# A convergent finite volume scheme for the stochastic barotropic
compressible Euler equations

Abhishek Chaudhary 22footnotemark: 2 Ujjwal Koley 33footnotemark: 3

###### Abstract

In this paper, we analyze a semi-discrete finite volume scheme for the three-
dimensional barotropic compressible Euler equations driven by a
_multiplicative_ Brownian noise. We derive necessary a priori estimates for
numerical approximations, and show that the Young measure generated by the
numerical approximations converge to a _dissipative measure–valued martingale_
solution to the stochastic compressible Euler system. These solutions are
probabilistically weak in the sense that the driving noise and associated
filtration are integral part of the solution. Moreover, we demonstrate
_strong_ convergence of numerical solutions to the regular solution of the
limit systems at least on the lifespan of the latter, thanks to the weak
(measure-valued)–strong uniqueness principle for the underlying system. To the
best of our knowledge, this is the first attempt to prove the convergence of
numerical approximations for the underlying system.

† Centre for Applicable Mathematics, Tata Institute of Fundamental Research

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

<EMAIL_ADDRESS>
$\ddagger$ Centre for Applicable Mathematics, Tata Institute of Fundamental
Research

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

<EMAIL_ADDRESS>
Keywords: Compressible fluids; Euler system; Finite volume schemes; Stochastic
forcing; Martingale solutions; Dissipative measure–valued solution;
Weak–strong uniqueness; Entropy stable fluxes; Convergence.

###### Contents

  1. 1 Introduction
    1. 1.1 Compressible Euler Equations
    2. 1.2 Numerical Schemes
    3. 1.3 Scope and Outline of the Paper
  2. 2 Preliminaries and Main Results
    1. 2.1 Analytic framework
      1. 2.1.1 Young measures, concentration defect measures
      2. 2.1.2 Convergence of arithmetic averages
    2. 2.2 Background on Stochastic framework
    3. 2.3 Stochastic compressible Euler equations
    4. 2.4 Measure-valued solutions
      1. 2.4.1 Dissipative measure-valued martingale solutions
    5. 2.5 Numerical scheme
      1. 2.5.1 Spatial discretization
      2. 2.5.2 Entropy stable flux and the scheme
    6. 2.6 Statements of main results
  3. 3 Stability of the Numerical Scheme
    1. 3.1 A priori estimates for the stochastic Euler system
  4. 4 Consistency of the Numerical Scheme
    1. 4.1 Consistency formulation of continuity and momentum equations
  5. 5 Proof of Theorem 2.14: Existence of Measure-Valued Solution
    1. 5.1 Compactness and almost sure representations
      1. 5.1.1 Passage to the limit
  6. 6 Weak-Strong Uniqueness Principle
  7. 7 Proof of Theorem 2.15: Convergence to dissipative solution
  8. 8 Proof of Theorem 2.16: Convergence to Regular Solution

##  1 Introduction

Most real world models involve a large number of parameters and coefficients
which cannot be exactly determind. Furthermore, there is a considerable
uncertainty in the source terms, initial or boundary data due to empirical
approximations or measuring errors. Therefore, study of PDEs with randomness
(stochastic PDEs) certainly leads to greater understanding of the actual
physical phenomenon. In this paper, we are interested in a stochastic variant
of the _compressible barotropic Euler system_ , a set of balance laws driven
by a _nonlinear multiplicative_ noise for mass density $\varrho$ and the bulk
velocity ${\bf u}$ describing the flow of isentropic gas, where the thermal
effects are neglected. The system of equations read

$\displaystyle{\rm d}\varrho+{\rm div}(\varrho{\bf u})\,{\rm d}t$
$\displaystyle=0,$ (1.1) $\displaystyle{\rm d}(\varrho{\bf u})+\left[{\rm
div}(\varrho{\bf u}\otimes{\bf u})+a\nabla\varrho^{\gamma}\right]\,{\rm d}t$
$\displaystyle=\Psi(\varrho,\varrho{\bf u})\,{\rm d}W$

Here $\gamma>1$ denotes the adiabatic exponent, $a>0$ is the squared
reciprocal of the Mach number (the ratio between average velocity and speed of
sound). The driving process $W$ is a cylindrical Wiener process defined on
some filtered probability space
$(\Omega,\mathbb{F},\\{\mathbb{F}_{t}\\}_{t\geq 0},\mathbb{P})$, and the noise
coefficient $\Psi$ is nonlinear and satisfies suitable growth assumptions (see
Subsection 2.2 for the complete list of assumptions). Note that
$(\varrho,\varrho{\bf u})\mapsto\Psi(\varrho,\varrho{\bf u})$ is a given
Hilbert space valued function signifying the _multiplicative_ nature of the
noise. We consider the stochastic compressible Euler equations (1.1)–(1.2) in
three spatial dimensions on a periodic domain i.e., on the torus
$\mathbb{T}^{3}$. The initial conditions are random variables

$\varrho(0,\cdot)=\varrho_{0},\ \varrho{\bf u}(0,\cdot)=(\varrho{\bf u})_{0},$
(1.2)

with sufficient spatial regularity to be specified later.

###  1.1 Compressible Euler Equations

The deterministic counterpart of the stochastic compressible Euler equations
(1.1)–(1.2) have received considerable attention and, in spite of monumental
efforts, satisfactory well-posedness results are still lacking. It is well-
known that the smooth solutions to deterministic counterpart of (1.1)–(1.2)
exists only for a finite lap of time, after which singularities may develop
for a generic class of initial data. Therefore, global-in-time (weak)
solutions must be sought in the class of discontinuous functions. But, weak
solutions may not be uniquely determind by their initial data and
admissibility conditions must be imposed to single out the physically correct
solution. However, the specification of such an admissibility criteria is
still open. Indeed, thanks to recent phenomenal work by De Lellis $\&$
Szekelyhidi [14, 15], and further investigated by Chiodaroli et. al. [13],
Feireisl [22], it is well understood that the compressible Euler equations is
desparetly ill-posed, due to the lack of compactness of functions satisfying
the equations. Even if the initial data is smooth, the global existence and
uniqueness of solutions can fail. Moreover, a quest for the existence of
global-in-time weak solutions to deterministic counterpart of (1.1)–(1.2) for
_general_ initial data remains elusive. Given this status quo, it is natural
to seek an alternative solution paradigm for compressible Euler system. To
that context, we recall the framework of _dissipative_ Young measure-valued
solutions in the context of compressible Navier–Stokes system, being first
introduced by Neustupa in [33], and subsequently revisited by Feireisl et. al.
in [21]. In a nutshell, these solutions are characterized by a parametrized
Young measure and a concentration Young measure in the total energy balance,
and they are defined globally in time.

The study of stochastic compressible Euler equations (1.1)–(1.2) is a
relatively new area of focus within the broarder field of stochastic PDEs, and
a satisfactory well/ill-posedness result is largely out of reach. However, we
want to emphasize that, to design efficient numerical schemes it is of
paramount importance to have prior knowledge about the existence of global-in-
time solutions for the underlying system of equations. Without such knowledge,
there is no way to establish whether or not the solution produced by a
numerical scheme is an approximation of the true solution. To that context,
let us first mention the work by Berthelin $\&$ Vovelle [2], where the authors
established the existence of a martingale solution for (1.1)–(1.2) in _one_
spatial dimension. Moreover, a recent work by Breit et. al. in [6] revealed
that ill-posedness issues for compressible Euler system driven by _additive_
noise, in the sense of [14, 22], persist even in the presense of a random
forcing. We mention that for compressible Euler equations driven by
_multiplicative_ noise, the existence of _dissipative measure-valued
martingale_ solutions was very recently established by Hofmanova et. al. in
[26] (see also [12] for the incompressible case). The authors have shown that
the existence can be obtained from a sequence of solutions of stochastic
Navier Stokes equations using tools from martingale theory and Young measure
theory.

###  1.2 Numerical Schemes

Parallel to mathematical efforts there has been a huge effort to derive
effective numerical schemes for deterministic fluid flow equations, and there
is a considerable body of literature dealing with the convergence of numerical
schemes for the specific problems in fluid mechanics represented through the
barotropic Euler system. In this context, we first mention the work by Karper
in [28] where he has established the convergence of a mixed finite element-
discontinuous Galerkin scheme to compressible Euler system under the
assumption $\gamma>3$. Subsequently, a series of works [18, 19, 20] by
Feireisl and his collaborators analyzed the convergence issues for several
different semi-discrete numerical schemes via the framework of dissipative
measure-valued solutions. Note that the concept of measure–valued solutions
introduced in Feireisl et. al. [19] (and also [26]) requires the solutions
generated by approximate sequences satisfying only the general energy bounds.
This is very different from many classical approach where the existence of
measure-valued solution is conditioned by mostly rather unrealistic
assumptions of boundedness of certain physical quantities and the
corresponding fluxes. Indeed, assuming only uniform lower bound on the density
and uniform upper bound on the energy they showed that the Lax-Friedrichs-type
finite volume schemes generate the dissipative measure–valued solutions to the
barotropic Euler equations. We also mention that the first numerical evidence
that indicated ill-posedness of the Euler system was presented by Elling [16].
Finally, we mention a series of recent works by Fjordholm et. al. [23, 24] in
the context of a general system of hyperbolic conservation laws, where they
proved the convergence of a semi-discrete entropy stable finite volume scheme
to the _measure-valued_ solutions under certain appropriate assumptions.

We remark that, despite the growing interest about the theory of stochastic
PDEs and the discretization of stochastic PDEs, the specific question about
numerical approximations of stochastic compressible Euler equations is
virtually untouched. In fact, the challenges related to numerical aspects of
(1.1) are manifold and mostly open, due to the presence of _multiplicative_
noise term in (1.1). Having said this, we mention that there are few results
available on stochastic _incompressible_ Euler equations. To that context,
concerning the convergence of the numerical methods, we mention the work of
Brzeźniak et. al. [11], where the scheme is based on finite elements combined
with implicit Euler method.

###  1.3 Scope and Outline of the Paper

The above discussions clearly highlight the lack of effective convergent
numerical schemes, for compressible fluid flow equations driven by a
_multiplicative_ Brownian noise, which are able to take the inherent
uncertainties into account, and are equipped with modules that quantify the
level of uncertainty. The challenges related to numerical aspects of the
underlying problems are mostly open and the research on this frontier is still
in its infancy. In fact, the main objective of this article is to lay down the
foundation for a comprehensive theory related to numerical methods for
(1.1)–(1.2). Although our work bears some similarities with recent wroks of
Fjordholm et. al [23, 24] on deteministic system of conservation laws, and
works of Feireisl et. al [18, 19, 20] on deterministic Euler systems, the main
novelty of this work lies in successfully handling the _multiplicative_ noise
term. Our problems need to invoke ideas from numerical methods for SDE and
meaningfully fuse them with available approximation methods for deterministic
problems. This is easier said than done as any such attempt has to capture the
noise-noise interaction as well. In the realm of stochastic conservation laws,
noise-noise interaction terms play a fundamental role to establish well-
posedness theory, for details see [3, 4, 5, 29, 30, 31, 32].

The main contributions of this paper are listed below:

  * (1)

We develop an appropriate mathematical framework of _dissipative measure-
valued martingale_ solutions to the stochastic compressible Euler system,
keeping in mind that this framework would allow us to establish weak (measure-
valued)–strong uniqueness principle. We remark that our solution framework
requires only natural energy bounds associated to approximate solutions.

  * (2)

We show that a Lax-Friedrichs-type numerical scheme for (1.1)–(1.2) generates
the _dissipative measure-valued martingale_ solutions to the stochastic
compressible Euler equations. With the help of the new framework based on the
theory of measure–valued solutions, we adapt the concept of
$\mathcal{K}$-convergence, first developed in the context of Young measures by
Balder [1] (see also Feireisl et. al. [20]), to show the pointwise convergence
of arithmetic averages (Cesaro means) of numerical solutions to a _dissipative
measure-valued martingale_ solution of the limit system (1.1)–(1.2).

  * (3)

When solutions of the limit continuous problem possess maximal regularity, by
making use of weak (measure-valued)–strong uniqueness principle, we show
_unconditional_ strong $L^{1}$-convergence of numerical approximations to the
regular solution of the limit systems.

A breif description of the organization of the rest of the paper is as
follows: we describe all necessary mathematical/technical framework and state
the main results in Section 2. Moreover, we introduce a Lax-Friedrichs-type
finite volume numerical scheme for the underlying system (1.1)–(1.2). Section
3 is devoted on deriving stability properties of the scheme, while Section 4
is focused on deriving suitable formulations of the continuity and momentum
equations, and exhibit consistency. In Section 5, we present a proof of
convergence of numerical solutions to a dissipative measure-valued martingale
solutions using stochastic compactness. Section 6 is devoted on deriving the
weak (measure-valued) – strong uniqueness principle by making use of a
suitable relative energy inequality. Section 7 uses the concept of
$\mathcal{K}$-convergence to exhibit the pointwise convergence of numerical
solutions. Finally, in Section 8, we make use of weak (measure-valued)–strong
uniqueness property to show the convergence of numerical approximations to the
solutions of stochastic compressible Euler system (1.1)–(1.2).

##  2 Preliminaries and Main Results

Here we first briefly recall some relevant mathematical tools which are used
in the subsequent analysis and then we state main results of this paper. To
begin, we fix an arbitrary large time horizon $T>0$. For the sake of
simplicity it will be assumed $a=1$, since its value is not relevant in the
present setting. Throughout this paper, we use the letter $C$ to denote
various generic constants that may change from line to line along the proofs.
Explicit tracking of the constants could be possible but it is highly
cumbersome and avoided for the sake of the reader. Let $\mathcal{M}_{b}(E)$
denote 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 $\gamma\in(0,1)$ be given, and $Z$ be a separable Hilbert space. Let
$W^{\gamma,2}(0,T;Z)$ denotes a $Z$-valued Sobolev space which is
characterized by its norm

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

Then we have following compact embedding result from Flandoli $\&$ Gatarek
[25, Theorem 2.2].

######  Lemma 2.1.

If $Z\subset\subset Y$ are two Banach spaces with compact embedding, and real
numbers $\gamma\in(0,1)$ satisfy $\gamma>1/2$, then the following embedding

$W^{\gamma,2}(0,T;Z)\subset\subset C([0,T];Y)$

is compact.

####  2.1.1 Young measures, concentration defect measures

In this subsection, we first briefly recall the notion of Young measures and
related results which have been used frequently in the text. For an excellent
overview of applications of the Young measure theory to hyperbolic
conservation laws, we refer to Balder [1]. Let us begin by assuming that
$(Z,\mathrm{M},\mu)$ is a sigma finite measure space. A Young measure from $Z$
into $\mathbb{R}^{M}$ is a weakly measurable function
$\mathcal{V}:Z\rightarrow\mathcal{P}(\mathbb{R}^{M})$ in the sense that
$x\rightarrow\mathcal{V}_{x}(A)$ is $\mathrm{M}$-measurable for every Borel
set $A$ in $\mathbb{R}^{M}$. In what follows, we make use of the following
generalization of the classical result on Young measures; for details, see [8,
Section 2.8].

######  Lemma 2.2.

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

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

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

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

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

$J(\cdot,\widetilde{\bf
W}_{n})\rightharpoonup\overline{J}\,\,\text{in}\,\,L^{p/q}(\mathcal{Q}),\,\,\text{where}\,\,\overline{J}(y)=\langle\mathcal{\widetilde{V}}^{\omega}_{(\cdot)};J(y,\cdot)\rangle:=\int_{\mathbb{R}^{M}}J(y,z)\,{\rm
d}\mathcal{\widetilde{V}}^{\omega}_{y}(z),\,\,\text{for
a.a.}\,\,y\in\mathcal{Q}.$

In literature, Young measure theory has been successfully exploited to extract
limits of bounded continuous functions. However, for our purpose, we need to
deal with typical functions $F$ for which we only know that

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

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

$\displaystyle\mbox{weak-* limit
in}\,\mathcal{M}_{b}(\mathcal{Q})\,\,\mbox{of}\,\,F({\bf
W}_{n})=\langle\mathcal{\widetilde{V}}^{\omega}_{y};F\rangle\,dy+F_{\infty},$

where $F_{\infty}\in\mathcal{M}_{b}(\mathcal{Q})$, and $F_{\infty}$ is called
_concentration defect measure (or concentration Young measure)_. We remark
that, a simple truncation analysis and Fatou’s lemma reveal that
$\mathbb{P}$-a.s.
$\|\langle\mathcal{\widetilde{V}}^{\omega}_{(\cdot)};F\rangle\|_{L^{1}(\mathcal{Q})}\leq
C$ and thus $\mathbb{P}$-a.s.
$\langle\mathcal{\widetilde{V}}^{\omega}_{y};F\rangle$ is finite for a.e.
$y\in\mathcal{Q}$. In what follows, regarding the concentration defect
measure, we shall make use of the following crucial lemma. For a proof of the
lemma modulo cosmetic changes, we refer to Feireisl et. al [21, Lemma 2.1].

######  Lemma 2.3.

Let $\\{{\bf W}_{n}\\}_{n>0}$, ${\bf
W}_{n}:\Omega\times\mathcal{Q}\rightarrow\mathbb{R}^{M}$ be a sequence
generating a Young measure $\\{\mathcal{V}^{\omega}_{y}\\}_{y\in\mathcal{Q}}$,
where $\mathcal{Q}$ 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

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

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.

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

Here ${\widetilde{F}},{\widetilde{G}}\in\mathcal{M}_{b}(\mathcal{Q})$ are
weak-$*$ limits of $\\{F({\bf W}^{n})\\}_{n>0}$, $\\{G({\bf W}^{n})\\}_{n>0}$
respectively in $\mathcal{M}_{b}(\mathcal{Q})$. Then $\mathbb{P}$-almost
surely $|F_{\infty}|\leq G_{\infty}$.

####  2.1.2 Convergence of arithmetic averages

Following Feireisl et. al. [20], we also show that the arithmetic averages of
numerical solutions converge pointwise to a generalized dissipative solution
of the compressible Euler system, as introduced in Hofmanova et. al. [26]. To
that context, we have the following result.

######  Proposition 2.4.

Let $(X,\mathcal{A},\mu)$ be a finite measure space, and ${\bf
U}_{n}\rightharpoonup{\bf U}$ weakly in $L^{1}(X;\mathbb{R}^{M})$. Then there
exists a subsequence $({\bf U}_{n_{k}})_{k\geq\,1}$ of sequence $({\bf
U}_{n})_{n\,\geq 1}$ such that

$\displaystyle\frac{1}{n}\sum_{k=1}^{n}{\bf U}_{n_{k}}\to{\bf U},\quad\mbox{
a.e. in }{X}.$

###### Proof.

Since the sequence $({\bf U}_{n})_{n\,\geq\,1}$ is uniformly bounded in
$L^{1}(X)$, thanks to Komlós theorem, there exists a subsequence $({\bf
U}_{n_{k}})_{k\,\geq\,1}$ and $\tilde{{\bf U}}\in L^{1}(X)$ such that

$\displaystyle\frac{1}{n}\sum_{k=1}^{n}{\bf U}_{n_{k}}\to\tilde{{\bf
U}},\quad\mbox{ a.e. in }{X}.$

Let us define ${\bf V}_{n}:=\frac{1}{n}\sum_{k=1}^{n}{\bf U}_{n_{k}}$. Since
${\bf U}_{n_{k}}$ is also converges weakly to ${\bf U}$, it implies that ${\bf
V}_{n}$ converges weakly to ${\bf U}$ in $L^{1}(X)$. So sequence ${V_{n}}$ is
uniformly integrable in $L^{1}(X)$. As consequence of Vitali’s convergence
theorem implies that ${\bf V}_{n}$ converges to $\tilde{\bf U}$ strongly in
$L^{1}(X)$. Therefore, uniqueness of weak limit implies that ${\bf
U}=\tilde{\bf U}$ in $L^{1}(X)$. This concludes the proof. ∎

###  2.2 Background on Stochastic framework

Here we briefly recapitulate some basics of stochastic calculus in order to
define the cylindrical Wiener process $W$ and the stochastic integral
appearing in (1.1). To that context, let
$(\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},\mathbb{P})$ be a stochastic
basis with a complete, right-continuous filtration. The stochastic process $W$
is a cylindrical $(\mathbb{F}_{t})$-Wiener process in a separable Hilbert
space $\mathfrak{W}$. 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 $(\mathbb{F}_{t})_{t\geq 0}$ and
$\\{e_{k}\\}_{k\geq 1}$ is an orthonormal basis of $\mathfrak{W}$. To give the
precise definition of the diffusion coefficient $\Psi$, 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
$\,\Psi(\varrho,{\bf m}):\mathfrak{W}\rightarrow L^{1}(\mathbb{T}^{3})$ be
defined as follows

$\Psi(\varrho,{\bf m})e_{k}=\Psi_{k}(\cdot,\varrho(\cdot),{\bf m}(\cdot)).$

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

$\displaystyle\Psi_{k}(\cdot,0,0)$ $\displaystyle=0$ (2.1)
$\displaystyle|\partial_{\varrho}\Psi_{k}|+|\nabla_{{\bf m}}\Psi_{k}|$
$\displaystyle\leq\beta_{k},\quad\sum_{k\geq 1}\beta_{k}<\infty.$ (2.2)

As usual, 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
$\Psi(\varrho,\varrho\bf u)$ belongs to
$L_{2}(\mathfrak{W};W^{-m,2}(\mathbb{T}^{3}))$, the space of Hilbert–Schmidt
operators from $\mathfrak{W}$ to $W^{-m,2}(\mathbb{T}^{3})$. Consequently,
if111Here $\mathcal{P}$ denotes the predictable $\sigma$-algebra associated to
$(\mathbb{F}_{t})$.

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

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

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

is a well-defined $(\mathbb{F}_{t})$-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 $\mathfrak{W}_{0}\supset\mathfrak{W}$ via

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

endowed with the norm

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

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

For the convergence of approximate solutions, it is necessary to secure strong
compactness (a.s. convergence) in the $\omega$-variable. For that purpose, we
need a version of Skorokhod representation theorem, so-called Skorokhod-
Jakubowski representations theorem. Note that classical Skorokhod theorem only
works for Polish spaces, but in our analysis path spaces are so-called quasi-
Polish spaces. In this paper, we use the following version of the Skorokhod-
Jakubowski theorem, taken from Brzeźniak et.al. [10].

######  Theorem 2.5.

Let $\mathcal{X}$ be a complete separable metric space and $\mathcal{Y}$ be a
topological space such that there is a sequence of continuous functions
$g_{n}:\mathcal{Y}\rightarrow\mathbb{R}$ that separates points of
$\mathcal{Y}$. Let $(\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},\mathbb{P})$
be a stochastic basis with a complete, right-continuous filtration and
$(\xi_{n})_{n\in\mathbb{N}}$ be a tight sequence of random variables in
$(\mathcal{Z},\mathcal{B}(\mathcal{X})\otimes\mathcal{M})$, where
$\mathcal{Z}=\mathcal{X}\times\mathcal{Y}$ and $\mathcal{Z}$ is equipped with
the topology induced by the canonical projections
$\Pi_{1}:\mathcal{Z}\rightarrow\mathcal{X}$ and
$\Pi_{2}:\mathcal{Z}\rightarrow\mathcal{Y}$. Note that $\mathcal{M}$ is the
$\sigma$-algebra generated by the sequence $\xi_{n}$, ${n\in\mathbb{N}}$.

Assume that there exists a random variable $\eta$ in $\mathcal{X}$ such that
$\mathbb{P}^{\Pi_{1}\circ\xi_{n}}=\mathbb{P}^{\eta}$. Then there exists a
subsequence $(\xi_{n_{k}})_{k\in\mathbb{N}}$ and random variables
$\tilde{\xi}_{k},\tilde{\xi}$ in $\mathcal{Z}$ for $k\in\mathbb{N}$ on a
common probability space
$(\tilde{\Omega},\tilde{\mathbb{F}},\tilde{\mathbb{P}})$ with

  * (a)

${\tilde{\mathbb{P}}}^{\tilde{\xi}_{k}}=\mathbb{P}^{\xi_{n_{k}}}$

  * (b)

$\tilde{\xi}_{k}\rightarrow\tilde{\xi}$ in $\mathcal{Z}$ almost surely for
$k\rightarrow\infty$.

  * (c)

$\Pi_{1}\circ\tilde{\xi}_{k}=\Pi_{1}\circ\tilde{\xi}$ almost surely.

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

######  Lemma 2.6.

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

$\displaystyle\mathbb{E}[|X(t)-X(s)|^{a}]\leq C|t-s|^{1+b}.$

Then there exists a continuous modification of $X$ and the paths of $X$ are
$c$-Hölder continuous for every $c\in[0,\frac{b}{a})$.

###  2.3 Stochastic compressible Euler equations

Since we aim at proving pointwise convergence of numerical solutions to the
regular solution of the limit system, using the weak (measure-valued)–strong
uniqueness principle for dissipative measure-valued solutions, we first recall
the notion of local strong pathwise solution for stochastic compressible Euler
equations, being first introduced in [7]. 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 2.7 (Local strong pathwise solution).

Let $\left(\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},\mathbb{P}\right)$ be
a stochastic basis with a complete right-continuous filtration. Let ${W}$ be
an $(\mathbb{F}_{t})$-cylindrical Wiener process and
$(\varrho_{0},\mathbf{v}_{0})$ be a $W^{m,2}(\mathbb{T}^{3})\times
W^{m,2}(\mathbb{T}^{3})$-valued $\mathbb{F}_{0}$-measurable random variable,
for some $m>7/2$, and let $\Psi$ satisfy (2.1) and (2.2). A triplet
$(\varrho,{\bf v},\mathfrak{t})$ is called a local strong pathwise solution to
the system (1.1)–(1.2) provided

  1. 1.

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

  2. 2.

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

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

  3. 3.

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

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

  4. 4.

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

$\begin{split}\varrho(t\wedge\mathfrak{t})&=\varrho_{0}-\int_{0}^{t\wedge\mathfrak{t}}{\rm
div}(\varrho{\bf v})\ \mathrm{d}s,\\\ (\varrho{\bf
v})(t\wedge\mathfrak{t})&=\varrho_{0}{\bf
v}_{0}-\int_{0}^{t\wedge\mathfrak{t}}{\rm div}(\varrho{\bf v}\otimes{\bf v})\
\mathrm{d}s-\int_{0}^{t\wedge\mathfrak{t}}a\nabla\varrho^{\gamma}\
\mathrm{d}s+\int_{0}^{t\wedge\mathfrak{t}}{\Psi}(\varrho,\varrho{\bf v})\ {\rm
d}W,\end{split}$

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

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

######  Definition 2.8 (Maximal strong pathwise solution).

Fix a stochastic basis with a cylindrical Wiener process and an initial
condition as in Definition 2.7. A quadruplet

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

is a maximal strong pathwise solution to system (1.1)–(1.2) provided

  1. 1.

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

  2. 2.

$(\mathfrak{t}_{R})_{R\in\mathbb{N}}$ is an increasing sequence of
$(\mathbb{F}_{t})$-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

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

  3. 3.

each triplet $(\varrho,{\bf v},\mathfrak{t}_{R})$, $R\in\mathbb{N}$, is a
local strong pathwise solution in the sense of Definition 2.7.

There are quite a few results available in the literature concerning the
existence of maximal pathwise solutions for various SPDE or SDE models, see
for instance [9, 17]. For compressible Euler equations, a specific work can be
found in Breit $\&$ Mensah in [7, Theorem 2.4].

######  Theorem 2.9.

Let $m>7/2$ and the coefficients $\Psi_{k}$ satisfy hypotheses (2.1), (2.2)
and let $(\varrho_{0},\mathbf{v}_{0})$ be an $\mathbb{F}_{0}$-measurable,
$W^{m,2}(\mathbb{T}^{3})\times W^{m,2}(\mathbb{T}^{3})$-valued random variable
such that $\varrho_{0}>0$ $\mathbb{P}$-a.s. Then there exists a unique maximal
strong pathwise solution, in the sense of Definition 2.8, $(\varrho,{\bf
v},(\mathfrak{t}_{R})_{R\in\mathbb{N}},\mathfrak{t})$ to problem (1.1)–(1.2)
with the initial condition $(\varrho_{0},{\bf v}_{0})$.

###  2.4 Measure-valued solutions

For the introduction of measure-valued solutions, it is convenient to work
with the following reformulation of the problem (1.1)–(1.2) in the
_conservative_ variables $\varrho$ and ${\bf m}=\varrho{\bf u}$:

$\displaystyle{\rm d}\varrho+{\rm div}\,{\bf m}\,{\rm d}t$ $\displaystyle=0,$
(2.3) $\displaystyle{\rm d}{\bf m}+\left[{\rm div}\bigg{(}\frac{{\bf
m}\otimes{\bf m}}{\varrho}\bigg{)}+\nabla p(\varrho)\right]\,{\rm d}t$
$\displaystyle=\Psi(\varrho,{\bf m})\,{\rm d}W.$ (2.4)

Note that, in general any uniformly bounded sequence in
$L^{1}(\mathbb{T}^{3})$ does not immediately imply weak convergence of it due
to the presence of oscillations and concentration effects. To overcome such a
problem, two kinds of tools are used:

  * (a)

Young measures: these are probability measures on the phase space and accounts
for the persistence of oscillations in the solution;

  * (b)

Concentration defect measures: these are measures on physical space-time,
accounts for blow up type collapse due to possible concentration points.

####  2.4.1 Dissipative measure-valued martingale solutions

Keeping in mind the previous discussion, we now introduce the concept of
dissipative measure–valued martingale solution to the stochastic compressible
Euler system. In what follows, let

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

be the phase space associated to the Euler system.

######  Definition 2.10 (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
$\big{[}\big{(}\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq
0},\mathbb{P}\big{)};\mathcal{V}^{\omega}_{t,x},W\big{]}$ is a dissipative
measure-valued martingale solution of (2.3)–(2.4), with initial condition
$\mathcal{V}^{\omega}_{0,x}$; if

  1. (01)

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

  2. (02)

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

  3. (03)

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

  4. (04)

the average density $\langle\mathcal{V}^{\omega}_{t,x};\varrho\rangle$
satisfies
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};\varrho\rangle(t,\cdot),\varphi\rangle\in
C[0,T]$ for any $\varphi\in C^{\infty}(\mathbb{T}^{3})$ $\mathbb{P}$-a.s., the
function
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};\varrho\rangle(t,\cdot),\varphi\rangle$
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. (05)

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
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};\textbf{m}\rangle(t,\cdot),\bm{\varphi}\rangle$
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. (06)

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

  7. (07)

the integral identity

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{\tau,x};\varrho\rangle\,\varphi\,dx-\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{0,x};\varrho\rangle\,\varphi\,dx=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{t,x};\textbf{m}\rangle\cdot\nabla_{x}\varphi\,dx\,dt$
(2.5)

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

  8. (08)

the integral identity

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{\tau,x};\textbf{m}\rangle\cdot\bm{\varphi}dx-\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{0,x};\textbf{m}\rangle\cdot\bm{\varphi}dx$
(2.6)
$\displaystyle\qquad=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\left[\left\langle\mathcal{V}^{\omega}_{t,x};\frac{\textbf{m}\otimes\textbf{m}}{\varrho}\right\rangle:\nabla_{x}\bm{\varphi}+\langle\mathcal{V}^{\omega}_{t,x};p(\varrho)\rangle\operatorname{div}_{x}\bm{\varphi}\right]dxdt$
$\displaystyle\qquad\qquad+\int_{\mathbb{T}^{3}}\bm{\varphi}\,\int_{0}^{\tau}\left\langle\mathcal{V}^{\omega}_{t,x};{\Psi}(\varrho,\textbf{m})\right\rangle\,{\rm
d}W\,dx+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\nabla_{x}\bm{\varphi}:d\mu_{m},$

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. (09)

there exists a real-valued martingale $M_{E}$, such that the following energy
inequality

$\displaystyle\begin{aligned} \mathrm{E}(t+)&\leq\mathrm{E}(s-)\\\
&\quad+\frac{1}{2}\int_{s}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\left\langle\mathcal{V}^{\omega}_{\tau,x};\varrho^{-1}|\Psi_{k}(\varrho,\textbf{m})|^{2}\right\rangle\bigg{)}\,d\tau+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}d\mu_{e}+\int_{s}^{t}dM_{E}\end{aligned}$
(2.7)

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

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

Here $P(\varrho):=\frac{\varrho^{\gamma}}{\gamma-1}$, $\mu_{e}\in
L^{\infty}_{w^{*}}\big{(}[0,T];\mathcal{M}_{b}({\mathbb{T}^{3}})\big{)}$,
$\mathbb{P}$-a.s., $\mathcal{D}\in L^{\infty}(0,T)$, $\mathcal{D}\geq 0$,
$\mathbb{P}$-a.s., with initial energy.

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

  10. (10)

there exists a constant $C>0$ such that

$\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\mu_{m}|+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\mu_{e}|\leq
C\int_{0}^{\tau}\mathcal{D}(t)dt,$ (2.8)

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

######  Remark 2.11.

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

$\displaystyle\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{t,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{D}(t)$
(2.9)
$\displaystyle\qquad\leq\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};\frac{1}{2}\frac{|\textbf{m}|^{2}}{\varrho}+P(\varrho)\right\rangle\,dx+\mathcal{D}(s)+\frac{1}{2}\int_{s}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\left\langle\mathcal{V}^{\omega}_{\tau,x};\varrho^{-1}|\Psi_{k}(\varrho,\textbf{m})|^{2}\right\rangle\bigg{)}\,d\tau$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}d\mu_{e}+\int_{s}^{t}dM^{2}_{E},\,\,\mathbb{P}-a.s.$

However, to establish weak (measure-valued)–strong uniqueness principle, we
require energy inequality to hold for _all_ $s,t\in(0,T)$. This can be
achieved following the argument depicted in Section 5.

######  Remark 2.12.

Note that the above solution concept slightly differs from the dissipative
measure-valued martingale solution concept introduced by Martina et. al. [26].
Indeed, the main difference lies in the successful identification of the
martingale term present in (2.6).

###  2.5 Numerical scheme

It is well known that standard finite difference, finite volume and finite
element methods have been very successful in computing solutions to system of
hyperbolic conservation laws, including deterministic compressible fluid flow
equations. Here we consider a semi-discrete finite volume scheme for the
stochastic compressible Euler equations (1.1)–(1.2). In what follows, drawing
preliminary motivation from the analysis depicted in [18, 19, 20], we describe
the finite volume numerical scheme which is later shown to converge in
appropriate sense. More precisely, we show that the sequence of numerical
solutions generate the Young measure that represents the dissipative measure-
valued martingale solution.

####  2.5.1 Spatial discretization

We begin by introducing some notation needed to define the semi-discrete
finite volume scheme. Throughout this paper, we reserve the parameter $h$ to
denote small positive numbers that represent the spatial discretizations
parameter of the numerical scheme. Note that, since we are working in a
periodic domain in $\mathbb{R}^{3}$, the relevant domain for the space
discretization is $[0,\ell]^{3},$ $\ell>0$. To this end, we introduce the
space discretization by finite volumes (control volumes). For that we need to
recall the definition of so called admissible meshes for finite volume scheme.

######  Definition 2.13 (Admissible mesh).

An admissible mesh $\mathcal{T}$ of $[0,\ell]^{3}$ is a family of disjoint
regular quadrilateral connected subset of $[0,\ell]^{3}$ satisfying the
following:

  * i)

$[0,\ell]^{3}$ is the union of the closure of the elements (called control
volume K) of $\mathcal{T}$, i.e.,
$[0,\ell]^{3}:=\cup_{K\in\mathcal{T}}\bar{K}$.

  * ii)

The common interface of any two elements of $\mathcal{T}$ is included in a
hyperplane of $[0,\ell]^{3}$.

  * iii)

There exists nonnegative constant $\alpha$ such that

$\displaystyle\begin{cases}\alpha h^{3}\leq|K|,\\\ |\partial
K|\leq\frac{1}{\alpha}h^{2},\quad\forall K\in\mathcal{T},\end{cases}$

where $h=\sup\big{\\{}\text{diam}(K):K\in\mathcal{T}\big{\\}}<+\infty$, $|K|$
denotes the $3$-dimensional Lebesgue measure of $K$, and $|\partial K|$
represents the $2$-dimensional Lebesgue measure of $\partial K$.

In the sequel, we denote the followings:

  * •

$\mathrm{E}_{K}$: the set of interfaces of the control volume $K$.

  * •

$\mathcal{N}(K)$: the set of control volumes neighbors of the control volume
$K$.

  * •

$\sigma_{K,L}$: the common interface between $K$ and $L$, for any
$L\in\mathcal{N}(K)$.

  * •

$\mathrm{E}$: the set of all the interfaces of the mesh $\mathcal{T}$.

  * •

$\mathbf{n}_{K}^{+}$: the unit normal vector to interface $\sigma_{K,L}$,
oriented from $K$ to $L$, for any $L\in\mathcal{N}(K)$.

  * •

$\mathbf{e}_{p}$: the unit basis vector in the $p$-th space direction,
$p=1,2,3.$ Note that in our case the mesh is a regular quadrilateral grid, and
thus $\mathbf{n}_{K}^{+}$ is parallel to $\mathbf{e}_{p}$, for some $p=1,2,3.$

Let $\mathcal{Y}(\mathcal{T})$ denote the space of piecewise constant
functions defined on admissible mesh $\mathcal{T}.$ For
$w_{h}\in\mathcal{Y}(\mathcal{T})$ we set $\displaystyle w_{K}\equiv
w_{h_{|_{K}}}.$ Then it holds that

$\displaystyle\int_{\mathbb{T}^{3}}w_{h}\,{\rm
d}{x}=h^{3}\sum_{K\in\mathcal{T}}w_{K}.$

The value of $W_{h}$ on the face $\sigma$ shall be denoted by ${W}_{\sigma},$
and analogously for faces $s\pm$ of cell $K$ in $\pm\mathbf{e}_{s}$ direction.
We also introduce a standard projection operator

$\displaystyle\Pi_{h}:L^{1}(\mathbb{T}^{3})\rightarrow\mathcal{Y}(\mathcal{T}),\quad(\Pi_{h}(\varphi))_{K}\vcentcolon=\frac{1}{h^{3}}\int_{K}{\varphi(x)\,dx}.$

For $w_{h},W_{h}\in X(\mathcal{T})$ we define the following discrete operators

$\displaystyle\left(\widetilde{\partial_{h}^{p}}{w_{h}}\right)_{K}$
$\displaystyle\vcentcolon=\frac{w_{L}-w_{J}}{2h},\
\left(\partial_{h}^{p+}{w_{h}}\right)_{K}\vcentcolon=\frac{w_{L}-w_{K}}{h},\
\left(\partial_{h}^{p-}{w_{h}}\right)_{K}\vcentcolon=\frac{w_{K}-w_{J}}{h},\quad
L=K+h\mathbf{e}_{p},J=K-h\mathbf{e}_{p},$
$\displaystyle\left(\partial_{h}^{p}{W_{h}}\right)_{K}$
$\displaystyle\vcentcolon=\frac{W_{\sigma,p+}-W_{\sigma,p-}}{h},\quad
p=1,2,3.$

The discrete Laplace and divergence operators are defined as follows

$\displaystyle\left(\Delta_{h}\,w_{h}\right)_{K}$
$\displaystyle\vcentcolon=\frac{1}{h^{2}}\sum_{L\in\mathcal{N}(K)}(w_{L}-w_{K})=\sum_{p=1}^{3}\left(\Delta^{s}_{h}\,w_{h}\right)_{K},$
$\displaystyle\left(\widetilde{\textnormal{div}_{h}}\,{\bf w}_{h}\right)_{K}$
$\displaystyle\vcentcolon=\sum_{p=1}^{3}\left(\widetilde{\partial_{h}^{p}}{w_{h}^{p}}\right)_{K},\quad\left(\textnormal{div}_{h}\,{\bf
W}_{h}\right)_{K}\vcentcolon=\sum_{p=1}^{3}\left(\partial_{h}^{p}{W_{h}^{p}}\right)_{K}.$

Furthermore, on the face $\sigma=K|L\in\mathrm{E}$ we define the jump and mean
value operators

$\displaystyle\left\llbracket
w_{h}\right\rrbracket:=w_{L}\mathbb{n}_{K}^{+}+w_{K}\mathbb{n}_{K}^{-},\quad(\overline{w_{h}})_{\sigma}\vcentcolon=\frac{w_{K}+w_{L}}{2},\quad
L=K+h\mathbf{e}_{p},\ p=1,2,3,$

respectively. Here $\mathbf{n}_{K}^{+},$
$\mathbf{n}_{K}^{-}\equiv\mathbf{n}_{L}^{+}$ denote the unit outer normal to
$K$ and $L,$ respectively. Finally, we introduce the mean value of
$w_{h}\in\mathcal{Y}(\mathcal{T})$ in cell $K$ in the direction of
$\mathbf{e}_{p}$ by

$\displaystyle(\widetilde{w_{h}})^{p}_{K}:=\frac{w_{L}+w_{J}}{2},\quad
L=K+h\mathbf{e}_{p},\ J=K-h\mathbf{e}_{p}.$

####  2.5.2 Entropy stable flux and the scheme

Note that constructing and analyzing numerical schemes for the deterministic
counterpart of the underlying system of equations (1.1)–(1.2) has a long
tradition. Usually the schemes are developed to satisfy certain additional
properties like entropy condition and kinetic energy stability which can be
important for turbulent flows. To that context, Tadmor [35] proposed the idea
of entropy conservative numerical fluxes which can then be combined with some
dissipation terms using entropy variables to obtain a scheme that respects the
entropy condition, i.e., the scheme must produce entropy in accordance with
the second law of thermodynamics. Such a flux is called _entropy stable flux_.

In order to introduce the finite volume numerical scheme for the underlying
system of equations, let us first recast the system of equations (2.3)–(2.4)
in the following form:

$\displaystyle{\rm d}{\bf U}(t)+\mathrm{div}f({\bf U})\,dt$
$\displaystyle=\mathbb{H}\big{(}\varrho,{\bf m}\big{)}\,{\rm d}W(t),$
$\displaystyle{\bf U}(t,0)$ $\displaystyle={\bf U}_{0},$

where we introduced the variables ${\bf U}=[\varrho,{\bf m}]$, $f({\bf
U})=[{\bf m},\frac{{\bf m}\otimes{\bf m}}{\varrho}+p(\varrho)\mathbb{I}]$, and
$\mathbb{H}(\varrho,{\bf m})=[0,\Psi(\varrho,{\bf m})]$.

We propose the following semi-discrete (in space) finite volume scheme
approximating the underlying system of equations (2.3)–(2.4)

$\displaystyle{\rm d}{\bf
U}_{K}(t)+\left(\widetilde{\textnormal{div}_{h}}\,{\bf
F_{h}}(t)\right)_{K}\,dt$ $\displaystyle=\mathbb{H}\big{(}\varrho_{K}(t),{\bf
m}_{K}(t)\big{)}\,{\rm d}W(t),\,\,t>0,\,K\in\mathcal{T},$ (2.10)
$\displaystyle{\bf U}_{K}(0)$ $\displaystyle=(\Pi_{h}({\bf
U}_{0}))_{K},\,\,K\in\mathcal{T}.$

Note that (2.10) is a stochastic differential equation in
$V:=L^{\gamma}(\mathbb{T}^{3})\times
L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})$. Let us now specify the
numerical flux ${\bf F_{h}}:={\bf F_{h}}({\bf U}_{K},{\bf U}_{L})$ associated
to the flux function $f$. Indeed, we want ${\bf F_{h}}$ to satisfy the
following properties:

  * (a)

(Consistency) The function ${\bf F_{h}}$ satisfies ${\bf F_{h}}(a,a)=f(a)$,
for all $a\in V$.

  * (b)

(Lipschitz continuity) There exist two constants $F_{1},F_{2}>0$ such that for
any $a,b,c\in V$, it holds that

$\displaystyle\big{|}{\bf F_{h}}(a,b)-{\bf F_{h}}(c,b)\big{|}\leq
F_{1}\big{|}a-c|,$ $\displaystyle\big{|}{\bf F_{h}}(a,b)-{\bf
F_{h}}(a,c)\big{|}\leq F_{2}\big{|}b-c|.$

  * (c)

(Entropy stability) The flux ${\bf F_{h}}$ is entropy stable.

Note that there are plethora of numerical fluxes available in literature
satisfying the above three conditions. However, to illustrate the main ideas,
we will consider a scheme with a Lax-Friedrichs-type numerical flux
$\mathbf{F}_{h}$ (which is entropy stable) whose value on a face $\sigma=K|L$
is given by

$\displaystyle{\mathbf{F}}_{\sigma}\vcentcolon=(\overline{\mathbf{f}({\bf
U}_{h})})_{\sigma}-{\lambda}_{\sigma}\left\llbracket{\bf
U}_{h}\right\rrbracket.$ (2.11)

Here the global diffusion coefficient is
$\displaystyle{\lambda}_{\sigma}\equiv\lambda\vcentcolon=\max_{K\in\mathcal{T}}\max_{s=1,\ldots,N}|\lambda^{s}({\bf
U}_{K})|$, while the local diffusion coefficient is
$\displaystyle{\lambda}_{\sigma}\vcentcolon=\max_{s=1,\ldots,N}\max(|\lambda^{s}({\bf
U}_{K})|,|\lambda^{s}({\bf U}_{L})|).$ Note that $\lambda^{s}$ is the $s-$th
eigenvalue of the corresponding Jacobian matrix $\mathbf{f}^{\prime}({\bf
U}_{h})$. We mention that we restrict ourselves to the case of constant
numerical viscosities. However, one can easily extend the results to local
diffusion case, as presented in [19]. Using the above notation, we introduce
the following semi-discrete finite volume scheme to approximate system
(2.3)–(2.4). The scheme can be written in the standard per cell finite volume
formulation for all $K\in\mathcal{T},$

$\displaystyle{\rm
d}\varrho_{K}(t)+\sum_{\sigma\in\mathrm{E}(K)}\frac{|\sigma|}{|K|}F_{h}(\varrho_{h}(t),{\bf
u}_{h}(t))_{K}=0,$ (2.12a) $\displaystyle{\rm d}{\bf
m}_{K}(t)+\sum_{\sigma\in\mathrm{E}(K)}\frac{|\sigma|}{|K|}\left({\bf
F}_{h}(\varrho_{h}(t),{\bf m}_{h}(t))_{K}+\overline{p_{K}}(t){\bf
n}\right)=\Psi\big{(}\varrho_{K}(t),{\bf m}_{K}(t)\big{)}\,{\rm d}W(t).$
(2.12b)

We can rewrite the above scheme (2.12) in the following explicit form

$\displaystyle{\rm d}{\varrho_{K}(t)}$
$\displaystyle+\left(\widetilde{\textnormal{div}_{h}}\,{\bf
m}_{h}(t)\right)_{K}-\lambda h\left(\Delta_{h}\,\varrho_{h}(t)\right)_{K}=0,$
(2.13a) $\displaystyle{\rm d}{{\bf m}_{K}(t)}$
$\displaystyle+\left(\widetilde{\textnormal{div}_{h}}\,\left(\frac{{\bf
m}_{h}(t)\otimes{\bf
m}_{h}(t)}{\varrho_{h}(t)}+p_{h}(t)\mathbb{I}\right)\right)_{K}-\lambda
h\left(\Delta_{h}\,{\bf m}_{h}(t)\right)_{K}=\Psi\big{(}\varrho_{K}(t),{\bf
m}_{K}(t)\big{)}\,{\rm d}W(t),\ t>0\ K\in\mathcal{T}.$ (2.13b)

Existence of numerical solutions. Note that the set of equations (2.13)
represent a system of stochastic differential equations. The discrete problem
(2.13) admits a unique (probabilistically) strong solution
$(\varrho_{h}(t),{\bf m}_{h}(t))$ for every $t\in(0,T)$. This follows from the
classical results on stochastic differential equations, thanks to positivity
of the density $\varrho_{K}$ (cf. Lemma 3.1) and Lipschitz continuity of the
fluxes and the noise coefficient. For more details, we refer to [19, Section
4].

###  2.6 Statements of main results

We now state main results of this paper. To begin with, regarding the
convergence of solutions of the numerical scheme, we have the following
theorem.

######  Theorem 2.14.

Suppose that the approximate solutions $\\{{\bf U}_{h}=(\varrho_{h}(t),{\bf
m}_{h}(t))\\}_{h>0}$ be generated by the scheme (2.13) for the stochastic
Euler system. Moreover, assume that

$0<\underline{\varrho}\leq\varrho_{h}\leq\tilde{\varrho},\quad|{\bf
m}_{h}|\leq\tilde{{\bf m}},\,\,\mbox{$\mathbb{P}$-a.s. uniformly
for}\,\,h\rightarrow 0,$

for some positive constants $\underline{\varrho},\tilde{\varrho}$, and
$\tilde{{\bf m}}$. Then $\\{{\bf U}_{h}\\}_{h>0}$ generates a dissipative
measure-valued martingale solution to the barotropic Euler system in the sense
of Definition 2.10.

Next, we make use of the $\mathcal{K}$-convergence in the context of Young
measures to conclude the following pointwise convergence of averages of
numerical solutions to a dissipative measure-valued martingale solution to
(2.3)–(2.4).

######  Theorem 2.15.

Suppose that the approximate solutions $\\{{\bf U}_{h}=(\varrho_{h}(t),{\bf
m}_{h}(t))\\}_{h>0}$ to (2.13) for the stochastic Euler system generate a
dissipative measure-valued martingale solution
$\big{[}\big{(}\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq
0},\mathbb{P}\big{)};\mathcal{V}^{\omega}_{t,x},W\big{]}$ in the sense of
Definition 2.10. Then following holds true,

there exists subsequence $\\{\mathbf{U}_{h_{k}}=(\varrho_{h_{k}}(t),{\bf
m}_{h_{k}}(t))\\}_{h_{k}\textgreater\,0}$ such that, $\mathbb{P}$-a.s.

$\varrho_{h_{k}}\to\langle{\mathcal{V}^{\omega}_{t,x}};\varrho\rangle\,\,\mbox{in}\,\,\,C_{w}([0,T],L^{\gamma}(\mathbb{T}^{3})),$
$\mathbf{m}_{h_{k}}\to\langle{\mathcal{V}^{\omega}_{t,x}};{\textbf{m}}\rangle\,\,\mbox{in}\,\,\,C_{w}([0,T],L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})).$

$\mathbb{P}$-a.s., there exists subsequece
$\\{\mathbf{U}_{h_{k}}=(\varrho_{h_{k}}(t),{\bf
m}_{h_{k}}(t))\\}_{h_{k}\textgreater\,0}$ such that

$\displaystyle\frac{1}{N}\sum_{k=1}^{N}\varrho_{h_{k}}$
$\displaystyle\to\langle{\mathcal{V}^{\omega}_{t,x}};\varrho\rangle,\ \mbox{as
$N\rightarrow\infty$ a.e. in}\,\,(0,T)\times\mathbb{T}^{3},$
$\displaystyle\frac{1}{N}\sum_{k=1}^{N}{\bf m}_{h_{k}}$
$\displaystyle\to\langle{\mathcal{V}^{\omega}_{t,x}};{\textbf{m}}\rangle,\
\mbox{as $N\rightarrow\infty$ a.e. in}\,\,(0,T)\times\mathbb{T}^{3}.$

Finally, making use of the weak (measure-valued)–strong uniqueness principle
(cf. Theorem 6.2), we prove the following result justifying the strong
convergence to the regular solution.

######  Theorem 2.16.

Suppose that the approximate solutions $\\{{\bf U}_{h}\\}_{h>0}$ to (2.13) for
the stochastic Euler system generate a dissipative measure-valued martingale
solution in the sense of Definition 2.10. In addition, let the Euler equations
(2.3)–(2.4) possess the unique strong (continuously differentiable) solution
$(\bar{{\bf
U}},(\mathfrak{t}_{R})_{R\in\mathbb{N}},\mathfrak{t})=([\bar{\varrho},\bar{{\bf
m}}],(\mathfrak{t}_{R})_{R\in\mathbb{N}},\mathfrak{t})$, emanating form the
initial data (1.2). Then $\mathbb{P}$-a.s.

$\displaystyle\varrho_{h}(\cdot\wedge\mathfrak{t}_{R})$
$\displaystyle\to\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R})\ \mbox{weakly-(*)
in}\ L^{\infty}(0,T;L^{\gamma}(\mathbb{T}^{3}))\ \mbox{and strongly in}\
L^{1}((0,T)\times\mathbb{T}^{3}),$ $\displaystyle{\bf
m}_{h}(\cdot\wedge\mathfrak{t}_{R})$ $\displaystyle\to\bar{{\bf
m}}(\cdot\wedge\mathfrak{t}_{R})\ \mbox{weakly-(*) in}\
L^{\infty}(0,T;L^{2\gamma/(\gamma+1)}(\mathbb{T}^{3}))\ \mbox{and strongly
in}\ L^{1}((0,T)\times\mathbb{T}^{3};\mathbb{R}^{3})).$

######  Remark 2.17.

Note that the results stated in Theorem 2.16 are unconditional provided that:

  * (1)

the limit system admits a smooth solution.

  * (2)

numerical approximations generate a dissipative measure-valued martingale
solution.

##  3 Stability of the Numerical Scheme

We show the stability of the numerical schemes defined in previous Section by
deriving a priori estimates.

###  3.1 A priori estimates for the stochastic Euler system

The approximate solutions resulting from scheme (2.13) enjoy the following
properties:

  * 1.

Conservation of mass

Multiplying the equation of continuity in (2.13) by $h^{3}$ for all
$K\in\mathcal{T}$, and integrating in time yields the total mass conservation,
i.e., $\mathbb{P}$-a.s.

$\int_{\mathbb{T}^{3}}\varrho_{h}(t,\cdot)\,{\rm
d}{x}=\int_{\mathbb{T}^{3}}\varrho_{h}^{0}\,{\rm d}{x},\ t\geq 0.$

  * 2.

Conditional positivity of numerical density

We show positivity of the density under an additional hypothesis on the
approximate velocity. We assume that $\mathbb{P}$-a.s.

${\bf u}_{h}\equiv\frac{{\bf m}_{h}(t)}{\varrho_{h}(t)}\in
L^{2}(0,T;L^{\infty}(\Omega)).$ (3.1)

Thus the first two equations of the numerical scheme for the Euler system
read,

$\displaystyle\frac{\rm d}{{\rm d}t}{\varrho_{K}(t)}$
$\displaystyle+\left(\widetilde{\textnormal{div}_{h}}\,(\varrho_{h}(t){\bf
u}_{h}(t))\right)_{K}-\lambda h\left(\Delta_{h}\,\varrho_{h}(t)\right)_{K}=0,$
(3.2a) $\displaystyle\frac{\rm d}{{\rm d}t}{(\varrho_{K}(t){\bf u}_{K}(t))}$
$\displaystyle+\left(\widetilde{\textnormal{div}_{h}}\,\big{(}\varrho_{h}(t)({\bf
u}_{h}(t)\otimes{\bf u}_{h}(t))+p_{h}(t)\mathbb{I}\big{)}\right)_{K}-\lambda
h\left(\Delta_{h}\,(\varrho_{h}(t){\bf u}_{h}(t))\right)_{K}$
$\displaystyle\qquad\qquad=\Psi\big{(}\varrho_{K}(t),\varrho_{K}(t){\bf
u}_{K}(t)\big{)}\,{\rm d}W(t),$ (3.2b)

equipped with the relevant initial conditions.

######  Lemma 3.1.

Let $\varrho_{h}(0)>0,$ and let the numerical solution $(\varrho_{h}(t),{\bf
u}_{h}(t)),$ $t>0$ satisfy the discrete continuity equation (3.2a), where we
assume ${\bf u}_{h}$ satisfies (3.1). Then $\mathbb{P}$-a.s.

$\displaystyle{\varrho}_{K}(t)>\underline{\varrho}>0,\quad t\in[0,T],\
K\in\mathcal{T}.$

###### Proof.

To establish the proof, one can follow [19] modulo cosmetic changes. The
details are left to the interested reader. ∎

Note that, under the hypothesis (3.1), setting ${\bf
m}_{h}\equiv\varrho_{h}{\bf u}_{h}$ and comparing (3.2a) with (2.13a), we
conclude that both formulations are equivalent.

  * 3.

Energy estimates

First observe that the positivity of $\varrho_{h}(t)$ implies that
$\mathbb{P}$-almost surely $\varrho_{h}\in L^{\infty}(0,T;L^{1}(\Omega))$.
Next, we show that the underlying entropy stable finite volume scheme (2.13)
produces the discrete entropy inequality. To see this, let us denote by

$\displaystyle\eta({{\bf U}}_{K})=\frac{1}{2}\frac{|{{\bf
m}}_{K}|^{2}}{{\varrho}_{K}}+P({\varrho}_{K}),$

where ${\bf U}_{K}(t)$ solves the equation (2.10). Now applying Itô formula to
the function $\eta({\bf U}_{K}(t))$ and using entropy stability properties of
numerical flux functions [35, Example 5.2], we get the discrete energy
inequality

$\displaystyle{\rm d}{\eta({\bf U}}_{K}(t))$
$\displaystyle+\left(\textnormal{div}_{h}\,\mathbf{Q}_{h}(t)\right)_{K}\,dt$
(3.3) $\displaystyle\leq\,\sum_{k=1}^{\infty}\Psi_{k}(\varrho_{K}(t),{\bf
m}_{K}(t))\cdot{\bf u}_{K}(t)\,{\rm
d}W_{k}(t)+\frac{1}{2}\sum_{k=1}^{\infty}\varrho_{K}(t)^{-1}|\Psi_{k}(\varrho_{K}(t),{\bf
m}_{K}(t))|^{2}\,{\rm d}t,$

where $\mathbf{Q}_{h}$ is a entropy stable flux. Since the numerical entropy
flux is conservative, i.e.,
$\displaystyle\sum_{K\in\mathcal{T}}\left(\textnormal{div}_{h}\,\mathbf{Q}_{h}\right)_{K}=0,$
the integral of (3.3) yields $\mathbb{P}$-a.s.

$\displaystyle\int_{\mathbb{T}^{3}}{\eta({\bf U}_{h}}(t))dx$
$\displaystyle\leq\,\int_{\mathbb{T}^{3}}{\eta({\bf
U}}_{h}(0))dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\Psi_{k}(\varrho_{h}(s),{\bf
m}_{h}(s))\cdot{\bf u}_{h}(s)\,dx\,{\rm d}W_{k}(s)$ (3.4)
$\displaystyle\qquad+\frac{1}{2}\int_{0}^{t}\int_{\mathbb{T}^{3}}\sum_{k=1}^{\infty}\varrho_{h}(s)^{-1}|\Psi_{k}(\varrho_{h}(s),{\bf
m}_{h}(s))|^{2}\,{\rm d}x\,{\rm d}s.$

We can apply the $p$-th power on both sides of (3.4), and then take
expectation to obtain usual energy bounds. In particular, we have following
uniform bounds

$\displaystyle\frac{{\bf m}_{h}}{\sqrt{\varrho_{h}}}$ $\displaystyle\in
L^{p}(\Omega;L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))),$ (3.5)
$\displaystyle\varrho_{h}$ $\displaystyle\in
L^{p}(\Omega;L^{\infty}(0,T;L^{\gamma}(\mathbb{T}^{3}))),$ (3.6)
$\displaystyle{\bf m}_{h}$ $\displaystyle\in
L^{p}(\Omega;L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3}))),$
(3.7)

######  Remark 3.2.

Note that above estimates are natural in the context of stochastic
compressible Euler equations.

Let $\psi\in C_{c}^{\infty}([0,T))$, $\psi\geq 0$. Applying Itô product
formula to the function $\eta({\bf U}_{K}(t))\psi(t)$, we have energy
inequality

$\displaystyle\begin{aligned}
-\int_{0}^{T}\partial_{t}\psi\int_{\mathbb{T}^{3}}&\bigg{[}\frac{1}{2}\frac{|{\bf
m}_{h}|^{2}}{\varrho_{h}}+\frac{\varrho^{\gamma}_{h}}{\gamma-1}\bigg{]}\,dx\,ds\leq\psi(0)\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|{\bf
m}_{h}(0)|^{2}}{\varrho_{h}(0)}+\frac{\varrho^{\gamma}_{h}(0)}{\gamma-1}\bigg{]}\,{\rm
d}{x}\\\
&\qquad+\sum_{k=1}^{\infty}\int_{0}^{T}\psi\bigg{(}\int_{\mathbb{T}^{3}}\Psi_{k}(\varrho_{h},{\bf
m}_{h})\cdot{\bf u}_{h}\,{\rm d}{x}\bigg{)}{\rm
d}W_{k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\varrho_{h}^{-1}|\Psi_{k}(\varrho_{h},{\bf
m}_{h})|^{2}\,dx\,{\rm d}s.\end{aligned}$

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

  * 4.

Additional estimates

Regarding the regularity estimates for the discrete numerical solution, we
have the following lemma.

######  Lemma 3.3.

The following relevant estimates with $\Gamma=\frac{2\gamma}{\gamma+1}$ hold
$\mathbb{P}$-almost surely:

$\displaystyle\varrho_{h}\in L^{\infty}(0,T;L^{\gamma}(\mathbb{T}^{3})),$
$\displaystyle\quad{\bf m}_{h}\in
L^{\infty}(0,T;L^{\Gamma}(\mathbb{T}^{3})),\quad\widetilde{\textnormal{div}_{h}}\,{\bf
m}_{h}\in L^{\infty}(0,T;W^{-1,\Gamma}(\mathbb{T}^{3})),$
$\displaystyle\Delta_{h}\varrho_{h}\in
L^{\infty}(0,T;W^{-2,\gamma}(\mathbb{T}^{3})).$

###### Proof.

Note that first two estimates are direct consequences of discrete energy
bounds. Next, we note that for any test function $\varphi\in
W^{1,\Gamma^{\prime}}(\mathbb{T}^{3})$

$\displaystyle\left<\widetilde{\textnormal{div}_{h}}\,{\bf
m}_{h}(t),\varphi\right>$
$\displaystyle=h^{3}{\sum_{K\in\mathcal{T}}\left(\widetilde{\textnormal{div}_{h}}\,{\bf
m}_{h}(t)\right)_{K}{(\Pi_{h}\varphi)}_{K}}$
$\displaystyle=-{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}m_{K}^{s}(t)\left(\int_{K}{\frac{\varphi(x+h\mathbf{e}_{s})-\varphi(x-h\mathbf{e}_{s})}{2h}}\,{\rm
d}{x}\right)}$
$\displaystyle=-{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}m_{K}^{s}(t)\left(\frac{1}{2h}\int_{K}{\int_{x+h\mathbf{e}_{s}}^{x-h\mathbf{e}_{s}}\varphi_{x}(\theta)\,d\theta}\,{\rm
d}{x}\right)}\leq\frac{1}{2h}{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}m_{K}^{s}(t)\left(\int_{K}{h\varphi_{x}}\,{\rm
d}{x}\right)}$
$\displaystyle\leq\frac{h^{1+1/\Gamma}}{2h}{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}m_{K}^{s}(t)\left(\int_{K}{\varphi^{\Gamma^{\prime}}_{x}\,dx}\,{\rm
d}{x}\right)^{\frac{1}{\Gamma^{\prime}}}}\leq C(\Gamma)\,\|{\bf
m}_{h}\|_{\gamma}\|\varphi\|_{W^{1,\Gamma^{\prime}}(\mathbb{T}^{3})}.$

This confirms the third estimate. A similar argument yields the result for the
discrete Laplacian. ∎

##  4 Consistency of the Numerical Scheme

In this section we show consistency of the entropy stable finite volume
scheme. In addition, we also exhibit consistency of the energy inequality.

###  4.1 Consistency formulation of continuity and momentum equations

We begin by multiplying the continuity equation (2.13a) by
$h^{3}{(\Pi_{h}\varphi)}_{K},$ with $\varphi\in C^{3}(\mathbb{T}^{3}),$ and
the momentum equation or (2.13b) by $h^{3}{(\Pi_{h}\bm{\varphi})}_{K},$ with
$\bm{\varphi}\in C^{3}(\mathbb{T}^{3};\mathbb{R}^{3})$. Then we sum the
resulting equations over $K\in\mathcal{T}$ and integrate in time. For time
derivatives in the continuity and momentum equations, it is straightforward to
observe that

$\displaystyle h^{3}\int_{0}^{T}{\sum_{K\in\mathcal{T}}{\rm
d}{{\varrho}_{K}(t)}\,{(\Pi_{h}\varphi)}_{K}}\,{\rm d}t$
$\displaystyle=\int_{0}^{T}{{\rm
d}{}\bigg{(}\int_{\mathbb{T}^{3}}\varrho_{h}(t)\varphi(x)\,{\rm
d}{x}\bigg{)}}\,{\rm
d}t=\left<\varrho_{h}(T),\varphi\right>-\left<\varrho_{h}(0),\varphi\right>,$
$\displaystyle h^{3}\int_{0}^{T}{\sum_{K\in\mathcal{T}}{\rm d}{{{\bf
m}}_{K}(t)}\cdot{(\Pi_{h}\bm{\varphi})}_{K}}\,{\rm d}t$
$\displaystyle=\int_{0}^{T}{{\rm d}{}\bigg{(}\int_{\mathbb{T}^{3}}{\bf
m}_{h}(t)\cdot\bm{\varphi}(x)\,{\rm d}{x}\bigg{)}}\,{\rm d}t=\left<{\bf
m}_{h}(T),\bm{\varphi}\right>-\left<{\bf m}_{h}(0),\bm{\varphi}\right>.$

To handle the convective terms, we shall make use of the discrete integration
by parts and the Taylor expansion. For the continuity equation, we have

$\displaystyle
h^{3}\int_{0}^{T}{\sum_{K\in\mathcal{T}}\left(\widetilde{\textnormal{div}_{h}}\,{\bf
m}_{h}(t)\right)_{K}{(\Pi_{h}\varphi)}_{K}}\,{\rm d}t$
$\displaystyle=-\int_{0}^{T}{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}m_{K}^{s}(t)\left(\int_{K}{\frac{\varphi(x+h\mathbf{e}_{s})-\varphi(x-h\mathbf{e}_{s})}{2h}}\,{\rm
d}{x}\right)}\,{\rm d}t=-\int_{0}^{T}{\int_{\mathbb{T}^{3}}{\bf
m}_{h}(t)\cdot\nabla\,\varphi(x)\,{\rm d}{x}}\,{\rm
d}t+\mathcal{R}_{1}(h,\varphi),$

where term $\mathcal{R}_{1}(h,\varphi)$ is estimated as follows

$\displaystyle\mathcal{R}_{1}(h,\varphi)\leq C(\varphi)h\left\lVert{{\bf
m}_{h}}\right\rVert_{L_{t}^{\infty}L^{1}_{x}},\,\mathbb{P}\,\,\mbox{a.s.}$
(4.1)

Similarly, for the convective term in the momentum equations, we have

$\displaystyle
h^{3}\int_{0}^{T}{\sum_{K\in\mathcal{T}}\left(\widetilde{\textnormal{div}_{h}}\,\bigg{(}\frac{{\bf
m}_{h}(t)\otimes{\bf
m}_{h}(t)}{\varrho_{h}(t)}+p_{h}(t)\mathbb{I}\bigg{)}\right)_{K}{(\Pi_{h}\bm{\varphi})}_{K}}\,{\rm
d}t$
$\displaystyle=-\int_{0}^{T}{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}\sum_{z=1}^{3}\bigg{(}\frac{m_{h}^{s}(t)m_{h}^{z}(t)}{\varrho_{h}(t)}+p_{h}(t)\bigg{)}\left(\int_{K}{\frac{\varphi^{z}(x+h\mathbf{e}_{s})-\varphi^{z}(x-h\mathbf{e}_{s})}{2h}}\,{\rm
d}{x}\right)}\,{\rm d}t$
$\displaystyle=-\int_{0}^{T}{\int_{\mathbb{T}^{3}}\bigg{(}\frac{{\bf
m}_{h}(t)\otimes{\bf
m}_{h}(t)}{\varrho_{h}(t)}+p_{h}(t)\mathbb{I}\bigg{)}\cdot\nabla\,\bm{\varphi}(x)\,{\rm
d}{x}}\,{\rm d}t+\mathcal{R}_{2}(h,\bm{\varphi}),$

where the term $\mathcal{R}_{2}(h,\bm{\varphi})$ is bounded by

$\displaystyle\mathcal{R}_{2}(h,\bm{\varphi})\leq
C(\bm{\varphi})h\Bigg{\\{}\left\lVert{\sqrt{\varrho_{h}(t)}{\bf
u}_{h}(t)}\right\rVert_{L_{t}^{\infty}L_{x}^{2}}+\left\lVert{p_{h}(t)}\right\rVert_{L_{t}^{\infty}L^{1}_{x}}\Bigg{\\}},\,\mathbb{P}\,\,\mbox{a.s.}$

Next, regarding the numerical diffusion term with global numerical diffusion
coefficients $\lambda$, we have

$\displaystyle
h^{4}\int_{0}^{T}{\lambda\sum_{K\in\mathcal{T}}\left(\Delta_{h}\,{\bf
U}_{h}(t)\right)_{K}{(\Pi_{h}\bm{\varphi})}_{K}}\,{\rm d}t$
$\displaystyle=h^{4}\int_{0}^{T}{\lambda\sum_{K\in\mathcal{T}}{{\bf
U}}_{K}(t)\left(\int_{K}{\sum_{s=1}^{N}\frac{\bm{\varphi}(x+h\mathbf{e}_{s})-2\bm{\varphi}(x)+\bm{\varphi}(x-h\mathbf{e}_{s})}{h^{2}}}\,{\rm
d}{x}\right)}\,{\rm d}t$
$\displaystyle=h^{3}\int_{0}^{T}{\lambda\int_{\mathbb{T}^{3}}{\bf
U}_{h}(t)\Delta{\bm{\varphi}(x)}\,{\rm d}{x}}\,{\rm
d}t+\mathcal{N}(h,\bm{\varphi}),$

where the term $\mathcal{N}(h,\bm{\varphi})$ is bounded by

$\displaystyle\mathcal{N}(h,\bm{\varphi})\leq C(\bm{\varphi})h\left\lVert{{\bf
U}_{h}}\right\rVert_{L_{t}^{\infty}L_{x}^{1}}\int_{0}^{T}{\lambda}\,{\rm
d}t,\,\mathbb{P}\,\,\mbox{a.s.}$

Finally, regarding the stochastic term, we have the following

$\displaystyle
h^{3}\int_{0}^{T}\sum_{K\in\mathcal{T}}\Psi\big{(}\varrho_{K}(t),{\bf
m}_{K}(t)\big{)}\,(\Pi_{h}\bm{\varphi})_{K}\,{\rm
d}W(t)=\int_{0}^{T}\langle\Psi(\varrho_{h},\mathbf{m}_{h}),\bm{\varphi}\rangle\,\mathrm{d}W(t)$

Let us summarize the consistency results derived in this section.

#### Consistency formulation for the stochastic Euler system

The consistency formulation of the numerical schemes (2.13) for the barotropic
Euler equations reads

  1. 1.

for all $\varphi\in C^{\infty}(\mathbb{T}^{3})$ and $\bm{\varphi}\in
C^{\infty}(\mathbb{T}^{3})$ we have $\mathbb{P}$-a.s. for all $t\in[0,T]$

$\displaystyle\langle\varrho_{h}(t),\varphi\rangle$
$\displaystyle=\langle\varrho_{h}(0),\varphi\rangle-\int_{0}^{t}\langle\mathbf{m}_{h},\nabla\varphi\rangle\mathrm{d}s+\lambda\,h^{3}\int_{0}^{t}\langle\varrho_{h}(t),\Delta\varphi\rangle\,ds+\mathcal{R}_{1}(h,\varphi)+\mathcal{N}_{1}(h,\varphi)$
(4.2) $\displaystyle\langle\mathbf{m}_{h}(t),\bm{\varphi}\rangle$
$\displaystyle=\langle\mathbf{m}_{h}(0),\bm{\varphi}\rangle-\int_{0}^{t}\bigg{\langle}\Big{(}\frac{{\bf
m}_{h}\otimes{\bf
m}_{h}}{\varrho_{h}}+p(\varrho_{h})\mathbb{I}\Big{)},\nabla\bm{\varphi}\bigg{\rangle}\,\mathrm{d}s+\lambda\,h^{3}\int_{0}^{t}\langle\mathbf{m}_{h}(t),\Delta\bm{\varphi}\rangle\,ds$
$\displaystyle+\int_{0}^{t}\langle\Psi(\varrho_{h},\mathbf{m}_{h}),\bm{\varphi}\rangle\,\mathrm{d}W+\mathcal{R}_{2}(h,\bm{\varphi})+\mathcal{N}_{2}(h,\bm{\varphi}).$
(4.3)

  2. 2.

the energy inequality

$\displaystyle\begin{aligned}
-\int_{0}^{T}\partial_{t}\psi\int_{\mathbb{T}^{3}}&\bigg{[}\frac{|{\bf
m}_{h}|^{2}}{\varrho_{h}}+P(\varrho_{h})\bigg{]}\,dx\,ds\leq\psi(0)\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|{\bf
m}_{h}(0)|^{2}}{\varrho_{h}(0)}+P(\varrho_{h}(0))\bigg{]}\,{\rm d}{x}\\\
&\qquad+\sum_{k=1}^{\infty}\int_{0}^{T}\psi\bigg{(}\int_{\mathbb{T}^{3}}\Psi_{k}(\varrho_{h},{\bf
m}_{h})\cdot{\bf u}_{h}\,{\rm d}{x}\bigg{)}{\rm
d}W_{k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\varrho_{h}^{-1}|\Psi_{k}(\varrho_{h},{\bf
m}_{h})|^{2}\,dx\,{\rm d}s.\end{aligned}$ (4.4)

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

Here

$\displaystyle\mathcal{R}_{1}(h,\varphi)$
$\displaystyle:=\int_{0}^{T}{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}m_{K}^{s}(t)\int_{K}{\left(\nabla\varphi-\frac{\varphi(x+h\mathbf{e}_{s})-\varphi(x-h\mathbf{e}_{s})}{2h}\right)}\,{\rm
d}{x}}\,{\rm d}t,$ (4.5) $\displaystyle\mathcal{R}_{2}(h,\bm{\varphi})$
$\displaystyle:=\int_{0}^{T}{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}\sum_{z=1}^{3}\bigg{(}\frac{m_{h}^{s}(t)m_{h}^{z}(t)}{\varrho_{h}(t)}+p_{h}(t)\bigg{)}\int_{K}{\left(\nabla\bm{\varphi}-\frac{\varphi^{z}(x+h\mathbf{e}_{s})-\varphi^{z}(x-h\mathbf{e}_{s})}{2h}\right)}\,{\rm
d}{x}}\,{\rm d}t,$ $\displaystyle\mathcal{N}_{1}(h,\varphi)$
$\displaystyle:=\lambda\,h\int_{0}^{T}{\sum_{K\in\mathcal{T}}{\varrho}_{K}(t)\int_{K}{\sum_{s=1}^{3}\left(\Delta\varphi-\frac{{\varphi}(x+h\mathbf{e}_{s})-2{\varphi}(x)+{\varphi}(x-h\mathbf{e}_{s})}{h^{2}}\right)}\,{\rm
d}{x}}\,{\rm d}t,$ $\displaystyle\mathcal{N}_{2}(h,\bm{\varphi})$
$\displaystyle:=\lambda\,h\int_{0}^{T}{\sum_{K\in\mathcal{T}}\sum_{s=1}^{3}m_{K}^{s}(t)\int_{K}{\left(\Delta\bm{\varphi}-\frac{\bm{\varphi}(x+h\mathbf{e}_{s})-2\bm{\varphi}(x)+\bm{\varphi}(x-h\mathbf{e}_{s})}{h^{2}}\right)}\,{\rm
d}{x}}\,{\rm d}t.$

##  5 Proof of Theorem 2.14: Existence of Measure-Valued Solution

We shall make use of the given a-priori estimates (3.5), (3.6), and (3.7) to
pass to the limit in the parameter $h$. In what follows, we begin by the
following compactness argument.

###  5.1 Compactness and almost sure representations

Note that, in general, securing a result of compactness in the probability
variable ($\omega$-variable) is a non-trivial task. To that context, to obtain
strong (a.s.) convergence in the $\omega$-variable, we make use of Skorokhod-
Jakubowski’s representation theorem (cf. [27]). We remark that the classical
Skorokhod representation theorem does not work in our setup since our path
spaces are not Polish spaces. To overcome this, we use Jakubowski version of
Skorokhod representation theorem [27] which works for quasi-Polish spaces.

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

$\displaystyle\mathcal{Y}_{\varrho}$
$\displaystyle=C_{w}([0,T];L^{\gamma}(\mathbb{T}^{3})),$
$\displaystyle\mathcal{Y}_{N}$ $\displaystyle=C([0,T];\mathbb{R}),$
$\displaystyle\mathcal{Y}_{\textbf{m}}$
$\displaystyle=C_{w}([0,T];L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})),$
$\displaystyle\mathcal{Y}_{W}$ $\displaystyle=C([0,T];\mathfrak{W}_{0}),$
$\displaystyle\mathcal{Y}_{\mathcal{V}}$
$\displaystyle=\big{(}L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4})),w^{*}\big{)}.$

Let us denote by $\mu_{\varrho_{h}}$, $\mu_{\textbf{m}_{h}}$, and
$\mu_{W_{h}}$ respectively, the law of $\varrho_{h}$, $\textbf{m}_{h}$, and
$W_{h}$ on the corresponding path space. Moreover, let
$\mu_{{\mathcal{V}}_{h}}$, and $\mu_{N_{h}}$ denote the law of
${\mathcal{V}}_{h}:=\delta_{[\varrho_{h},\textbf{m}_{h}]}$, and
$N_{h}:=\sum_{k\geq 1}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf
u}_{h}\cdot{\Psi_{k}}(\varrho_{h},\textbf{m}_{h})\ dx\,\mathrm{d}W$ on the
corresponding path spaces. Finally, let $\mu^{h}$ denotes joint law of all the
variables on $\mathcal{Y}$. To proceed further, it is necessary to establish
tightness of $\\{\mu^{h};\,h\in(0,1)\\}$. To this end, we observe that
tightness of $\mu_{W_{h}}$ is immediate. So we show tightness of other
variables.

######  Proposition 5.1.

The sets $\\{\mu_{\varrho_{h}};\,h\in(0,1)\\}$, and
$\\{\mu_{\textbf{m}_{h}};\,h\in(0,1)\\}$ are tight on path spaces
$\mathcal{Y}_{\varrho}$, $\mathcal{Y}_{\mathbf{u}}$, and
$\mathcal{Y}_{\textbf{m}}$ respectively.

###### Proof.

Proof of this proposition is straightforward, by making use of the a priori
bounds given in Lemma 3.3. For the details of this proof, we refer to [8]. ∎

######  Proposition 5.2.

The set $\\{\mu_{{\mathcal{V}}_{h}};\,h\in(0,1)\\}$ is tight on the path space
$\mathcal{Y}_{\mathcal{V}}$.

###### 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

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

The proof is complete. ∎

######  Proposition 5.3.

The set $\\{\mu_{N_{h}};\,h\in(0,1)\\}$ is tight on the path space
$\mathcal{Y}_{N}$.

###### Proof.

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

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

and the Kolmogorov continuity criterion i.e., Lemma 2.6 applies. This in
particular implies that, for some $\alpha>1$

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

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

Combining all the informations obtained from Proposition 5.1, Proposition 5.2,
and Proposition 5.3, we conclude that

######  Corollary 5.4.

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

At this point, we are ready to apply Jakubowski-Skorokhod representation
theorem (see also Brzezniak et. al. [10]) to extract a.s convergence on a new
probability space. In what follows, passing to a weakly convergent subsequence
$\mu^{h}$ (and denoting by $\mu$ the limit law) we infer the following result:

######  Proposition 5.5.

There exists a subsequence $\mu^{h}$ (not relabelled), a probability space
$(\widetilde{\Omega},\widetilde{\mathbb{F}},\widetilde{\mathbb{P}})$ with
$\mathcal{Y}$-valued Borel measurable random variables
$(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h},\widetilde{W}_{h},\widetilde{N}_{h},{\mathcal{\widetilde{V}}}_{h})$,
$h\in(0,1)$, and $(\widetilde{\varrho},\widetilde{\bf
m},\widetilde{W},\widetilde{N},{\mathcal{\widetilde{V}}})$ such that

  1. (1)

the law of $(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h},\widetilde{W}_{h},\widetilde{N}_{h},{\mathcal{\widetilde{V}}}_{h})$ is
given by $\mu^{h}$, $h\in(0,1)$,

  2. (2)

the law of $(\widetilde{\varrho},\widetilde{\bf
m},\widetilde{W},\widetilde{N},{\mathcal{\widetilde{V}}})$, denoted by $\mu$,
is a Radon measure,

  3. (3)

$(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h},\widetilde{W}_{h},\widetilde{N}_{h},{\mathcal{\widetilde{V}}}_{h})$
converges $\,\widetilde{\mathbb{P}}$-almost surely to
$(\widetilde{\varrho},\widetilde{\bf
m},\widetilde{\textbf{u}},\widetilde{W},\widetilde{N},{\mathcal{\widetilde{V}}})$
in the topology of $\mathcal{Y}$, i.e.,

$\displaystyle\widetilde{\varrho}_{h}\rightarrow\widetilde{\varrho}\,\,\text{in}\,\,C_{w}([0,T];L^{\gamma}(\mathbb{T}^{3})),\quad$
$\displaystyle\widetilde{\textbf{m}}_{h}$
$\displaystyle\rightarrow\widetilde{\textbf{m}}\,\,\text{in}\,\,C_{w}([0,T];L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})),$
$\displaystyle\widetilde{N}_{h}\rightarrow\widetilde{N}\,\,\text{in}\,\,C([0,T];\mathbb{R}),\quad$
$\displaystyle\widetilde{W}_{h}$
$\displaystyle\rightarrow\widetilde{W}\,\,\text{in}\,\,C([0,T];\mathfrak{W}_{0})),$
$\displaystyle{\mathcal{\widetilde{V}}}_{h}\rightarrow{\mathcal{\widetilde{V}}}\,\,\text{weak-$*$
in}\,\,L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4})).$

  4. (4)

For every $h$, we have $\widetilde{W}_{h}=\widetilde{W}$ $\mathbb{P}$-a.s.

  5. (5)

For Carathéodory functions
$\underline{H}=\underline{H}(t,x,\varrho,\textbf{m})$ and
$\overline{H}=\overline{H}(t,x,\varrho,\textbf{m})$, 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

$\displaystyle|\underline{H}(t,x,\varrho,\textbf{m})|$ $\displaystyle\leq
1+|\varrho|^{p}+|\textbf{m}|^{q},$
$\displaystyle|\overline{H}(t,x,\varrho,\textbf{m})|$
$\displaystyle=\mathcal{O}\Big{(}1+|\varrho|^{\gamma}+|\textbf{m}|^{\frac{2\gamma}{\gamma+1}}\Big{)},$

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

$\displaystyle\underline{H}(\widetilde{\varrho}_{h},\widetilde{\textbf{m}}_{h})\rightarrow\overline{\underline{H}(\widetilde{\varrho},\widetilde{\textbf{m}})}\,\,\text{in}\,\,L^{r}((0,T)\times\mathbb{T}^{3}),\,\,\text{for
all}\,\,1<r\leq\frac{\gamma}{p}\wedge\frac{2\gamma}{q(\gamma+1)},$
$\displaystyle\overline{H}(\widetilde{\varrho}_{h},\widetilde{\textbf{m}}_{h})\rightharpoonup\left\langle\mathcal{\widetilde{V}}_{(\cdot,\cdot)};\overline{H}(\widetilde{\varrho}_{h},\widetilde{\textbf{m}}_{h})\right\rangle
dxdt+\widetilde{\mu}_{\overline{H}},\,\,\text{ weak-$*$
in}\,\,L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$

where $\mu_{\overline{H}}$ is the concentration defect measure associated to
the function $\overline{H}$.

###### Proof.

Proof of the items $(1)$, $(2)$, and $(3)$ directly follow from Jakubowski-
Skorokhod representation theorem [27], while item $(4)$ follows from [27], and
[10]. Finally, item $(5)$ follows from Lemma 2.2, and Lemma 2.3. ∎

####  5.1.1 Passage to the limit

We shall now make use of the above convergences to pass to the limit in
approximate equations (4.2)–(4.3), and the energy inequality (4.4). To that
context, let us first show that the approximations
$\widetilde{\varrho}_{h},\widetilde{u}_{h}$ solve equations (4.2)–(4.3) on the
new probability space
$(\widetilde{\Omega},\widetilde{\mathbb{F}},\widetilde{\mathbb{P}})$. Note
that, since $(\varrho_{h},{\bf m}_{h},N_{h})$ are random variables with values
in $C([0,T];L^{\gamma}(\mathbb{T}^{3}))\times
C([0,T];L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3}))\times
C([0,T];\mathbb{R})$. By [36, Lemma A.3] and [34, Corollary A.2],
$(\widetilde{\varrho}_{h},\widetilde{\bf m}_{h},\widetilde{N}_{h})$ are also
random variables with values in $C([0,T],L^{\gamma}(\mathbb{T}^{3}))\times
C([0,T],L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3}))\times
C([0,T];\mathbb{R})$. Let $(\widetilde{\mathbb{F}}_{t}^{h})$ be the
$\widetilde{\mathbb{P}}$-augmented canonical filtration of the process
$(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h},\widetilde{W},\widetilde{N}_{h})$, that is

$\begin{split}\widetilde{\mathbb{F}}_{t}^{h}&=\sigma\big{(}\sigma\big{(}\mathbf{r}_{t}\widetilde{\varrho}_{h},\,\mathbf{r}_{t}\widetilde{\bf
m}_{h},\,\mathbf{r}_{t}\widetilde{W},\mathbf{r}_{t}\widetilde{N}_{h}\big{)}\cup\big{\\{}N\in\widetilde{\mathbb{F}};\;\widetilde{\mathbb{P}}(N)=0\big{\\}}\big{)},\quad
t\in[0,T],\\\ \end{split}$

where we denote by $\mathbf{r}_{t}$ the operator of restriction to the
interval $[0,t]$ acting on various path spaces. Let us remark that by assuming
that the initial filtration $(\mathbb{F}_{t})$ is the one generated by $W$, by
[36, Lemma A.6], one can consider
$(\widetilde{\mathbb{F}}_{t}^{h})=(\widetilde{\mathbb{F}_{t}})$ is the
filtration generated by $\widetilde{W}$.

######  Proposition 5.6.

For every $h\in(0,1)$,
$\big{(}(\widetilde{\Omega},\widetilde{\mathbb{F}},(\widetilde{\mathbb{F}}_{t}),\widetilde{\mathbb{P}}),\widetilde{\varrho}_{h},\widetilde{\mathbf{u}}_{h},\widetilde{W}\big{)}$
is a finite energy weak martingale solution to (4.2)–(4.3) with the initial
law $\Lambda_{h}$.

###### Proof.

Proof of the above proposition directly follows form the Theorem 2.9.1 of the
monograph by Breit et. al. [8]. ∎

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 $\varphi\in C^{\infty}(\mathbb{T}^{3})$ and $\bm{\varphi}\in
C^{\infty}(\mathbb{T}^{3})$ we have $\mathbb{P}$-a.s. for all $t\in[0,T]$

$\displaystyle\langle\widetilde{\varrho}_{h}(t),\varphi\rangle$
$\displaystyle=\langle\widetilde{\varrho}_{h}(0),\varphi\rangle-\int_{0}^{t}\langle\widetilde{\bf
m}_{h},\nabla\varphi\rangle\mathrm{d}s+\lambda\,h^{3}\int_{0}^{t}\langle\widetilde{\varrho}_{h}(t),\Delta\varphi\rangle\,ds+\mathcal{\widetilde{R}}_{1}(h,\varphi)+\mathcal{\widetilde{N}}_{1}(h,\varphi)$
(5.1) $\displaystyle\langle\widetilde{\bf m}_{h}(t),\bm{\varphi}\rangle$
$\displaystyle=\langle\widetilde{\bf
m}_{h}(0),\bm{\varphi}\rangle-\int_{0}^{t}\bigg{\langle}\bigg{(}\frac{\widetilde{\bf
m}_{h}\otimes\widetilde{\bf
m}_{h}}{\widetilde{\varrho}_{h}}+\widetilde{p}_{h}\mathbb{I}\bigg{)},\nabla\bm{\varphi}\bigg{\rangle}\,\mathrm{d}s+\lambda\,h^{N}\int_{0}^{t}\langle\widetilde{\mathbf{m}}_{h}(t),\Delta\bm{\varphi}\rangle\,ds$
$\displaystyle+\int_{0}^{t}\langle\Psi(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h}),\bm{\varphi}\rangle\,\mathrm{d}\widetilde{W}_{h}+\mathcal{\widetilde{R}}_{2}(h,\bm{\varphi})+\mathcal{\widetilde{N}}_{2}(h,\bm{\varphi}),$
(5.2)

where
$\mathcal{\widetilde{R}}_{1}(h,\varphi),\mathcal{\widetilde{R}}_{2}(h,\bm{\varphi}),\mathcal{\widetilde{N}}_{1}(h,\varphi)$,
and $\mathcal{\widetilde{N}}_{2}(h,\bm{\varphi})$ are defined similarly as in
(4.5), in the new probability space.

  * •

the energy inequality holds

$\displaystyle\begin{aligned}
&-\int_{0}^{T}\partial_{t}\psi\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|\widetilde{\bf
m}_{h}|^{2}}{\widetilde{\varrho}_{h}}+\frac{\widetilde{\varrho}_{h}^{\gamma}}{\gamma-1}\bigg{]}\,dx\,ds\leq\psi(0)\int_{\mathbb{T}^{3}}\bigg{[}\frac{1}{2}\frac{|\widetilde{\bf
m}_{h}(0)|^{2}}{\widetilde{\varrho}_{h}(0)}+\frac{\widetilde{\varrho}_{h}^{\gamma}(0)}{\gamma-1}\bigg{]}\,{\rm
d}{x}\\\
&\qquad\qquad+\sum_{k=1}^{\infty}\int_{0}^{T}\psi\bigg{(}\int_{\mathbb{T}^{3}}\Psi_{k}(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h})\cdot\widetilde{\bf u}_{h}\,{\rm d}{x}\bigg{)}{\rm
d}\widetilde{W}_{h,k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\widetilde{\varrho}_{h}^{-1}|\Psi_{k}(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h})|^{2}\,dx\,{\rm d}s.\end{aligned}$ (5.3)

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

Next we would like to pass to the limit in $h$ in (5.1), (5.2), and (5.3). To
do this, we first recall that a-priori estimates (3.5)–(3.7) continue to hold
for the new random variables. Thus, making use of the item $(5)$ of
Proposition 5.5, we conclude that $\widetilde{\mathbb{P}}$-a.s.,

$\displaystyle\widetilde{\varrho}_{h}\rightharpoonup\langle{\mathcal{\widetilde{V}}^{\omega}_{t,x}};\widetilde{\varrho}\rangle,\,\,\text{weakly
in}\,\,L^{\gamma}((0,T)\times\mathbb{T}^{3}),$ (5.4)
$\displaystyle\widetilde{\textbf{m}}_{h}\rightharpoonup\langle{\mathcal{\widetilde{V}}^{\omega}_{t,x}};\widetilde{\textbf{m}}\rangle,\,\,\text{weakly
in}\,\,L^{\frac{2\gamma}{\gamma+1}}((0,T)\times\mathbb{T}^{3}).$ (5.5)

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

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

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

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

Note that, collacting all the previous informations, we can pass to the limit
in equation (5.1) to get $\mathbb{P}$-a.s.

$\displaystyle\int_{\mathbb{T}^{3}}\langle\widetilde{\mathcal{V}}^{\omega}_{\tau,x};\widetilde{\varrho}\rangle\,\varphi\,\,{\rm
d}{x}-\int_{\mathbb{T}^{3}}\langle\widetilde{\mathcal{V}}^{\omega}_{0,x};\widetilde{\varrho}\rangle\,\varphi\,\,{\rm
d}{x}=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\langle\widetilde{\mathcal{V}}^{\omega}_{s,x};\widetilde{\textbf{m}}\rangle\cdot\nabla_{x}\varphi\,dx\,{\rm
d}s$

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

Next, we move onto the martingale term
$\widetilde{M}_{h}:=\int_{0}^{t}\langle\Psi(\widetilde{\varrho}_{h},\widetilde{\bf
m}_{h}),\bm{\varphi}\rangle\,\mathrm{d}\widetilde{W}_{h}$ coming from the
momentum equation. Note that thanks to compact embedding given in Lemma 2.1,
we conclude that for each $t$, $\tilde{M}_{h}(t)\rightarrow\tilde{M}(t)$,
$\mathbb{P}$-a.s. in the topology of $W^{-m,2}(\mathbb{T}^{N})$. However, we
are interested in identifying $\tilde{M}(t)$. Indeed, we may apply item $(5)$
of Proposition 5.5, to the composition
$\Psi_{k}(\widetilde{\rho}_{h},\widetilde{\bf m}_{h})$, $k\in\mathbb{N}$. This
gives

$\Psi_{k}(\widetilde{\rho}_{h},\widetilde{\bf
m}_{h})\rightharpoonup\Big{\langle}{\mathcal{\widetilde{V}}}^{\omega}_{t,x};\Psi_{k}(\widetilde{\rho},\widetilde{\bf
m})\Big{\rangle}\,\,\mbox{weakly in}\,\,L^{q}((0,T)\times\mathbb{T}^{3}),$

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

$\displaystyle\widetilde{\mathbb{E}}\Big{[}\int_{0}^{T}\|\Psi(\widetilde{\rho}_{h},\widetilde{\bf
m}_{h})\|^{2}_{L_{2}(\mathcal{U};W^{-m,2})}\,dt\Big{]}\leq\widetilde{\mathbb{E}}\Big{[}\int_{0}^{T}(\widetilde{\rho}_{h})_{\mathbb{T}^{3}}\int_{\mathbb{T}^{3}}(\widetilde{\rho}_{h}+\widetilde{\rho}_{h}|\widetilde{\bf
u}_{h}|^{2})\,dx\,dt\Big{]}\leq c(r).$

This implies that

$\Psi_{k}(\widetilde{\rho},\widetilde{\bf
m})\rightharpoonup\Big{\langle}\mathcal{\widetilde{V}}^{\omega}_{t,x};\Psi_{k}(\widetilde{\rho},\widetilde{\bf
m})\Big{\rangle}\,\,\mbox{weakly
in}\,\,L^{2}(\Omega\times[0,T];W^{-m,2}(\mathbb{T}^{3})).$

Note that the Itô integral

$I_{t}:\varphi\rightarrow\int_{0}^{t}\varphi(s){\rm d}\widetilde{W}(s)$

is a linear and continuous (hence weakly continuous) map from
$L^{2}(\Omega\times[0,T];W^{-m,2}(\mathbb{T}^{3}))$ to
$L^{2}(\Omega;W^{-m,2}(\mathbb{T}^{3}))$. Therefore, we can make use of weak
continuity of Itô integral, and item $(4)$ of Proposition 5.5, to conclude
$I_{t}(\Psi_{k}(\widetilde{\rho}_{h},\widetilde{\bf m}_{h}))$ converges weakly
to
$I_{t}\big{(}\langle{\mathcal{\widetilde{V}}}^{\omega}_{t,x};\Psi_{k}(\widetilde{\rho},\widetilde{\bf
m})\rangle\big{)}$ in $L^{2}(\Omega;W^{-m,2}(\mathbb{T}^{3}))$. Collectting
all above informations, we can conclude that

$\displaystyle\int_{\Omega}\Big{(}\int_{\mathbb{T}^{3}}\langle\widetilde{\mathcal{V}}^{\omega}_{\tau,x};\widetilde{\textbf{m}}\rangle\cdot\bm{\varphi}\,dx-\int_{\mathbb{T}^{3}}\langle\widetilde{\mathcal{V}}^{\omega}_{0,x};\widetilde{\textbf{m}}\rangle\cdot\bm{\varphi}\,{\rm
d}x\Big{)}\alpha(\omega)\,{\rm d}\tilde{\mathbb{P}}(\omega)$ (5.6)
$\displaystyle\qquad=\int_{\Omega}\Big{(}\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\left[\left\langle\widetilde{\mathcal{V}}^{\omega}_{t,x};\frac{\widetilde{\textbf{m}}\otimes\widetilde{\textbf{m}}}{\widetilde{\varrho}}\right\rangle:\nabla{\bm{\varphi}}+\langle\widetilde{\mathcal{V}}^{\omega}_{t,x};p(\widetilde{\varrho})\rangle\operatorname{div}\bm{\varphi}\right]dxdt$
$\displaystyle\qquad\qquad+\int_{\mathbb{T}^{3}}\bm{\varphi}\,\int_{0}^{\tau}\left\langle\widetilde{\mathcal{V}}^{\omega}_{t,x};\Psi(\widetilde{\varrho},\widetilde{\textbf{m}})\right\rangle\,{\rm
d}\widetilde{W}\,dx+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\nabla\bm{\varphi}:d(\widetilde{\mu}_{C}+\widetilde{\mu}_{P}\mathbb{I})\Big{)}\alpha(\omega)\,{\rm
d}\widetilde{\mathbb{P}}(\omega),$

holds for all $\tau\in[0,T)$, for all $\alpha\in L^{2}(\widetilde{\Omega})$
and for all $\bm{\varphi}\in C^{\infty}(\mathbb{T}^{3};\mathbb{R}^{3})$. Since
$C^{\infty}(\mathbb{T}^{3})$ is separable space with sup norm, above equality
(5.6) implies that $\mathbb{P}$-a.s.

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

holds for all $\tau\in[0,T)$, and for all $\bm{\varphi}\in
C^{\infty}(\mathbb{T}^{3};\mathbb{R}^{3})$. where
$(\widetilde{\mu}_{C}+\widetilde{\mu}_{P}\mathbb{I})\in
L^{\infty}_{w^{*}}\big{(}[0,T];\mathcal{M}_{b}({\mathbb{T}^{3}}),\tilde{\mathbb{P}}$-a.s.,
is tensor-valued measure. Therefore we conclude that (2.5)-(2.6) holds.

Regarding the convergence of martingale term $\widetilde{N}_{h}$, appearing in
the energy inequality, we have following proposition.

######  Proposition 5.7.

For each $t$, $\widetilde{N}_{h}(t)\rightarrow\widetilde{N}(t)$ in
$\mathbb{R}$, $\mathbb{P}$-a.s., and $\widetilde{N}(t)$ is a real valued
square-integrable martingale.

###### Proof.

Note that, thanks to Proposition 5.5, we have the information
$\widetilde{N}_{h}\rightarrow\widetilde{N}$ $\mathbb{P}$-a.s. in
$C([0,T];\mathbb{R})$. To conclude that $\widetilde{N}(t)$ is a martingale, We
have to show that, $\mathbb{P}$-a.s.

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

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

$\widetilde{\mathbb{E}}\Big{[}\mathcal{I}_{A}\big{(}\widetilde{N}(t)-\widetilde{N}(s)\big{)}\Big{]}=0,$

Now using the fact that $\widetilde{N}_{h}(t)$ is a martingale, we know that

$\widetilde{\mathbb{E}}\Big{[}\mathcal{I}_{A}\big{(}\widetilde{N}_{h}(t)-\widetilde{N}_{h}(s)\big{)}\Big{]}=0,$

for all $A\in\mathcal{F}_{s}$. Note that for all $t\in[0,T]$
$\widetilde{N}_{h}(t)$ is uniform bounded in $L^{2}(\widetilde{\Omega})$, with
the help of Vitali’s convergence theorem, we can pass to the limit in $h$ to
conclude that $\widetilde{N}(t)$ is a martingale. ∎

######  Lemma 5.8.

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

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

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

###### Proof.

Following deterministic argument we can conclude that $\widetilde{\mu}_{E}$
dominates defect measures $\widetilde{\mu}_{C},\widetilde{\mu}_{P}$. To show
the dominance of $\widetilde{\mu}_{E}$ over $\widetilde{\mu}_{D}$, observe
that by virtue of hypotheses (2.1), (2.2), the function

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

and as such dominated by the total energy

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

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

To conclude (2.7), we proceed as follows. First note that we can pass to limit
in $h\to 0$ in (5.3) to obtain the following energy inequality in the new
probablity space.

$\displaystyle\begin{aligned}
-\int_{0}^{T}\partial_{t}\psi\bigg{[}\int_{\mathbb{T}^{3}}&\bigg{\langle}\widetilde{\mathcal{V}}_{t,x}^{\omega};\frac{|\widetilde{\bf
m}|^{2}}{\widetilde{\varrho}}+P(\widetilde{\varrho})\bigg{\rangle}dx+\widetilde{\mathcal{D}}(s)\bigg{]}\,ds\leq\psi(0)\int_{\mathbb{T}^{3}}\bigg{[}\bigg{\langle}\widetilde{\mathcal{V}}_{t,x}^{\omega};\frac{1}{2}\frac{|\widetilde{\bf
m}|^{2}}{\widetilde{\varrho}}+P(\widetilde{\varrho}(0))\bigg{\rangle}\bigg{]}\,{\rm
d}{x}\\\ &\qquad+\frac{1}{2}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}{\rm
d}\tilde{\mu}_{{D}}+\int_{0}^{T}\psi\,{\rm
d}\widetilde{N}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\psi\int_{\mathbb{T}^{3}}\bigg{\langle}\widetilde{\mathcal{V}}_{t,x};{\widetilde{\varrho}}^{-1}|\Psi_{k}(\widetilde{\varrho},\widetilde{\bf
m})|^{2}\bigg{\rangle}\,dx\,{\rm d}s.\end{aligned}$ (5.7)

holds $\mathbb{P}$-a.s., for all $\psi\in
C_{c}^{\infty}([0,T)),\,\psi\,\geq\,0.$ Fix any $s$ and $t$ such that
$0\,\textless\,s\,\textless\,t\,\textless\,T$. For any $r\,\textgreater\,0$
with $0\,\textless\,s-r\textless\,t+r\,\textless\,T$, let $\psi_{r}$ be a
Lipschitz fucntion that is linear on $[s-r,s]$ or $[t,t+r]$ and satiesfies

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

Then, $\psi_{r}$ is an admissible test fuction in (5.7), via a standard
regularization argument. From (5.7) with $\psi_{r}$ as test fuction, we have
$\mathbb{P}$-a.s, for all $t\in[0,T]$

$\displaystyle\frac{1}{r}\int_{t}^{t+r}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\frac{1}{2}\frac{|{\widetilde{\bf
m}}|^{2}}{{\widetilde{\varrho}}}+P(\widetilde{\varrho})\right\rangle\,{\rm
d}{x}+\mathcal{\widetilde{D}}(s)\bigg{)}{\rm d}\tau\,$
$\displaystyle\qquad\leq\,\frac{1}{r}\int_{t-r}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\widetilde{V}}^{\omega}_{\tau,x};\frac{1}{2}\frac{|\widetilde{{\bf
m}}|^{2}}{{\widetilde{\varrho}}}+P(\widetilde{\varrho})\right\rangle\,{\rm
d}{x}+\mathcal{\widetilde{D}}(s)\bigg{)}{\rm
d}\tau+\frac{1}{2}\int_{s-r}^{t+r}\psi_{r}(\tau)\int_{\mathbb{T}^{3}}\left\langle\widetilde{\mathcal{V}}^{\omega}_{\tau,x};{\widetilde{\varrho}^{-1}|{\Psi_{k}(\widetilde{\varrho},\widetilde{\mathbf{m}})}|^{2}}\right\rangle\,{\rm
d}{x}$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{s-r}^{t+r}\int_{\mathbb{T}^{3}}\psi_{r}(\tau){\rm
d}\widetilde{\mu}_{D}(x,\tau)+\int_{s-r}^{t+r}\psi_{r}(\tau){\rm
d}\widetilde{N}$ (5.8)

Now letting limit as $r\to 0^{+}$ in (5.1.1), then we have
$\mathbb{P}$-a.s,for all $t\in[0,T]$

$\displaystyle\lim_{r\to
0^{+}}\frac{1}{r}\int_{t}^{t+r}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\widetilde{V}}^{\omega}_{\tau,x};\frac{1}{2}\frac{|\widetilde{{\bf
m}}|^{2}}{\varrho}+P(\widetilde{\varrho})\right\rangle\,{\rm
d}{x}+\mathcal{\widetilde{D}}(s)\bigg{)}{\rm d}\tau\,$
$\displaystyle\qquad\leq\,\lim_{r\to
0^{+}}\frac{1}{r}\int_{t-r}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\widetilde{V}}^{\omega}_{\tau,x};\frac{1}{2}\frac{|{{\widetilde{\bf
m}}|^{2}}}{\widetilde{\varrho}}+P(\widetilde{\varrho})\right\rangle\,{\rm
d}{x}+\mathcal{\widetilde{D}}(s)\bigg{)}{\rm
d}\tau+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{\tau,x};{\widetilde{\varrho}^{-1}|{\Psi_{k}(\widetilde{\varrho},\widetilde{\mathbf{m}})}|^{2}}\right\rangle\,{\rm
d}{x}\,{\rm d}\tau$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}{\rm
d}\widetilde{\mu}_{D}(x,\tau)+\int_{s}^{t}{\rm d}\widetilde{N}$

Thus we conclude that (2.7) holds. If $s=0$, then we need different test
function to conclude result. In this case we take

$\psi_{r}(\tau)=\begin{cases}1,&\text{if}\,\,\tau\in[0,t]\\\
\text{linear},&\text{if}\,\,\tau\in[t,t+r]\\\ 0,&\text{Otherwise}.\end{cases}$

and apply the same argument as before.

##  6 Weak-Strong Uniqueness Principle

In this section, we establish pathwise weak (measure-valued)–strong uniqueness
principle for dissipative measure-valued martingale solutions. In what
follows, we first introduce the relative energy functional which plays a
pivotal role in the proof of weak (measure-valued)–strong uniqueness
principle. In the context of compressible Euler equations, relative energy
functional reads

$\displaystyle\mathfrak{E}^{\prime}_{\text{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(t):$
$\displaystyle=\int_{\mathbb{T}^{3}}\bigg{\langle}{\mathcal{V}^{\omega}_{t,x}};\frac{1}{2}{\frac{|\textbf{m}|^{2}}{\varrho}}+P(\varrho)\bigg{\rangle}{\rm
d}x-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\textbf{m}}\big{\rangle}\cdot\mathbf{Q}{\rm
d}x+\frac{1}{2}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}|\mathbf{Q}|^{2}{\rm
d}x$
$\displaystyle\qquad-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}P^{\prime}(s)\,{\rm
d}x-\int_{\mathbb{T}^{3}}[P^{\prime}(s)s-P(s)]{\rm d}x+\mathcal{D}(t).$ (6.1)

In view of the energy inequality (LABEL:energy_001), it is clear that the
above energy functional (6) is defined for all
$t\in[0,T]\setminus\mathcal{A}$, where the set $\mathcal{A}$, may depends on
$\omega$, has Lebesgue measure zero. We also define relative energy function
for all $t\in\mathcal{A}$ as follows

$\displaystyle\mathfrak{E}^{\prime\prime}_{\text{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(t):$
$\displaystyle=\lim_{r\to
0}\int_{t}^{t+r}\bigg{[}\int_{\mathbb{T}^{3}}\bigg{\langle}{\mathcal{V}^{\omega}_{s,x}};\frac{1}{2}{\frac{|\textbf{m}|^{2}}{\varrho}}+P(\varrho)\bigg{\rangle}{\rm
d}x+\mathcal{D}(s)\bigg{]}{\rm
d}s-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\textbf{m}}\big{\rangle}\cdot\mathbf{Q}{\rm
d}x$
$\displaystyle\qquad+\frac{1}{2}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}|\mathbf{Q}|^{2}{\rm
d}x-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\varrho\big{\rangle}P^{\prime}(s)\,{\rm
d}x-\int_{\mathbb{T}^{3}}[P^{\prime}(s)s-P(s)]{\rm d}x+\mathcal{D}(t).$ (6.2)

Using relative energy functionals (6)-(6), we define relative energy
functional for all time $t\in[0,T]$ as follows

$\displaystyle\mathfrak{E}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(t):=\begin{cases}\mathfrak{E}^{\prime}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(t),&\text{if}\,\,t\in[0,T]\setminus\mathcal{A}\\\
\mathfrak{E}^{\prime\prime}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(t),&\text{if}\,\,t\in\mathcal{A}\end{cases}$
(6.3)

With the help of the above definition of relative energy functional, we are
now in a position to derive the following relative energy inequality.

######  Proposition 6.1 (Relative Energy Inequality).

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

${\rm d}s=s_{1}{\rm d}t+s_{2}{\rm d}W,$ ${\rm d}\mathbf{Q}=\mathbf{Q}_{1}{\rm
d}t+\mathbf{Q}_{2}{\rm d}W$

with

$s\in C([0,T],W^{1,q}(\mathbb{T}^{3})),\,\,\mathbf{Q}\in
C([0,T];W^{1,q}(\mathbb{T}^{3}))\,\,\,\,\mathbb{P}-\text{a.s.}$
$\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|s\|_{W^{1,q}(\mathbb{T}^{3})}^{2}\bigg{]}^{q}+\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\mathbf{Q}\|_{W^{1,q}}^{2}\bigg{]}^{q}\,\leq\,C\,\,\text{for
all}\,\,\,2\leq\,q\,\textless\,\infty,$
$0\textless\,r_{1}\leq\,s(t,x)\leq\,r_{2}\,\,\,\mathbb{P}-\text{a.s.}$

Moreover, $s_{i},\mathbf{Q}_{i}$ for $i=1,2$, satisfy

$s_{1},\mathbf{Q}_{1}\in
L^{q}(\Omega;L^{q}(0,T;W^{1,q}(\mathbb{T}^{3})))\,\,\,\,s_{2},\mathbf{Q}_{2}\in
L^{2}(\Omega;L^{2}((0,T);L_{2}(\mathfrak{U};L^{2}(\mathbb{T}^{3}))),$
$\bigg{(}\sum_{k\geq\,1}|s_{2}(e_{k})|^{q}\bigg{)}^{1/q},\,\bigg{(}\sum_{k\geq\,1}|\mathbf{Q}_{2}(e_{k})|^{q}\bigg{)}^{1/q}\in
L^{q}(\Omega;L^{q}(0,T;L^{q}(\mathbb{T}^{3})))$

Then the following relative energy inequality holds $\mathbb{P}$-a.s., for all
$t\in[0,T]$

$\displaystyle\mathfrak{E}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(t)\leq\,\mathfrak{E}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(0)+\mathcal{M}_{RE}(t)+\int_{0}^{t}\mathfrak{R}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(\tau)\,{\rm
d}\tau$ (6.4)

where

$\displaystyle\begin{aligned}
\mathfrak{R}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,s,\mathbf{Q}\big{)}(t)&=\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\mathbf{Q}-\mathbf{m}\big{\rangle}\cdot[\mathbf{Q}_{1}+\nabla\mathbf{Q}\cdot\mathbf{Q}]{\rm
d}x+\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\frac{(\mathbf{m}-\varrho\mathbf{Q})\otimes(\varrho\mathbf{Q}-\mathbf{m})}{\varrho}\big{\rangle}:\nabla\mathbf{Q}{\rm
d}x\\\
&\qquad+\int_{\mathbb{T}^{3}}[(s-\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}P^{\prime\prime}(s)s_{1}+\nabla_{x}P^{\prime}(s)\cdot(s\mathbf{Q}-\big{\langle}\mathcal{V}_{t,x}^{\omega});\mathbf{m}\big{\rangle})]{\rm
d}x\\\
&\qquad+\int_{\mathbb{T}^{3}}[p(s)-\big{\langle}\mathcal{V}_{t,x}^{\omega};p(s)\big{\rangle}]{\rm
div}(\mathbf{Q}){\rm
d}x+\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\bigg{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\bigg{|}\frac{\Psi_{k}(\varrho,\mathbf{m})}{\varrho}-\mathbf{Q}_{2}(e_{k})\bigg{|}^{2}\bigg{\rangle}{\rm
d}x\\\
&\qquad+\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}P^{\prime\prime}(s)|s_{2}(e_{k})|^{2}\,{\rm
d}x+\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}p^{\prime\prime}(s)|s_{2}(e_{k})|^{2}\,{\rm
d}x\\\ &\qquad-\int_{\mathbb{T}^{3}}\nabla\mathbf{Q}:{\rm
d}\mu_{m}+\frac{1}{2}\int_{\mathbb{T}^{3}}d\mu_{e}.\end{aligned}$

Here $\mathcal{M}_{RE}$ is a real valued square integrable matingale.

###### Proof.

The proof of this proposition is a consequence of generalized Itô formula,
which is similar to the Lemma 4.1 in [26]. However, strictly speaking, the
proof given in [26] is based on a slightly different notion of dissipative
measure-valued martingale solutions. Therefore, for the sake of completness,
we briefly mention the proof. Note that given condtions on stochastic process
in this propostion allows us to apply Itô formula to compute
$\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\mathbf{m}\big{\rangle}\cdot\mathbf{Q}{\rm
d}x$. The result is

$\displaystyle{\rm
d}\bigg{(}\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\mathbf{m}\big{\rangle}\cdot\mathbf{Q}{\rm
d}x\bigg{)}=\int_{\mathbb{T}^{3}}\bigg{(}\big{\langle}\mathcal{V}_{t,x}^{\omega};\mathbf{m}\big{\rangle}\cdot\mathbf{Q}_{1}+\bigg{\langle}\mathcal{V}_{t,x}^{\omega};\frac{\mathbf{m}\otimes\mathbf{m}}{\varrho}\bigg{\rangle}:\nabla\mathbf{Q}+\big{\langle}\mathcal{V}_{t,x}^{\omega};p(\varrho)\big{\rangle}{\rm
div}\mathbf{Q}\bigg{)}{\rm d}x{\rm d}t$
$\displaystyle\qquad+\sum_{k\,\geq\,1}\int_{\mathbb{T}^{3}}\mathbf{Q}_{2}(e_{k})\cdot\big{\langle}\mathcal{V}_{t,x}^{\omega};\Psi_{k}(\varrho,\mathbf{m})\big{\rangle}{\rm
d}x{\rm d}t+\int_{\mathbb{T}^{3}}\nabla\mathbf{Q}:{\rm d}\mu_{m}{\rm
d}t+\int_{\mathbb{T}^{3}}\mathbf{Q}\cdot\big{\langle}\mathcal{V}_{t,x}^{\omega};\Psi(\varrho,\mathbf{m})\big{\rangle}{\rm
d}W$
$\displaystyle\qquad+\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\mathbf{m}\big{\rangle}\cdot\mathbf{Q}_{2}{\rm
d}x{\rm d}W.$ (6.5)

Similary, we get

$\displaystyle{\rm
d}\bigg{(}\int_{\mathbb{T}^{3}}\frac{1}{2}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}|\mathbf{Q}|^{2}{\rm
d}x\bigg{)}$
$\displaystyle=\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\mathbf{m}\big{\rangle}\cdot\nabla\mathbf{Q}\cdot\mathbf{Q}{\rm
d}x{\rm
d}t+\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}\mathbf{Q}\cdot\mathbf{Q_{1}}{\rm
d}x{\rm d}t$ (6.6)
$\displaystyle\qquad+\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}|\mathbf{Q}_{2}(e_{k})|^{2}{\rm
d}x{\rm
d}t+\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}\mathbf{Q}\cdot\mathbf{Q}_{2}{\rm
d}x{\rm d}W,$

and

$\displaystyle{\rm
d}\bigg{(}\int_{\mathbb{T}^{3}}\big{(}P^{\prime}(s)s-P(s)\big{)}{\rm
d}x\bigg{)}=\int_{\mathbb{T}^{3}}p^{\prime}(s)s_{1}{\rm d}x{\rm
d}t+\frac{1}{2}\sum_{k\,\geq\,1}\int_{\mathbb{T}^{3}}p^{\prime\prime}(s)|s_{2}(e_{k})|^{2}\,{\rm
d}x{\rm d}t+\int_{\mathbb{T}^{3}}p^{\prime}(s)s_{2}{\rm d}x{\rm d}W,$ (6.7)

and

$\displaystyle{\rm
d}\bigg{(}\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}P^{\prime}(s){\rm
d}x\bigg{)}$
$\displaystyle=\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\mathbf{m}\big{\rangle}\cdot\nabla_{x}P^{\prime}(s){\rm
d}x{\rm
d}t+\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}P^{\prime\prime}(s)s_{1}{\rm
d}x{\rm d}t$
$\displaystyle\qquad+\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}P^{\prime\prime}(s)|s_{2}(e_{k})|^{2}{\rm
d}x{\rm
d}t+\int_{\mathbb{T}^{3}}\big{\langle}\mathcal{V}_{t,x}^{\omega};\varrho\big{\rangle}P^{\prime\prime}(s)s_{2}{\rm
d}x{\rm d}W.$ (6.8)

Now we can combine (6.1)-(6.1) with (2.7) and define $\mathcal{M}_{RE}$ be the
sum of all martingale terms which come from (6.1)-(6.1) and (2.7) to obtain
(6.4). ∎

With the help of the Proposition 6.1, we now briefly describe the proof of the
weak (measure-valued)–strong uniqueness principle. For detials of the proof,
we refer to [8, Chapter 6] and [26].

######  Theorem 6.2 (Weak-Strong Uniqueness).

Let $\big{[}\big{(}\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq
0},\mathbb{P}\big{)};\mathcal{V}^{\omega}_{t,x},W\big{]}$ be a dissipative
measure-valued martingale solution to the system (1.1)–(1.2). On the same
stochastic basis $\big{(}\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq
0},\mathbb{P}\big{)}$, 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 a.e. $t\in[0,T]$ $\mathcal{D}(t\wedge\mathfrak{t}_{R})=0$,
$\mathbb{P}$-a.s., and $\mathbb{P}-\mbox{a.s.,}$

$\mathcal{V}^{\omega}_{t\wedge\mathfrak{t}_{R},x}=\delta_{\bar{\varrho}(t\wedge\mathfrak{t}_{R},x),(\bar{\varrho}\bar{\bf
u})(t\wedge\mathfrak{t}_{R},x)},\,\,\mbox{for a.e.
}(t,x)\in(0,T)\times\mathbb{T}^{3}.$ (6.9)

###### Proof.

Since $(\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R}),\bar{\mathbf{u}})$ is the
strong pathwise solution to stystem (1.1)-(1.2), so we can replace
$(s,\mathbf{Q})$ by
$(\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R}),\bar{\mathbf{u}})$ in the
relative energy inequality (6.4). Then we have $\mathbb{P}$-a.s., for all
$t\in[0,T]$,

$\displaystyle\mathfrak{E}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}(t\wedge\mathfrak{t}_{R})\leq\,\mathfrak{E}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}(0)+\mathcal{M}_{RE}(t\wedge\mathfrak{t}_{R})+\int_{0}^{t\wedge\mathfrak{t}_{R}}\mathfrak{R}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}(s){\rm
d}s,$ (6.10)

where
$\mathfrak{R}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}$
is given by (LABEL:reminder) after replacing $(s,\mathbf{Q})$ by
$(\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R}),\bar{\mathbf{u}})$. Following
[26] and [8, Chapter 6], one can verify that

$\displaystyle\int_{0}^{t\wedge\mathfrak{t}_{R}}\mathfrak{R}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}(s)\,{\rm
d}s\leq\,c(R)\int_{0}^{t\wedge\mathfrak{t}_{R}}\big{(}\mathfrak{E}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}(s)\,{\rm
d}s.$ (6.11)

In light of (6.10) and (6.11), a straightforward consequence of Gronwall’s
lemma yields, for all $t\in[0,T]$

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

Since initial data are same for both solutions, right hand side of above
inequality equals to zero. Therefore it implies that for all $t\in[0,T]$

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

This also implies that

$\lim_{r\to
0}\int_{t}^{t+r}\mathbb{E}\big{[}\mathfrak{E}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}(s\wedge\mathfrak{t}_{R})\big{]}{\rm
d}s=0.$

In view of a priori estimates, a usual Lebesgue point argument, and
application of Fubini’s theorem reveals that for a.e. $t\in[0,T]$,

$\mathbb{E}\big{[}\mathfrak{E}^{\prime}_{\mathrm{mv}}\big{(}\varrho,\mathbf{m}\,\big{|}\,\bar{\varrho},\bar{\mathbf{u}}\big{)}(t\wedge\mathfrak{t}_{R})\big{]}=0$

Since the defect measure $\mathcal{D}\geq\,0$, we have for a.e. $t\in[0,T]$,
$\mathcal{D}(t\wedge\mathfrak{t}_{R})=0$, $\mathbb{P}$-a.s. Moreover,
$\mathbb{P}-\mbox{a.s.}$

$\mathcal{V}^{\omega}_{t\wedge\mathfrak{t}_{R},x}=\delta_{\bar{\varrho}(t\wedge\mathfrak{t}_{R},x),(\bar{\varrho}\bar{\bf
u})(t\wedge\mathfrak{t}_{R},x)},\,\,\mbox{for a.e.
}(t,x)\in(0,T)\times\mathbb{T}^{3}.$

This finishes the proof of the theorem. ∎

##  7 Proof of Theorem 2.15: Convergence to dissipative solution

In view of the Proposition 5.5 and convergence results given by (5.4)-(5.5),
we conclude that there is subsequence $\\{(\varrho_{h_{k}}(t),{\bf
m}_{h_{k}}(t))\\}_{h_{k}>0}$ such that $\mathbb{P}$-a.s,

$\varrho_{h_{k}}\to\langle{\mathcal{V}^{\omega}_{t,x}};\varrho\rangle\,\,\mbox{in}\,\,\,C_{w}([0,T],L^{\gamma}(\mathbb{T}^{3})),$
$\mathbf{m}_{h_{k}}\to\langle{\mathcal{V}^{\omega}_{t,x}};{\textbf{m}}\rangle\,\,\mbox{in}\,\,\,C_{w}([0,T],L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})).$

For the pointwise converegnce of numerical approximations, we can make use of
Proposition 2.4. Indeed, we obtain $\mathbb{P}$-a.s., there exists a
subsequece $\\{(\varrho_{h_{k}}(t),{\bf m}_{h_{k}}(t))\\}_{h_{k}>0}$ such that

$\displaystyle\frac{1}{N}\sum_{k=1}^{N}\varrho_{h_{k}}$
$\displaystyle\to\langle{\mathcal{V}^{\omega}_{t,x}};\varrho\rangle,\ \mbox{as
$N\rightarrow\infty$ a.e. in}\,\,(0,T)\times\mathbb{T}^{3},$
$\displaystyle\frac{1}{N}\sum_{k=1}^{N}{\bf m}_{h_{k}}$
$\displaystyle\to\langle{\mathcal{V}^{\omega}_{t,x}};{\textbf{m}}\rangle,\
\mbox{as $N\rightarrow\infty$ a.e. in}\,\,(0,T)\times\mathbb{T}^{3}.$

##  8 Proof of Theorem 2.16: Convergence to Regular Solution

We have proven that the numerical solutions $\\{{\bf U}_{h}\\}_{h>0}$ to
(2.13) for the stochastic Euler system converges to the dissipative
measure–valued martingake solution, in the sense of Definition 2.10. Employing
the corresponding weak (measure-valued)–strong uniqueness results (cf. Theorem
6.2), we can show the strong convergence of numerical approximations to the
strong solution of the system on its lifespan.

First note that, Proposition 5.5 and Theorem 6.2 gives the required weak-$*$
convergence. Indeed, from Proposition 5.5, we have $\mathbb{P}$-a.s.,

$\varrho_{h}(\cdot\wedge\mathfrak{t}_{R})\to\langle{\mathcal{V}^{\omega}_{t,x}};\varrho\rangle(\cdot\wedge\mathfrak{t}_{R})\,\,\mbox{in}\,\,\,C_{w}([0,T],L^{\gamma}(\mathbb{T}^{3})),$
$\mathbf{m}_{h}(\cdot\wedge\mathfrak{t}_{R})\to\langle{\mathcal{V}^{\omega}_{t,x}};{\textbf{m}}\rangle(\cdot\wedge\mathfrak{t}_{R})\,\,\mbox{in}\,\,\,C_{w}([0,T],L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^{3})).$

Combination of above convergence and Theorem 6.2 gives the required weak-*
convergence. For the proof of strong convergence of density and momentum in
$L^{1}(\mathbb{T}^{3})$, note that from Proposition 5.5, Theorem 6.2, energy
bounds (3.5)-(3.7) and using the fact limit Young measure of any subsequence
$(\delta_{\varrho_{h_{k}}(\cdot\wedge\mathfrak{t}_{R}),\mathbf{m}_{h_{k}}(\cdot\wedge\mathfrak{t}_{R})})_{k\geq\,1}$
is
$\delta_{\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R}),\bar{\mathbf{m}}(\cdot\wedge\mathfrak{t}_{R})}$,
we have $\mathbb{P}$-a.s., sequence of young measure converges to dirac Young
measure, i.e. $\mathbb{P}$-a.s.

$\delta_{\varrho_{h}(\cdot\wedge\mathfrak{t}_{R}),\mathbf{m}_{h}(\cdot\wedge\mathfrak{t}_{R})}\rightarrow\delta_{\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R}),\bar{\mathbf{m}}(\cdot\wedge\mathfrak{t}_{R})},\,\,\text{weak-$*$
in}\,\,L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{4}))$

By theory of Young measure [1, Proposition 4.16], it implies that,
$\mathbb{P}$-a.s. $\varrho_{h}(\cdot\wedge\mathfrak{t}_{R})$, ${\bf
m}_{h}(\cdot\wedge\mathfrak{t}_{R})$ converges to
$\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R}),\bar{\mathbf{m}}(\cdot\wedge\mathfrak{t}_{R})$
in measure respectively. Note that, $\mathbb{P}$-a.s. sequence
$(\varrho_{h}(\cdot\wedge\mathfrak{t}_{R})$, ${\bf
m}_{h}(\cdot\wedge\mathfrak{t}_{R}))$ is uniformly integrable and converges in
measure, therefore Vitali’s convergence theorem implies that $\mathbb{P}$-a.s,

$\displaystyle\varrho_{h}(\cdot\wedge\mathfrak{t}_{R})$
$\displaystyle\to\bar{\varrho}(\cdot\wedge\mathfrak{t}_{R})\ \mbox{ strongly
in}\ L^{1}((0,T)\times\mathbb{T}^{3}),$ $\displaystyle{\bf
m}_{h}(\cdot\wedge\mathfrak{t}_{R})$
$\displaystyle\to\bar{\mathbf{m}}(\cdot\wedge\mathfrak{t}_{R})\ \mbox{
strongly in}\ L^{1}((0,T)\times\mathbb{T}^{3};\mathbb{R}^{3})).$

This finishes the proof of the theorem.

## Acknowledgements

U.K. acknowledges the support of the Department of Atomic Energy, Government
of India, under project no.$12$-R$\&$D-TFR-$5.01$-$0520$, and India SERB
Matrics grant MTR/$2017/000002$.

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