathcal{V}^{\omega,\varepsilon}_{t,x}}$ and $v$, for all
$\varepsilon>0$. Indeed, this can be done thanks to the fact that the unique
strong solution $v$ can be constructed on any given stochastic basis and is
adapted to the Brownian motion which is indepependent of $\varepsilon$. In
other words, for every filtration $(\mathfrak{F}_{{\varepsilon},t})_{t\geq 0}$
the strong solution to the incompressible Euler equation is adapted to that
filtration, since it is adapted to the brownian motion and the brownian motion
is adapted to $(\mathfrak{F}_{{\varepsilon},t})_{t\geq 0}$.

###  5.1 Solutions of the incompressible Euler system

Let us assume that we are given the stochastic basis
$\left(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P}\right)$ and
the Wiener process $W$ identified in the beginning of this manuscript.

######  Definition 5.1.

Let $\left(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P}\right)$
be a stochastic basis with a complete right-continuous filtration, let $W$ be
an $(\mathfrak{F}_{t})_{t\geq 0}$-cylindrical Wiener process. A stochastic
process ${\bf v}$ with a stopping time $\mathfrak{t}$ is called a (local)
strong solution to the Euler system (5.4), (5.5) provided

  * •

the velocity ${\bf v}\in C([0,T];W^{3,2}(\mathbb{T}^{3};\mathbb{R}^{3}))$,
$\mathbb{P}$-a.s. is $(\mathfrak{F}_{t})_{t\geq 0}$-adapted,

$\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|{\bf
v}(t,\cdot)\|_{W^{3,2}(\mathbb{T}^{3};\mathbb{R}^{3})}^{p}\bigg{]}<\infty,\,\,\mbox{for
all}\,\,1\leq p<\infty;$

  * •

There holds $\mathbb{P}$-a.s.

$\begin{split}{\rm div}_{x}{\bf v}&=0,\\\ {\bf v}(t\wedge\mathfrak{t})&={\bf
v}(0)-\int_{0}^{t\wedge\mathfrak{t}}{{\bf P}_{H}}\left[{\bf
v}\cdot\nabla_{x}{\bf v}\right]\,{\rm d}t+\int_{0}^{t\wedge\mathfrak{t}}{\bf
P}_{H}\left[{\mathbb{G}}(1,{\bf v})\right]\,{\rm d}W,\end{split}$ (5.6)

a.e. in $(0,T)\times\mathbb{T}^{3}$. Here ${\bf P}_{H}$ denotes the standard
Helmholtz projection onto the space of solenoidal functions.

Regarding the local-in-time existence of strong solutions to the stochastic
Euler system, under certain restrictions imposed on the forcing coefficients
$\mathbb{G}$, we refer to the recent work by Glatt-Holtz and Vicol [30,
Theorem 4.3]. Here, as mentioned before, we assume a very simple form of
$\mathbb{G}$, namely as in (5.3). One of the advantage of such a choice is
that the pressure $\Pi$ can be computed explicitly from (5.6). Indeed noting
that

${\bf P}_{H}\left[{\mathbb{G}}(1,{\bf v})\right]={\mathbb{G}}(1,{\bf v}),$

we get

$\nabla_{x}\Pi=-{\bf P}^{\perp}_{H}[{\bf v}\cdot\nabla_{x}{\bf
v}]=-\nabla_{x}\Delta^{-1}{\rm div}_{x}({\bf v}\otimes{\bf v}).$ (5.7)

Accordingly, the second equation in (5.6) reads

${\bf v}(t\wedge\mathfrak{t})={\bf
v}(0)-\int_{0}^{t\wedge\mathfrak{t}}\left[{\bf v}\cdot\nabla_{x}{\bf
v}\right]\,{\rm d}t-\int_{0}^{t\wedge\mathfrak{t}}\nabla_{x}\Pi\ \,{\rm
d}t+\int_{0}^{t\wedge\mathfrak{t}}{\mathbb{G}}(1,{\bf v})\,{\rm d}W.$ (5.8)

###  5.2 Main result

We now state the main result of this section related to the rescaled
stochastic compressible Euler system (5.1)–(5.2).

######  Theorem 5.2.

Let $\mathbb{G}$ be given as in (5.3), and the initial data
$\varrho_{0,\varepsilon}$, $(\varrho{\bf u})_{0,\varepsilon}$, and ${\bf
v}_{0}$ be given such that $\mathbb{P}$-a.s.

$\Bigg{\\{}\varrho_{0,\varepsilon},(\varrho{\bf u})_{0,\varepsilon}\in
L^{\gamma}(\mathbb{T}^{3})\times
L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3};\mathbb{R}^{3})\ \Big{|}\
\varrho_{0,\varepsilon}\geq\underline{\varrho}>0,\
\frac{|\varrho_{0,\varepsilon}-1|}{\varepsilon}\leq\delta(\varepsilon),\
|(\varrho{\bf u})_{0,\varepsilon}-{\bf
v}_{0}|\leq\delta(\varepsilon)\Bigg{\\}},$

where

$\delta(\varepsilon)\to 0\ \mbox{as}\ \varepsilon\to 0,$

and where ${\bf v}_{0}$ is an $\mathfrak{F}_{0}$-measurable random variable,

$\displaystyle{\bf v}_{0}\in W^{3,2}(\mathbb{T}^{3};R^{3}),\ {\rm div}_{x}{\bf
v}_{0}=0,\quad\text{$\mathbb{P}$-a.s.},$ $\displaystyle\mathbb{E}\left[\|{\bf
v}_{0}\|_{W^{3,2}(\mathbb{T}^{3};R^{3})}^{p}\right]<\infty,\,\,\mbox{for
all}\,\,1\leq p<\infty.$

Let $\big{[}\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{\varepsilon,t})_{t\geq
0},\mathbb{P}\big{)};{\mathcal{V}^{\omega,\varepsilon}_{t,x}},W\big{]}$ be a
dissipative measure-valued martingale solution to the system (5.1)–(5.2),
satisfying the compatibility condition (2.10), and suppose $({\bf
v},\mathfrak{t})$ defined on the same probability space
$\big{(}\Omega,\mathfrak{F},(\mathfrak{F}_{\varepsilon,t})_{t\geq
0},\mathbb{P}\big{)}$ is a unique local strong solution of the Euler system
(5.4)–(5.5) driven by the same cylindrical Wiener process $W$ in the sense of
Definition 5.1. Then as $\varepsilon\to 0$, $\mathbb{P}$-a.s.

$\mathcal{D}^{\varepsilon}\to 0\ \mbox{in}\ L^{\infty}(0,T),$
$\operatorname*{ess\,sup}_{t\in(0,T)}\mathbb{E}\Bigg{[}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{1}{2}\varrho\left|\frac{{\bf
m}}{\varrho}-{\bf
v}(t,x)\right|^{2}+\frac{1}{\varepsilon^{2}}\Big{(}P(\varrho)-P^{\prime}(1)(\varrho-1)-P(1)\Big{)}\right>\,{\rm
d}{x}\,(t\wedge\tau)\Bigg{]}\to 0$

We remark that the above theorem asserts that the probability measures
${\mathcal{V}^{\omega}_{t,x}}$ shrink to their expected value as
$\varepsilon\to 0$, where the latter are characterized by the constant value
$1$ for the density and the solution ${\bf v}$ of the incompressible system.
The situation considered in the above theorem corresponds to the so called
well-prepared data. However, one can extend these results to the case of ill-
prepared data (refer to Masmoudi [37], for the related deterministic results)
with additional technical computations.

###  5.3 Proof of Theorem 5.2

Let us first assume that ${\bf v}$, with a stopping time $\mathfrak{t}$, is a
local strong solution of the stochastic Euler system (5.6). For each $M>0$,
let us define

$\tau_{M}:=\inf\bigg{\\{}t\in[0,T]:\|\nabla_{x}v(t)\|_{L^{\infty}(\mathbb{T}^{3})}>M\bigg{\\}},$

be another stopping time. Thanks to the existence theorem [30, Theorem 4.3],
we assume, without loss of generality, that $\tau_{M}\leq\mathfrak{t}$.
Following (4.2), for ${\mathcal{V}^{\omega,\varepsilon}_{t,x}}$ \- the
dissipative measure-valued solution of the rescaled system - we denote for
a.e. $t\in[0,T]$

$\mathcal{E}_{\mathrm{mv}}^{\varepsilon,1}\left(\varrho,{\bf m}\Big{|}1,{\bf
v}\right)(t)=\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{1}{2}\varrho\left|\frac{{\bf
m}}{\varrho}-{\bf
v}(t,x)\right|^{2}+\frac{1}{\varepsilon^{2}}\Big{(}P(\varrho)-P^{\prime}(1)(\varrho-1)-P(1)\Big{)}\right>\,{\rm
d}{x}+\mathcal{D}^{\varepsilon}(t),$

the relative energy functional associated to $1$, ${\bf v}$. Similarly,
following (4.3) and (4.4), we define
$\mathcal{E}_{\mathrm{mv}}^{\varepsilon,2}\left(\varrho,{\bf m}\Big{|}1,{\bf
v}\right)$, and $\mathcal{E}_{\mathrm{mv}}^{\varepsilon}\left(\varrho,{\bf
m}\Big{|}1,{\bf v}\right)$ appropriately.

####  5.3.1 Relative energy inequality

As the quantities $r=1$, ${\bf U}(\tau)={\bf v}(\tau\wedge\tau_{M})$ enjoy the
required regularity, they can be used as test functions in the relative
entropy inequality. Therefore, we conclude

$\displaystyle\mathcal{E}_{\mathrm{mv}}^{\varepsilon}$
$\displaystyle\left(\varrho,{\bf m}\ \Big{|}\ 1,{\bf
v}\right)(\tau\wedge\tau_{M})\leq\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{0,x}};\frac{1}{2}\varrho\left|\frac{{\bf
m}}{\varrho}-{\bf
v}_{0}(x)\right|^{2}+\frac{1}{\varepsilon^{2}}\Big{(}P(\varrho)-P^{\prime}(1)(\varrho-1)-P(1)\Big{)}\right>\,{\rm
d}{x}$ (5.9)
$\displaystyle+\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\left[\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho{\bf
v}(t,x)-{\bf m}\right>\cdot D^{d}_{t}{\bf
v}+\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};(\varrho{\bf v}(t,x)-{\bf
m})\otimes\frac{{\bf m}}{\varrho}\right>:\nabla_{x}{\bf v}\right]\,{\rm
d}{x}\,{\rm d}t$
$\displaystyle-\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\nabla_{x}{\bf
v}:{\rm d}\mu^{\varepsilon}_{m}+M_{RE}(\tau\wedge\tau_{M})-M_{RE}(0)$
$\displaystyle+\frac{1}{2}\sum_{k\geq
1}\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho\Big{|}\frac{1}{{\varrho}}{{\bf
G}_{k}}(\varrho,\varrho{\bf u})-{\bf G}_{k}(1,{\bf
v})\Big{|}^{2}\Bigg{\rangle}\,dx\ \,{\rm
d}t+\frac{1}{2}\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}d\mu^{\varepsilon}_{e}.$

As the initial data are well-prepared, we get

$\mathbb{E}\bigg{[}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{0,x}};\frac{1}{2}\varrho\left|\frac{{\bf
m}}{\varrho}-{\bf
v}_{0}(x)\right|^{2}+\frac{1}{\varepsilon^{2}}\Big{(}P(\varrho)-P^{\prime}(1)(\varrho-1)-P(1)\Big{)}\right>\,{\rm
d}{x}\bigg{]}\leq\delta(\varepsilon),\,\delta(\varepsilon)\to 0\ \mbox{as}\
\varepsilon\to 0.$ (5.10)

In addition, since the compatibility condition (2.10) is satisfied for all
$\varepsilon$, we deduce

$\frac{1}{2}\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}d\mu^{\varepsilon}_{e}-\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\nabla_{x}{\bf
v}:{\rm d}\mu^{\varepsilon}_{m}\leq
CM\int_{0}^{\tau\wedge\tau_{M}}\mathcal{D}^{\varepsilon}\ \,{\rm d}t$ (5.11)

Next, motivated by the specific form of $\mathbb{G}(1,{\bf v})$ introduced in
(5.3), and thanks to the boundedness of the sum $\sum_{k\geq 1}|L_{k}|^{2}$,

$\displaystyle\sum_{k\geq 1}$
$\displaystyle\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho\Big{|}\frac{1}{{\varrho}}{{\bf
G}_{k}}(\varrho,\varrho{\bf u})-{\bf G}_{k}(1,{\bf
v})\Big{|}^{2}\Bigg{\rangle}\,dx$ (5.12)
$\displaystyle\qquad\qquad=\sum_{k\geq
1}\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho\left|({\bf
u}-{\bf
v})H_{k}\right|^{2}\Bigg{\rangle}\,dx\leq\,C\,\mathcal{E}^{\varepsilon}_{\mathrm{mv}}\left(\varrho,{\bf
m}\ \Big{|}\ 1,{\bf v}\right).$

We proceed further, and claim that

$\begin{split}\int_{0}^{\tau\wedge\tau_{M}}&\int_{\mathbb{T}^{3}}\left[\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho{\bf
v}(t,x)-{\bf m}\right>\cdot D^{d}_{t}{\bf
v}+\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};(\varrho{\bf v}(t,x)-{\bf
m})\otimes\frac{{\bf m}}{\varrho}\right>:\nabla_{x}{\bf v}\right]\,{\rm
d}{x}\,{\rm d}t\\\ &\qquad\leq
CM\int_{0}^{\tau\wedge\tau_{M}}\mathcal{E}^{\varepsilon}_{\mathrm{mv}}\left(\varrho,{\bf
m}\ \Big{|}\ 1,{\bf v}\right)\,\,{\rm d}t.\end{split}$ (5.13)

In order to justify (5.13), we first start by writing

$\begin{split}&\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};(\varrho{\bf
v}(t,x)-{\bf m})\otimes\frac{{\bf m}}{\varrho}\right>:\nabla_{x}{\bf v}\,{\rm
d}{x}\\\
&=\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};(\varrho{\bf
v}(t,x)-{\bf m})\otimes\frac{{\bf m}-\varrho{\bf
v}}{\varrho}\right>:\nabla_{x}{\bf v}\,{\rm
d}{x}+\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho{\bf
v}(t,x)-{\bf m}\right>\cdot{\bf v}\cdot\nabla_{x}{\bf v}\,{\rm
d}{x},\end{split}$

where obviously,

$\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};(\varrho{\bf
v}(t,x)-{\bf m})\otimes\frac{{\bf m}-\varrho{\bf
v}}{\varrho}\right>:\nabla_{x}{\bf v}\,{\rm d}{x}\leq
CM\,\mathcal{E}^{\varepsilon}_{\mathrm{mv}}\left(\varrho,{\bf m}\ \Big{|}\
1,{\bf v}\right).$

Moreover, as ${\bf v}$ fulfills equation (5.5), we observe that justifying
(5.13) reduces to showing

$\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};{\bf
m}-\varrho{\bf v}(t,x)\right>\cdot\nabla_{x}\Pi\,{\rm d}{x}\,{\rm d}t\leq
C\int_{0}^{\tau\wedge\tau_{M}}\mathcal{E}^{\varepsilon}_{\mathrm{mv}}\left(\varrho,{\bf
m}\ \Big{|}\ 1,{\bf v}\right)\,\,{\rm d}t.$ (5.14)

To see (5.14), we deduce from the density equation that

$\begin{split}\int_{0}^{\tau\wedge\tau_{M}}&\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};{\bf
m}\right>\cdot\nabla_{x}\Pi\,{\rm d}{x}\,{\rm d}t\\\
&\qquad=\left[\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho\right>\Pi\,{\rm
d}{x}\right]_{t=0}^{t=\tau\wedge\tau_{M}}{-\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathcal{T}^{N}}\nabla_{x}\Pi\cdot{\rm
d}\mu^{\varepsilon}_{c}}\\\
&\qquad\qquad=\varepsilon\left[\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{\varrho-1}{\varepsilon}\right>\Pi\,{\rm
d}{x}\right]_{t=0}^{t=\tau\wedge\tau_{M}}-\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathcal{T}^{N}}\nabla_{x}\Pi\cdot{\rm
d}\mu^{\varepsilon}_{c}.\end{split}$ (5.15)

Similarly, we may use the incompressibility condition ${\rm div}_{x}{\bf v}=0$
to obtain

$\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\varrho{\bf
v}(t,x)\right>\cdot\nabla_{x}\Pi\,{\rm d}{x}\,{\rm
d}t=\varepsilon\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{\varrho-1}{\varepsilon}\right>{\bf
v}\cdot\nabla_{x}\Pi\,{\rm d}{x}\,{\rm d}t$ (5.16)

Note that $\Pi$ and $\nabla_{x}\Pi$ are bounded continuous in
$[0,T]\times\mathbb{T}^{3}$. Now observe that the right-most integral in
(5.15) can be controlled by the dissipation defect
$\mathcal{D}^{\varepsilon}$. In order to control the other term on the right
side of (5.15) and the term on the right side of (5.16), we evoke again
relative energy inequality, this time for $r=1$, ${\bf U}=0$ obtaining

$\begin{split}&\mathbb{E}\left[\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\left[\frac{1}{2}\varrho|{\bf
u}|^{2}+\frac{1}{\varepsilon^{2}}\left(P(\varrho)-P^{\prime}(1)(\varrho-1)-P(1)\right)\right]\Bigg{\rangle}\,dx(\tau\wedge\tau_{M})+\mathcal{D}^{\varepsilon}(\tau\wedge\tau_{M})\right]\\\
&\qquad\leq\mathbb{E}\left[\int_{\mathbb{T}^{3}}\Bigg{\langle}{\mathcal{V}^{\omega,\varepsilon}_{0,x}};\left[\frac{1}{2}\varrho|{\bf
u}|^{2}+\frac{1}{\varepsilon^{2}}\left(P(\varrho)-P^{\prime}(1)(\varrho-1)-P(1)\right)\right]\Bigg{\rangle}\,dx(0)\right].\end{split}$

Thus, if the right-hand side of the above inequality is bounded uniformly for