# On weak-strong uniqueness for stochastic equations of incompressible fluid
flow

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

###### Abstract

We introduce a novel concept of _dissipative measure-valued martingale
solution_ to the stochastic Euler equations describing the motion of an
_inviscid incompressible_ fluid. These solutions are characterized by a
parametrized Young measure and a concentration defect measure in the total
energy balance. Moreover, they are weak in the probablistic sense i.e., the
underlying probablity space and the driving Wiener process are intrinsic part
of the solution. In a significant departure from the existing literature, we
first exhibit the relative energy inequality for the incompressible Euler
equations driven by a _multiplicative_ noise, and then demonstrate pathwise
weak-strong uniqueness principle. Finally, we also provide a sufficient
condition, á la Prodi [31] $\&$ Serrin [33], for the uniqueness of weak
martingale solutions to stochastic Naiver–Stokes system in the class of finite
energy weak martingale solutions.

† Centre for Applicable Mathematics, Tata Institute of Fundamental Research

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

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

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

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

###### Contents

  1. 1 Introduction
  2. 2 Technical Framework and Main Results
    1. 2.1 Analytic framework
      1. 2.1.1 Young measures, concentration defect measures
    2. 2.2 Basics of stochastic framework
    3. 2.3 Stochastic incompressible Euler equations
    4. 2.4 Stochastic incompressible Navier–Stokes equations
    5. 2.5 Measure-valued martingale solutions
      1. 2.5.1 Dissipative measure-valued martingale solutions
    6. 2.6 Statements of main results
  3. 3 Proof of Theorem 2.17
    1. 3.1 A priori Bounds
    2. 3.2 Tightness and almost sure representations
      1. 3.2.1 Passing to the limit
  4. 4 Weak-Strong Uniqueness Principle for Euler System
    1. 4.1 Relative Energy Inequality (Euler System)
    2. 4.2 Proof of Theorem 2.18
  5. 5 Weak-Strong Uniqueness for Navier–Stokes System
    1. 5.1 Proof of Theorem 2.19

##  1 Introduction

In this paper, we introduce a notion of _dissipative measure-valued solution_
for the stochastically forced system of the _incompressible Euler system_
describing by the velocity vector field ${\bf u}$ of a fluid and the scalar
pressure field $p$. The system of equations reads

$\begin{cases}{\rm d}{\bf u}(t,x)+\left[{\rm div}({\bf u}(t,x)\otimes{\bf
u}(t,x))+\nabla p(t,x)\right]\,{\rm d}t=\mathcal{G}({\bf u}(t,x))\,{\rm
d}W(t),&\quad\mbox{in}\,\,\Pi_{T},\\\ {\rm div}\,{\bf
u}(t,x)=0,&\quad\mbox{in}\,\,\Pi_{T},\\\ {\bf u}(0,x)={\bf
u}_{0}(x),&\quad\mbox{in}\,\,\mathbb{T}^{3},\end{cases}$ (1.1)

where $\Pi_{T}:=\mathbb{T}^{3}\times(0,T)$ with $T>0$ fixed, $u_{0}$ is the
given random initial function with sufficient spatial regularity to be
specified later. Let
$\big{(}\Omega,\mathbb{F},\mathbb{P},(\mathbb{F}_{t})_{t\geq 0}\big{)}$ be a
stochastic basis, where $\big{(}\Omega,\mathbb{F},\mathbb{P}\big{)}$ is a
probability space and $(\mathbb{F}_{t})_{t\geq 0}$ is a complete filtration
with the usual assumptions. We assume that $W$ is a cylindrical Wiener process
defined on the probability space $(\Omega,\mathbb{F},\mathbb{P})$, and the
coefficient $\mathcal{G}$ is generally nonlinear and satisfies suitable growth
assumptions (see Section 2 for the complete list of assumptions). In
particular, the map ${\bf u}\mapsto\mathcal{G}({\bf u})$ is a Hilbert space
valued function signifying the _multiplicative_ nature of the noise.

The Euler equations (1.1) are the classical model for the motion of an
inviscid, incompressible fluid. The presence of stochastic terms in the
governing equations accounts for numerical, physical and empirical
uncertainties in various real life applications. The theory for deterministic
counterpart of (1.1) has experienced a substancial progress in past decade,
and has reached some level of maturity thanks to pioneering work by De Lellis
$\&$ Szekelyhidi [15, 16]. In a nutshell, these results show that for suitable
(large class of) initial data there exists _infinitely many_ weak solutions,
even if the solution satisfies an _entropy condition_. Indeed, it was Scheffer
[32] (see also [34]) who first constructed a nontrival weak solution of the
two dimensional incompressible Euler equation with compact support in time. In
other words, recent results by De Lellis $\&$ Szekelyhidi, and others [9, 10,
12] suggest that non-uniqueness of weak solutions to incompressible Euler
equations in several space dimensions is a fact of life. However, for
_general_ initial data, the existence of global-in-time weak solutions is
still unknown. In quest for a global-in-time solution, we recall the framework
of _measure-valued_ solutions, as introduced by Diperna $\&$ Majda [18] (see
also [17]) for the incompressible Euler equations. Despite being a weaker
notion of solution, thanks to the work by Brenier et. al. [8], it is well-
known that these measure-valued solutions of incompressible Euler equations
enjoy a remarkable _weak-strong uniqueness_ property. Moreover, recent work by
Szekelyhidi $\&$ Wiedemann [36] confirms that the notions of weak solutions
and measure-valued solutions coincide for incompressible Euler equations.

In the stochastic set-up, existence of _pathwise_ strong solutions (defined
upto a stopping time) for incompressible Euler equations (1.1) driven by a
_multiplicative noise_ , in a three dimensional smooth bounded domain, was
established by Glatt-Holtz $\&$ Vicol [21]. Moreover, a stochastic variant of
deterministic results, as developed by Buckmaster $\&$ Vicol in [9, 10], for
the stochastic incompressible Euler system have been recently established by
Hofmanova. et. al. in [23]. In fact, by making use of method of convex
integration, they have constructed infinitely many solutions to the
incompressible Euler system with a random forcing. Note that, although
measure-valued solution for the deterministic counterpart of (1.1) has a long
and intense history (see Lions [28]), their formulation (in the multiplicative
case) seems rather intricating in the stochastic setting. Indeed, there has
been a number of attemps to define a suitable notion of measure-valued
solutions for the stochastic incompressible Euler equations driven by
_additive_ noise, starting from the work of Kim [25], Breit $\&$ Moyo [5], and
most recently by Hofmanova et. al. [23], where the authors introduced a class
of dissipative solutions which allowed them to demostrate weak-strong
uniqueness property and non-uniqueness of solutions in law. However, none of
the above mentioned framework can be applied to (1.1) since the driving noise
is _multiplicative_ in nature.

In this paper, our first objective is to introduce a novel framework of
_dissipative measure-valued_ solutions for incompressible Euler equations
(1.1) driven by a _multiplicative_ stochastic perturbation of Nemytskii type,
which will facilitate to demonstrate weak-strong uniquness principle. These
solutions are measure-valued solutions of (1.1) augmented with an appropriate
form of energy inequality. The main advantage of this class of solutions is
that, for any finite energy initial data, they can be shown to exist globally-
in-time. The existence of such solutions is addressed through the well known
vanishing viscosity method with the help of a stochastic compactness argument
combining Prohorov theorem, and Skorokhod-Jakubowski [24] representation
theorem. In other words, measure-valued solutions for (1.1) is obtained
through a sequence of solutions of Navier–Stokes system subject to stochastic
forcing given by

$\displaystyle{\rm d}{\bf u}_{\varepsilon}+\left[{\rm div}({\bf
u}_{\varepsilon}\otimes{\bf u}_{\varepsilon})+\nabla p\right]\,dt$
$\displaystyle={\varepsilon}\,\Delta{\bf u}\,dt+\mathcal{G}({\bf
u}_{\varepsilon})\,{\rm d}W,$ (1.2) $\displaystyle{\rm div}\,{\bf
u}_{\varepsilon}\,$ $\displaystyle=0.$

There is a vast literature on the mathematical theory for stochastic
perturbations of Navier–Stokes equations (1.2), being first initiated by
Flandoli $\&$ Gatarek in [21] (see also [14]), where the global-in-time
existence of _weak martingale solutions_ is shown. From the PDE standpoint,
these solutions are weak, i.e., derivatives only exists in the sense of
distributions, and from probabilistic point of view, these solutions are also
weak in the sense that the driving noise and associated filtration are part of
the solution.

The primary difficulty in comparison to existing works, for example [5, 25],
lies in the successful identification (in the limit) of martingale terms
involving nonlinear noise coefficient, which in turn plays a pivotal role in
the proof of weak (measure-valued)-strong uniqueness principle. Indeed, since
the martingale solutions of approximate equations (1.2) are not unique,
associated filtration depends on the approximation parameter $\varepsilon$,
and passing to the limit in martingale terms seems delicate. We can only
establsih that the limit object is a martinagle, without knowing its explicit
structure. However, a key observation reveals that only a knowledge about the
cross variation of a martingale solution with a strong solution is sufficient
to exhibit weak-strong uniqueness property. This observation is encoded in the
Definition 2.15, by stipulating the correct cross variation between a
martingale measure-valued solution and a given smooth process. The cross
variation term resembles “noise-noise” interaction term appears in “doubling
of variables” argument, à la Kružkov, see [2, 3, 4, 26, 27]. Recently, a
concept of dissipative measure-valued solutions to stochastic compressible
Euler equations has been introduced by Hofmanova et. al. in [22], and used to
prove convergence of a numerical scheme for stochastic compressible Euler
equations by Chaudhury et. al. in [11]. However, the present work differs
significantly from [22, 11], due to the inherent challenges posed by the
divergence-free condition associated to (1.1).

Our second objective is to display a sufficient condition for the uniqueness
of weak martingale solutions to stochastic Naiver–Stokes equations (1.2) in
class of finite energy weak martingale solutions. Note that the uniqueness of
(Leray-Hopf) weak solutions for deterministic counterpart of (1.2) is an
outstanding open problem. Therefore, there has been a lot of growing interest
in searching for sufficient conditions for uniqueness of weak solutions,
starting with the celebrated works by Prodi [31] and Serrin [33]. Due to the
limitations of general uniqueness result in the stochastic set-up also, we
follow Prodi $\&$ Serrin (see also Wiedemann [37]), to provide a sufficient
condition for uniqueness by exploiting weak-strong uniqueness property of
solutions to stochastic incompressible Navier–Stokes equations (1.2).

A brief description of the organization of the rest of this paper is as
follows: Section 2 outlines all relevant underlying mathematical/technical
framework, and list of conditions imposed on noise coefficient. We then
describe solution concepts and display main results. In Section 3, we
establish a priori estimates and demonstrate convergence of approximate
solutions, using stochastic compactness, to show existence of dissipative
measure-valued martingale solutions to (1.1). In light of a suitable relative
energy inequality for incompressible Euler system (1.1), we establish weak
(measure-valued) – strong uniqueness principle for (1.1) in Section 4, while
in Section 5, we provide a sufficient condition for uniqueness of weak
martingale solutions to stochastic incompressible Naiver–Stokes equations
(1.2), under additional conditions on weak martingale solutions.

##  2 Technical Framework and Main Results

In this section, we recapitulate some of the relevant mathematical tools to be
used in the subsequent analysis and state main results of this paper. To begin
with, we fix an arbitrary large time horizon $T>0$. Throughout this paper we
use the letters C, K etc. to denote various generic constants independent of
approximation parameters, which may change line to line along the proofs.
Explicit tracking of the constants could be possible but it is cumbersome and
avoided for the sake of the reader.

###  2.1 Analytic framework

Let us denote the Sobolev space $H^{s}(\mathbb{T}^{3})$, for $s\in\mathbb{R}$,
as the set of tempered distributions for which the norm

$\|u\|^{2}_{H^{s}(\mathbb{T}^{3})}:=\int_{\mathbb{T}^{3}}\big{(}1+|\xi|^{2}\big{)}^{s}|\hat{u}(\xi)|^{2}\,{\rm
d}\xi,$

is finite. Here $\hat{u}$ denotes the Fourier transform of $u$. Let
$C_{\text{div}}^{\infty}(\mathbb{T}^{3})$ and
$L_{\text{div}}^{2}(\mathbb{T}^{3})$ be the spaces of infinitely
differentiable $3$-dimensional vector fields $u$ on $\mathbb{T}^{3}$
satisfying $\nabla\cdot\,u=0$, and closure of
$C_{\text{div}}^{\infty}(\mathbb{T}^{3})$ with respect to $L^{2}$-norm
respectively. In other words,

$\displaystyle C_{\text{div}}^{\infty}(\mathbb{T}^{3})$
$\displaystyle:=\Big{\\{}\bm{\varphi}\in
C^{\infty}(\mathbb{T}^{3}):\nabla\cdot\bm{\varphi}=0\Big{\\}},$ $\displaystyle
L_{\text{div}}^{2}(\mathbb{T}^{3})$ $\displaystyle:=\Big{\\{}\bm{\varphi}\in
L^{2}(\mathbb{T}^{3}):\nabla\cdot\bm{\varphi}=0\Big{\\}}.$

In a similar fashion, we denote by $H_{\text{div}}^{\alpha}(\mathbb{T}^{3})$
the closure of $C_{\text{div}}^{\infty}(\mathbb{T}^{3})$ in
$H^{\alpha}(\mathbb{T}^{3};\mathbb{R}^{3})$, for $\alpha\,\geq\,0$.
Identityfying $L^{2}_{\rm div}(\mathbb{T}^{3})$ with its dual space
$(L^{2}_{\rm div}({\mathbb{T}^{3}}))^{\prime}$ and identfying $(L^{2}_{\rm
div}({\mathbb{T}^{3}}))^{\prime}$ with a subspace of
$H^{-\alpha}(\mathbb{T}^{3})$(the dual space of $H^{\alpha}(\mathbb{T}^{3})$),
we have $H_{\rm div}^{\alpha}(\mathbb{T}^{3})\subset L^{2}_{\rm
div}(\mathbb{T}^{3})\subset H_{\rm div}^{-\alpha}(\mathbb{T}^{3})$, and we can
denote the dual pairing between $H^{\alpha}_{\rm div}$ and $H^{-\alpha}_{\rm
div}$ by $\langle\cdot,\cdot\rangle$ when no confusion may arise, see [20].

Moreover, we set $D(A):=H_{\rm div}^{2}(\mathbb{T}^{3})$, and define the
linear operator $A:D(A)\subset L^{2}_{\rm div}(\mathbb{T}^{3})\to L^{2}_{\rm
div}(\mathbb{T}^{3})$ by $Au=-\Delta u$. We then define the bilinear operator
$B(u,v):H^{1}_{\rm div}\times H^{1}_{\rm div}\to H^{-1}_{\rm div}$ as

$\langle
B(u,v),z\rangle:=\int_{\mathbb{T}^{3}}z(x)\cdot(u(x)\cdot\nabla)v(x)dx,\,\,\mbox{for
all}\,\,z\in H^{1}_{\rm div}(\mathbb{T}^{3}).$

Note that the bilinear operator $B$ can be extended to a continuous operator

$B:L^{2}_{\rm div}(\mathbb{T}^{3})\times L^{2}_{\rm div}(\mathbb{T}^{3})\to
D(A^{-\alpha})=H^{-2\alpha}_{\rm div}$

for certain $\alpha\,\textgreater\,1$, for details consult [20]. A
straightforward computation using incompressiblity condition reveals that

$\displaystyle\langle B(u,v),z\rangle=-\langle B(u,z),v\rangle=-\langle
u\otimes v,\nabla z\rangle$ (2.1)

for all $u,v\in H^{1}_{\rm div}(\mathbb{T}^{3})$ and $z\in
C_{\text{div}}^{\infty}(\mathbb{T}^{3})$.

An important consequence of elliptic theory is the existence of the Helmholtz
decomposition. It allows to decompose any vector-valued function in
$L^{2}(\mathbb{T}^{3};\mathbb{R}^{3})$ into a divergence free part and a
gradient part. In other words, we have the following decomposition

$L^{2}(\mathbb{T}^{3})=L^{2}_{\rm div}(\mathbb{T}^{3})\oplus(L^{2}_{\rm
div}(\mathbb{T}^{3}))^{\perp},$

where we denote

$(L^{2}_{\rm div}(\mathbb{T}^{3}))^{\perp}:=\Big{\\{}\mathbf{u}\in
L^{2}(\mathbb{T}^{3};\mathbb{R}^{3})\,|\,\mathbf{u}=\nabla\psi,\,\psi\in
H^{1}(\mathbb{T}^{3};\mathbb{R})\Big{\\}}$

The Helmholtz decomposition is defined by

$\mathbf{u}=\mathcal{P}_{H}\mathbf{u}+\mathcal{Q}_{H}\mathbf{u},\,\,\text{for
any }\,\mathbf{u}\in L^{2}(\mathbb{T}^{3}),$

where $\mathcal{P}_{H}$ denotes the projection operator from
$L^{2}(\mathbb{T}^{3})$ to $L^{2}_{\rm div}(\mathbb{T}^{3})$, and
$\mathcal{Q}_{H}:=Id-\mathcal{P}_{H}$ denotes the projection operator from
$L^{2}(\mathbb{T}^{3})$ to $(L^{2}_{\rm div}(\mathbb{T}^{3}))^{\perp}$. Note
that this decomposition is orthogonal with respect to
$L^{2}(\mathbb{T}^{3})$-inner product. By property of projection operator
$\mathcal{P}_{H}$, we have for $\mathbf{u}\in L^{2}(\mathbb{T}^{3})$

$\displaystyle\langle\mathcal{P}_{H}\mathbf{u},\psi\rangle=\langle\mathbf{u},\psi\rangle,\,\,\text{for
all}\,\,\psi\in C_{\text{div}}^{\infty}(\mathbb{T}^{3}),$ (2.2)

Finally, we recall a compact embedding result from Flandoli $\&$ Gatarek [20,
Theorem 2.2]. To state the result, let us first denote $K$ to be a separable
Hilbert space. Given $q\,\textgreater\,1$, $\gamma\in(0,1)$, let
$W^{\gamma,q}(0,T;K)$ denotes a $K$-valued Sobolev space which is
characterized by its norm

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

The following compact embedding result follows from Flandoli $\&$ Gatarek [20,
Theorem 2.2].

######  Lemma 2.1.

If $H_{1}\subset H_{2}$ are two Banach spaces with compact embedding, and real
numbers $\gamma\in(0,1)$, $q\,\textgreater\,1$ satisfy $\gamma q>1$, then the
embedding

$W^{\gamma,q}(0,T;H_{1})\subset C([0,T];H_{2})$

is compact.

####  2.1.1 Young measures, concentration defect measures

In this subsection, we recall the notion of Young measures and related results
used in this manuscript. For a detail overview on this topic, we refer to
Balder [1]. To begin with, let us denote by $\mathcal{M}_{b}(F)$, the space of
bounded Borel measures on a generic set $F$ equipped with the norm given by
the total variation of measures. It is well-known that it is the dual space to
the space of continuous functions vanishing at infinity $C_{0}(F)$ with
respect to the supremum norm. Moreover, let us denote by $\mathcal{P}(F)$, the
space of probability measures on $F$.

To define Young measure, let us first fix a sigma finite measure space
$(Y,\mathcal{N},\mu)$. A Young measure from $Y$ into $\mathbb{R}^{P}$ is a
weakly measurable function
$\mathcal{V}:Y\rightarrow\mathcal{P}(\mathbb{R}^{P})$. In other words, the map
$y\rightarrow\mathcal{V}_{y}(S)$ is $\mathcal{N}$-measurable for every Borel
set $S$ in $\mathbb{R}^{P}$. For our purpose, let us recall the following
probabilistic generalization of the well-known classical result on Young
measures. For a proof, we refer to the monograph by Breit et. al. [7, Section
2.8].

######  Lemma 2.2.

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

$\mathbb{E}\Big{[}\|{\bf V}_{m}\|^{p}_{L^{p}(\mathcal{D})}\Big{]}\leq
C,\,\,\text{for a certain}\,\,p\in(1,\infty).$

Then, on the complete probability space
$\big{(}[0,1],\overline{\mathcal{B}[0,1]},\mathcal{L}_{\mathbb{R}}\big{)}$,
there exists a new subsequence $(\tilde{\bf V}_{m})_{m\in\mathbb{N}}$ (not
relabeled) and a family
${\\{\mathcal{\tilde{V}}^{\omega}_{x}\\}}_{x\in\mathcal{D}}$ of parametrized
random probability measures on $\mathbb{R}^{P}$. These random probability
measures can be treated as random variables taking values in
$\big{(}L_{w^{*}}^{\infty}(\mathcal{D};\mathcal{P}(\mathbb{R}^{P})),w^{*}\big{)}$.
Moreover, the random variables ${\bf V}_{m}$ has the same law as $\tilde{\bf
V}_{m}$, i.e. ${\bf V}_{m}\sim_{d}\tilde{\bf V}_{m},$ and the following
property holds: given any Carathéodory function
$H=H(x,Z),x\in\mathcal{D},Z\in\mathbb{R}^{P}$, such that

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

implies $\mathcal{L}_{\mathbb{R}}$ almost surely,

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

Usually, Young measure theory plays an important role while extracting limits
of bounded continuous functions. However, in our context, we have to deal with
nonlinear function $H$ which may not be continuous, but instead, enjoys the
following bound:

$\mathbb{E}\Big{[}\|H({\bf V}_{m})\|^{p}_{L^{1}(\mathcal{D})}\Big{]}\leq
C,\,\,\text{for a certain}\,\,p\in(1,\infty),\,\mbox{uniformly in }m.$

In fact, in this situation, it is customary to embed the function space
$L^{1}(\mathcal{D})$ into the space of bounded Radon measures
$\mathcal{M}_{b}(\mathcal{D})$ to characterize the limit object. Indeed, we
can infer that $\mathbb{P}$-a.s.

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

where $H_{\infty}\in\mathcal{M}_{b}(\mathcal{D})$, and $H_{\infty}$ is called
_concentration defect measure (or concentration Young measure)_. It is worth
noting that, $\mathbb{P}$-a.s.
$\langle\mathcal{\tilde{V}}^{\omega}_{x};H\rangle$ is finite for a.e.
$x\in\mathcal{D}$, thanks to a classical truncation error analysis and Fatou’s
lemma which yield
$\|\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot)};H\rangle\|_{L^{1}(\mathcal{D})}\leq
C$, $\mathbb{P}$ almost surely. For our purpose, we shall repeatedly use the
following crucial lemma concerning the concentration defect measure. A proof
of this lemma can be furnished, modulo cosmetic changes, using the same
arguments given by Feireisl et. al [19, Lemma 2.1].

######  Lemma 2.3.

Let $\\{{\bf V}_{m}\\}_{m>0}$, ${\bf
V}_{m}:\Omega\times\mathcal{D}\rightarrow\mathbb{R}^{P}$ be a sequence
generating a Young measure $\\{\mathcal{V}^{\omega}_{y}\\}_{y\in\mathcal{D}}$,
where $\mathcal{D}$ is a measurable set in $\mathbb{R}^{Q}\times(0,T)$. Let
$H^{1}:\mathbb{R}^{P}\rightarrow[0,\infty)$ be a continuous function such that

$\sup_{m>0}\mathbb{E}\Big{[}\|H^{1}({\bf
V}_{m})\|^{p}_{L^{1}(\mathcal{D})}\big{]}<+\infty,\,\text{for a
certain}\,\,p\in(1,\infty),$

and let $H^{2}$ be a continuous function such that

$H^{2}:\mathbb{R}^{P}\rightarrow\mathbb{R},\quad|H^{2}(\bm{z})|\leq
H^{1}(\bm{z}),\mbox{ for all }\bm{z}\in\mathbb{R}^{P}.$

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

${H^{1}_{\infty}}:={\widetilde{H^{1}}}-\langle\mathcal{\tilde{V}}^{\omega}_{y},H^{1}(\textbf{v})\rangle\,dy,\quad{H^{2}_{\infty}}:={\widetilde{H^{2}}}-\langle\mathcal{\tilde{V}}^{\omega}_{y},H^{2}(\textbf{v})\rangle\,dy.$

Here ${\widetilde{H^{1}}},{\widetilde{H^{2}}}\in\mathcal{M}_{b}(\mathcal{D})$
are weak-$*$ limits of $\\{H^{1}({\bf V}^{m})\\}_{m>0}$, $\\{H^{2}({\bf
V}^{m})\\}_{m>0}$ respectively in $\mathcal{M}_{b}(\mathcal{D})$. Then
$|H^{2}_{\infty}|\leq H^{1}_{\infty}$ almost surely.

###  2.2 Basics of stochastic framework

Here we briefly recall some aspects of the theory of stochastic analysis which
are pertinent to the present work. We start by fixing a stochastic basis
$(\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},\mathbb{P},W)$ with a complete,
right-continuous filtration. Here $(\Omega,\mathbb{F},\mathbb{P})$ is a
complete probability space, and the stochastic process $W$ is a cylindrical
$(\mathbb{F}_{t})$-Wiener process defined on an auxiliary separable Hilbert
space $\mathfrak{U}$. It is formally given by the expansion

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

where the elements $\\{W_{k}\\}_{k\geq 1}$ are a sequence of mutually
independent one dimensional standard Brownian motions relative to
$(\mathbb{F}_{t})_{t\geq 0}$ and $\\{e_{k}\\}_{k\geq 1}$ is a complete
orthonormal basis of $\mathfrak{U}$. To define the stochastic integral
featured in (1.1), with nonlinear diffusion coefficient $\mathcal{G}$, assume
that $\mathbf{u}:\Omega\times[0,T]\Omega\to L^{2}(\mathbb{T}^{3})$ is
predictable, and let $\,\mathcal{G}({\bf u}):\mathfrak{U}\rightarrow
L^{2}(\mathbb{T}^{3})$ be defined as follows

$\mathcal{G}({\bf u})e_{k}=\mathbf{G}_{k}({\bf u}(\cdot)).$

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

$\displaystyle\sum_{k\geq 1}|\mathbf{G}_{k}({\bf u})|^{2}\leq D_{0}(1+|{\bf
u}|^{2}),$ (2.3) $\displaystyle\sum_{k\geq 1}|\mathbf{G}_{k}({\bf
u})-\mathbf{G}_{k}({\bf v})|^{2}\leq D_{1}|{\bf u}-\textbf{v}|^{2},$ (2.4)

for some real constants $D_{0}$, and $D_{1}$. Then the stochastic integral

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

is a well-defined $(\mathbb{F}_{t})$-martingale taking values in
$L^{2}(\mathbb{T}^{3})$. Finally, since $W(t)=\sum_{k\geq 1}e_{k}W_{k}(t)$
does not converge in $\mathfrak{U}$, we define the auxiliary space
$\mathfrak{U}_{0}\supset\mathfrak{U}$ via

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

according to the norm

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

Observe that the embedding $\mathfrak{U}\hookrightarrow\mathfrak{U}_{0}$ is
Hilbert–Schmidt. Moreover, the trajectories of the Brownian motion $W$ are
almost surely in $C([0,T];\mathfrak{U}_{0})$, thanks to a standard martingale
argument.

In order to state the existence of pathwise strong solution for stochastic
incompressible Euler equations, we next describe the conditions imposed on the
diffusion coefficient $\mathcal{G}$. For details, we refer to the paper by
Glatt-Holtz $\&$ Vicol [21]. Although the below mentioned conditions appear to
be rather involved, but they cover many realistic stochastic models. In what
follows, let $L_{2}$ denotes the usual space of Hilbert-Schmidt operators, and
for $p\geq 2,m\geq 0$, define

$\displaystyle\mathbb{L}^{m,p}=\bigg{\\{}\sigma:\mathbb{T}^{N}\to\,L_{2}\,\Big{|}\,\sigma_{k}(\cdot)=\sigma(\cdot)e_{k}\in
W^{m,p},\,\,\text{and}\sum_{|\beta|\,\leq\,m}\int_{\mathbb{T}^{3}}|\partial^{\beta}\sigma_{k}|^{p}_{L_{2}}\,{\rm
d}x\,\textless\,\infty\bigg{\\}},$

which is a Banach space endowed with the norm

$\displaystyle\|\sigma\|_{\mathbb{L}^{m,p}}:=\sum_{|\beta|\leq\,m}\int_{\mathbb{T}^{3}}|\partial^{\beta}\sigma|_{L_{2}}^{p}\,{\rm
d}x=\sum_{|\beta|\leq\,m}\int_{\mathbb{T}^{3}}\bigg{(}\sum_{k\geq
1}|\partial^{\beta}\sigma_{k}|^{2}\bigg{)}^{p/2}\,{\rm d}x.$

Consider any pair of Banach spaces $X,Y$. For an increasing, locally bounded
function $\gamma(\cdot)\geq 1$, we denote the space of locally bounded maps

$\displaystyle\text{Bnd}_{u,\text{loc}}(X,Y):=\bigg{\\{}{\mathcal{G}}\in
C(X;Y):\|{\mathcal{G}}(x)\|_{Y}\leq\gamma(\|x\|_{L^{\infty}})(1+\|x\|_{X}),\,\forall\,x\in
X\bigg{\\}}.$

In addition, we also define the space of locally Lipschitz functions,

$\displaystyle\text{Lip}_{u,\text{loc}}(X,Y)=\bigg{\\{}{\mathcal{G}}\in\text{Bnd}_{u,\text{loc}}(X,Y):\|\mathcal{G}(x)-\mathcal{G}(y)\|_{Y}\leq\,\gamma\Big{(}\|x\|_{L^{\infty}}+\|y\|_{L^{\infty}}\Big{)}\,\|x-y\|_{X},\forall\,x,y\in
X\bigg{\\}}$

For the statement of local pathwise existence result (cf. Theorem 2.9), we
shall fix $p\geq 2$ and an integer $m\,\textgreater\,{3}/{p}+1$, and suppose
that

$\displaystyle\mathcal{G}\in\text{Lip}_{u,\text{loc}}(L^{p},$
$\displaystyle\mathbb{L}^{0,p})\cap\text{Lip}_{u,\text{loc}}(W^{m+1,p},\mathbb{L}^{m+1,p})\cap\text{Lip}_{u,\text{loc}}(W^{m+5,2},\mathbb{L}^{m+5,2}).$
(2.5)

Relating to the convergence of approximate solutions, strong convergence in
$\omega$ variable plays a pivotal role. To that context, we need Skorokhod
embedding theorem, delivering a new probability space and new random
variables, with the same laws as the original ones, converging almost surely.
However, for technical reasons, we have to use a modified version of classical
Skorokhod embedding theorem [30, Corollary 2] which is stated below.

######  Theorem 2.4.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $H_{1}$ be a
separable metric space. Let $H_{2}$ be a quasi-polish space (i.e., there is a
sequence of continuous functions $h_{n}:H_{2}\to[-1,1]$ that separates points
of $H_{2}$). Assume that $\mathcal{B}(H_{1})\otimes\mathcal{H}_{2}$ is a sigma
algebra associated with the product space $H_{1}\times H_{2}$, where
$\mathcal{H}_{2}$ is the sigma algebra generated by the sequence of continuous
functions $\\{h_{n}\\}_{n=1}^{\infty}$. Let $U_{n}:\Omega\to H_{1}\times
H_{2}$, $n\in\mathbb{N}$, be a family of random variables, such that the
sequence $\\{\mathcal{L}aw(U_{n}):n\in\mathbb{N}\\}$ is weakly convergent on
$H_{1}\times H_{2}$. For $k=1,2$, let $\pi_{i}:H_{1}\times H_{2}$ be the
projection onto $H_{i}$, i.e.,

$U=(U_{1},U_{2})\in H_{1}\times H_{2}\mapsto\pi_{i}(U)=U_{i}\in H_{i}.$

Finally, let us assume that there exists a random variable $X:\Omega\to H_{1}$
such that
$\mathcal{L}aw(\pi_{1}(U_{n}))=\mathcal{L}aw(X),\,\forall\,n\in\mathbb{N}$.
Then, there exists a probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$, a family of
$H_{1}\times H_{2}$-valued random variables
$\\{\tilde{U}_{n}:n\in\mathbb{N}\\}$, on
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ and a random
variable $\tilde{U}:\tilde{\Omega}\to H_{1}\times H_{2}$ such that

  1. (a)

$\mathcal{L}aw(\tilde{U}_{n})=\mathcal{L}aw(U_{n}),\,\forall\,n\in\mathbb{N};$

  2. (b)

$\tilde{U}_{n}\to\tilde{U}\,\text{in}\,H_{1}\times
H_{2},\,\mathbb{P}-\text{a.s.}$

  3. (c)

$\pi_{1}(\tilde{U}_{n})(\tilde{w})=\pi_{1}(\tilde{U})(\tilde{w}),\,\forall\,\tilde{w}\in\tilde{\Omega}.$

Finally, we recall the celebrated Kolmogorov continuity thereom related to the
existence of continuous modifications of stochastic processes.

######  Lemma 2.5.

Let $Z={\\{Z(t)\\}}_{t\in[0,T]}$ be a real-valued stochastic process defined
on a complete filtered 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 $s,t\in[0,T]$,

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

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

###  2.3 Stochastic incompressible Euler equations

As we mentioned before, we are primarily interested in establishing weak
(measure-valued)–strong uniqueness principle for dissipative measure-valued
solutions to (1.1). Since such an argument requires the existence of strong
solution, therefore, we first recall the notion of local strong pathwise
solution for stochastic incompressible Euler equations. We remark that such a
solution can be constructed on any given stochastic basis, that is, solutions
are probablistically strong, and satisfies the underlying equation (1.1)
pointwise (not only in the sense of distributions), that is, solutions are
srtong from the PDE standpoint. Existence of such a solution was first
established by Glatt-Holtz $\&$ Vicol in [21].

######  Definition 2.6 (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, and ${W}$ be
an $(\mathbb{F}_{t})$-cylindrical Wiener process. Let $\mathbf{u}_{0}$ be a
$W^{m,p}(\mathbb{T}^{3})$-valued $\mathbb{F}_{0}$-measurable random variable,
and let $\mathcal{G}$ satisfy (2.5). Then $({\bf u},\mathfrak{t})$ is said to
be a local strong pathwise solution to the system (1.1) provided

  1. (a)

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

  2. (b)

the velocity ${\bf u}$ is a $W_{\rm div}^{m,p}(\mathbb{T}^{3})$-valued
$(\mathbb{F}_{t})$-predictable measurable process satisfying

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

  3. (c)

for all $t\,\geq\,0$,

$\begin{split}{\bf u}(t\wedge\mathfrak{t})&={\bf
u}_{0}-\int_{0}^{t\wedge\mathfrak{t}}\mathcal{P}_{H}({\bf u}\cdot\nabla{\bf
u})\mathrm{d}s+\int_{0}^{t\wedge\mathfrak{t}}\mathcal{P}_{H}{\mathcal{G}}({\bf
u})\ {\rm d}W.\end{split}$ (2.6)

######  Remark 2.7.

By the property of Projection operator (2.2), we can recast the item $(c)$ in
Definition 2.6 as follows:

  1. $(c^{\prime})$

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

$\begin{split}\langle{\bf u}(t\wedge\mathfrak{t}),\psi\rangle&=\langle{\bf
u}_{0},\psi\rangle-\int_{0}^{t\wedge\mathfrak{t}}\langle{\bf u}\cdot\nabla{\bf
u},\psi\rangle\mathrm{d}s+\int_{0}^{t\wedge\mathfrak{t}}\langle{\mathcal{G}}({\bf
u}),\psi\rangle\ {\rm d}W,\end{split}$ (2.7)

for all ${\psi}\in C_{\text{div}}^{\infty}(\mathbb{T}^{3})$.

It is evident that classical solutions require spatial derivatives of the
velocity field ${\bf u}$ to be continuous $\mathbb{P}$-a.s. This motivates the
following definition.

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

Fix an initial condition, and a complete stochastic basis with a cylindrical
Wiener process as in Definition 2.6. Then a triplet

$({\bf u},(\tau_{L})_{L\in\mathbb{N}},\mathfrak{t})$

is said to be a maximal strong pathwise solution to system (1.1) provided

  1. (a)

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

  2. (b)

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

$\sup_{t\in[0,\tau_{L}]}\|{\bf u}(t)\|_{W^{1,\infty}(\mathbb{T}^{3})}\geq
L\quad\text{on}\quad[\mathfrak{t}<T];$ (2.8)

  3. (c)

each pair $({\bf u},\tau_{L})$, $L\in\mathbb{N}$, is a local strong pathwise
solution in the sense of Definition 2.6.

In view of the above definitions, we are now in a position to state relevant
existence theorems. For a proof, we refer to the work by Glatt-Holtz $\&$
Vicol [21].

######  Theorem 2.9 (Local existence for nonlinear multiplicative noise).

Let $\left(\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},\mathbb{P}\right)$ be
a stochastic basis with a complete right-continuous filtration. Suppose that
$p\geq 2$, and $m\,\textgreater\,3/p+1$. Let ${W}$ be an
$(\mathbb{F}_{t})$-cylindrical Wiener process and $\mathbf{u}_{0}$ be a
$W^{m,p}(\mathbb{T}^{3})$-valued $\mathbb{F}_{0}$-measurable random variable,
and let $\mathcal{G}$ satisfy (2.5). Then there exists a unique maximal strong
pathwise solution $({\bf u},(\tau_{L})_{L\in\mathbb{N}},\mathfrak{t})$ of
(1.1) in the sense of Definition 2.8.

###  2.4 Stochastic incompressible Navier–Stokes equations

There is a large and intense literature concerning the incompressible
Navier–Stokes equations driven by noise, starting with the work by Flandoli
et. al [20] where the authors proved the existence of weak martingale
solutions to (1.2). As expected, these solutions are weak in both analytical
and probabilistic sense. However, to prove existence of dissipative measure-
valued solutions for stochastic Euler equations, we first need to introduce
the concept of _finite energy weak martingale_ solutions to (1.2). Note that,
such solutions exist globally in time, and the time-evolution of the energy
for such solutions can be controlled in terms of its initial state.

######  Definition 2.10 (Finite energy weak martingale solution).

Let $\Lambda_{\varepsilon}$ be a Borel probability measure on
$L_{\text{div}}^{2}(\mathbb{T}^{3})$. Then
$\big{[}\big{(}\Omega_{\varepsilon},\mathbb{F}_{\varepsilon},(\mathbb{F}_{{\varepsilon},t})_{t\geq
0},\mathbb{P}_{\varepsilon}\big{)};\mathbf{u}_{\varepsilon},W_{\varepsilon}\big{]}$
is a weak martingale solution of (1.2) if

  1. (a)

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

  2. (b)

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

  3. (c)

the velocity field $\mathbf{u}_{\varepsilon}$ is $L^{2}_{\rm
div}(\mathbb{T}^{3})$-valued progressively measurable process and
$\mathbb{P}-$a.s.

$\displaystyle{\bf u}(\cdot,\omega)\in C([0,T];H^{-2\alpha}_{\rm
div}(\mathbb{T}^{3}))\cap L^{\infty}(0,T;L^{2}_{\rm div}(\mathbb{T}^{3}))\cap
L^{2}(0,T;H^{1}_{\rm div}(\mathbb{T}^{3}))$

  4. (d)

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

  5. (e)

for all $\bm{\varphi}\in H^{2\alpha}_{\rm div}(\mathbb{T}^{3})$, we have

$\displaystyle\langle\mathbf{u}_{\varepsilon}(t),\bm{\varphi}\rangle$
$\displaystyle=\langle\mathbf{u}_{\varepsilon}(0),\bm{\varphi}\rangle-\int_{0}^{t}\langle
B(\mathbf{u}_{\varepsilon}(s),\mathbf{u}_{\varepsilon}(s)),\bm{\varphi}\rangle\,\mathrm{d}s+{\varepsilon}\,\int_{0}^{t}\langle\Delta\mathbf{u}_{\varepsilon}(s)\,,\bm{\varphi}\rangle\mathrm{d}s+\int_{0}^{t}\langle\mathcal{P}_{H}\mathcal{G}(\mathbf{u}_{\varepsilon}),\bm{\varphi}\rangle\,\mathrm{d}W$
(2.9)

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

  6. (f)

the energy inequality

$\displaystyle-\int_{0}^{T}\partial_{t}\phi\int_{\mathbb{T}^{3}}\frac{1}{2}|{\bf
u}_{\varepsilon}|^{2}\,{\rm d}x\,{\rm
d}t+{\varepsilon}\,\int_{0}^{T}\phi\int_{\mathbb{T}^{3}}|\nabla_{x}{\bf
u}_{\varepsilon}|^{2}\,{\rm d}{x}\,{\rm d}t$ (2.10)
$\displaystyle\leq\phi(0)\int_{\mathbb{T}^{3}}\frac{1}{2}{|\mathbf{u}_{\varepsilon}(0)|^{2}}+\sum_{k=1}^{\infty}\int_{0}^{T}\phi\bigg{(}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}\mathbf{G}_{k}({\bf
u}_{\varepsilon})\cdot{\bf u}_{\varepsilon}\,{\rm d}{x}\bigg{)}{\rm
d}W_{k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\phi\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}\mathbf{G}_{k}({\bf
u}_{\varepsilon})|^{2}\,{\rm d}t$

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

######  Remark 2.11.

Note that, in view of Skorohod [35], it is possible to consider,

$\Big{(}\Omega_{\varepsilon},\mathbb{F}_{\varepsilon},\mathbb{P}_{\varepsilon}\Big{)}=\Big{(}[0,1],\mathcal{B}([0,1]),\mathcal{L}_{\mathbb{R}}\Big{)},$

for every $\varepsilon$. Moreover, we may assume the existence of a common
Wiener space $W$ for all $\varepsilon$, thanks to a classical compactness
argument applied to any chosen subsequence
${\\{\varepsilon_{n}\\}}_{n\in\mathbb{N}}$ at once. However, it is worth
noticing that, it may not be possible obtain a filtration which is independent
of $\varepsilon$, due to lack of pathwise uniqueness for the underlying
system.

######  Remark 2.12.

By using property (2.1)-(2.2), we can recast the item $(e)$ in Definition 2.10
as

  1. $(e^{\prime})$

For all $\bm{\varphi}\in C_{\text{div}}^{\infty}(\mathbb{T}^{3})$,

$\displaystyle\langle\mathbf{u}_{\varepsilon}(t),\bm{\varphi}\rangle=\langle\mathbf{u}_{\varepsilon}(0),\bm{\varphi}\rangle$
$\displaystyle+\int_{0}^{t}\langle\mathbf{u}_{\varepsilon}\otimes\mathbf{u}_{\varepsilon}(s),\nabla_{x}\bm{\varphi}\rangle\,\mathrm{d}s$
(2.11)
$\displaystyle-{\varepsilon}\,\int_{0}^{t}\langle\nabla_{x}\mathbf{u}_{\varepsilon}(s)\,,\nabla_{x}\bm{\varphi}\rangle\mathrm{d}s+\int_{0}^{t}\langle\mathcal{G}(\mathbf{u}_{\varepsilon}),\bm{\varphi}\rangle\,\mathrm{d}W$

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

Regarding the existence of finite energy martingale solutions, one may follow
the arguments given by Flandoli et. al. [20] to obtain the following result.

######  Theorem 2.13 (Existence of martingale solution for Naiver Stokes).

Assume that $\Lambda_{\varepsilon}$ is a Borel probability measure on
$L_{\rm{div}}^{2}(\mathbb{T}^{3})$ such that the following moment estimate

$\int_{L_{\rm div}^{2}(\mathbb{T}^{3})}{\|\mathbf{u}\|}^{p}_{L_{{\rm
div}}^{2}(\mathbb{T}^{3})}\,\mathrm{d}\Lambda_{\varepsilon}(\mathbf{u})<\infty,$

holds for all $1\leq p<\infty$. Moreover, assume that (2.3) and (2.4) hold.
Then there exists a finite energy weak martingale solution of (1.2) in the
sense of Definition 2.10 with initial law $\Lambda_{\varepsilon}$.

###### Proof.

Existence proof for a weak martingale solution follows from the work of
Flandoli et. al. [20]. To prove the energy inequality, one may simply apply
Itô formula to obtain item (f) of Definition 2.10. Indeed, this is very
similar to the recent works on compressible fluids, see Breit et. al. [7]. The
details are left to the interested reader. ∎

######  Remark 2.14.

(A different form of energy inequality) It is well-known that, one can
establish weak-strong uniqueness principle only in the class of dissipative
weak solutions, i.e., weak solutions satisfying an appropriate energy
inequality. To that context, we make use of a different form of energy
inequality for the proof of weak-strong uniqueness related to incompressible
Navier–Stokes equations. First observe that, in view of a standard cut-off
argument applied to (2.10), energy inequality holds for a.e.
$0\leq\,s\,\textless\,t\,\in(0,T):$

$\displaystyle\int_{\mathbb{T}^{3}}\frac{1}{2}|{\bf
u}_{\varepsilon}(t)|^{2}\,{\rm
d}x+{\varepsilon}\,\int_{s}^{t}\int_{\mathbb{T}^{3}}|\nabla_{x}{\bf
u}_{\varepsilon}|^{2}\,{\rm d}{x}\,{\rm d}s$ (2.12)
$\displaystyle\quad\leq\int_{\mathbb{T}^{3}}\frac{1}{2}{|\mathbf{u}_{\varepsilon}(s)|^{2}}\,{\rm
d}x+\sum_{k=1}^{\infty}\int_{s}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}\mathbf{G}_{k}({\bf
u}_{\varepsilon})\cdot{\bf u}_{\varepsilon}\,{\rm d}{x}\bigg{)}{\rm
d}W_{k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{s}^{t}\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}\mathbf{G}_{k}({\bf
u}_{\varepsilon})|^{2}\,{\rm d}s.$

It follows from (2.12) that the limits

$\operatorname*{ess\,lim}_{\tau\to
s^{+}}\int_{\mathbb{T}^{3}}\frac{1}{2}|u(\tau)|^{2}\,{\rm
d}x,\,\,\,\operatorname*{ess\,lim}_{\tau\to
t^{-}}\int_{\mathbb{T}^{3}}\frac{1}{2}|u(\tau)|^{2}\,{\rm d}x$

exist $\mathbb{P}$-a.s. for a.a. $0\,\leq\,s\,\leq\,t\,\leq\,T$ including
$s=0$. Finally, in view to the weak lower-semicontinuity of convex
functionals, we have for any $t\in[0,T)\,\,\mathbb{P}$-a.s.

$\liminf_{\tau\to t^{-}}\int_{\mathbb{T}^{3}}\frac{1}{2}|u(\tau)|^{2}\,{\rm
d}x\,\geq\,\int_{\mathbb{T}^{3}}\frac{1}{2}|u(t)|^{2}\,{\rm d}x$

By making use of the above informations, relative energy inequality (2.12) can
be rewritten as

$\displaystyle\begin{aligned} &\int_{\mathbb{T}^{3}}\frac{1}{2}|{\bf
u}_{\varepsilon}(t)|^{2}\,{\rm
d}x+{\varepsilon}\,\int_{0}^{t}\int_{\mathbb{T}^{3}}|\nabla_{x}{\bf
u}_{\varepsilon}|^{2}\,{\rm d}{x}\,{\rm d}s\\\
&\quad\leq\int_{\mathbb{T}^{3}}\frac{1}{2}{|\mathbf{u}_{\varepsilon}(0)|^{2}}\,{\rm
d}x+\sum_{k=1}^{\infty}\int_{0}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}\mathbf{G}_{k}({\bf
u}_{\varepsilon})\cdot{\bf u}_{\varepsilon}\,{\rm d}{x}\bigg{)}{\rm
d}W_{k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{t}\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}\mathbf{G}_{k}({\bf
u}_{\varepsilon})|^{2}\,{\rm d}s\end{aligned}$ (2.13)

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

###  2.5 Measure-valued martingale solutions

In general, in view of the energy inequality (2.13), solutions to
incompressible Navier–Stokes equations only has a uniform energy bound,
usually $L^{2}(\mathbb{T}^{3})$. However, such a priori bound does not
guarantee weak convergence of nonlinear terms ${\bf u}\otimes{\bf
u},\textbf{G}^{2}({\bf u})\in L^{1}(\mathbb{T}^{3})$, due to the presence of
oscillations and concentration effects. In this scenario, one can only
identify weak limits (corresponding to nonlinear terms) as a combination of
Young measure and concentration measure.

Note that the Young measures, which are probability measures on the phase
space, capture oscillations in the solution. On the other hand concentration
defect measures, which are measures on physical space-time, accounts for blow
up type collapse due to possible concentration points. In what follows, we use
two different forms of concentration defect measures. To illustrate the
difference, we consider the following situation:

  * •

Let $v_{\varepsilon}$ converges weakly in $L^{2}(\mathbb{T}^{3})$ to a
function $v$, and we assume that
$\int_{\mathbb{T}^{3}}|v_{\varepsilon}|^{2}\,dx\leq C$.

Inspired by Banach-Alaoglu theorem, one can define a defect measure - which is
a non-negative Radon measure, as

$\mu_{1}:=\mbox{Weak -*}\lim_{\varepsilon\rightarrow
0}\Big{(}|v_{\varepsilon}|^{2}-|v|^{2}\Big{)}\in\mathcal{M}^{+}_{b}(\Pi_{T}).$

In a similar fashion, by making use of Young measure theory, one can define
another defect measure $\mu_{2}$ as

$\mu_{2}:=\mbox{Weak -*}\lim_{\varepsilon\rightarrow
0}\Big{(}|v_{\varepsilon}|^{2}-\langle\mathcal{V}_{t,x}(\lambda);|\lambda|^{2}\rangle\Big{)}\in\mathcal{M}^{+}_{b}(\Pi_{T}).$

It is well-known that $\mu_{1}$ is too large to describe concentration effects
in a useful way, while the main advantage of the concentration defect measure
$\mu_{2}$ is that it allows to describe weak limits in terms of Young measure.
It is easy to see that, thanks to Hölder inequality, $\mu_{2}\leq\mu_{1}$. We
shall make use of both forms of concentration defect measures below to define
a notion of measure valued solution.

####  2.5.1 Dissipative measure-valued martingale solutions

Keeping in mind the previous discussion, we are ready to introduce the concept
of dissipative measure–valued martingale solution to the stochastic
compressible Euler system. In what follows, let $\mathcal{S}=\mathbb{R}^{3}$
be the phase space associated to the incompressible Euler system.

######  Definition 2.15 (Dissipative measure-valued martingale solution).

Let $\Lambda$ be a Borel probability measure on
$L_{\text{div}}^{2}(\mathbb{T}^{3})$. 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 (1.1), with initial condition
$\mathcal{V}^{\omega}_{0,x}$; if

  1. (a)

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

  2. (b)

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

  3. (c)

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

  4. (d)

the average velocity $\langle\mathcal{V}^{\omega}_{t,x};{\bf u}\rangle$
satisfies, for any $\bm{\varphi}\in C_{\text{div}}^{\infty}(\mathbb{T}^{3})$,
$t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};{\bf
u}\rangle(t,\cdot),\bm{\varphi}\rangle\in C[0,T]$, $\mathbb{P}$-a.s., the
function $t\mapsto\langle\langle\mathcal{V}^{\omega}_{t,x};{\bf
u}\rangle(t,\cdot),\bm{\phi}\rangle$ is progressively measurable, and for any
$\bm{\varphi}\in C^{1}(\mathbb{T}^{3})$,

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{t,x};{\bf
u}\rangle\cdot\nabla_{x}\bm{\varphi}\,dx=0$

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

$\displaystyle\mathbb{E}\,\bigg{[}\sup_{t\in(0,T)}\|\langle\mathcal{V}^{\omega}_{t,x};{\bf
u}\rangle(t,\cdot)\|_{L_{\rm div}^{2}(\mathbb{T}^{3})}^{p}\bigg{]}<\infty$

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

  5. (e)

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

  6. (f)

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

$\displaystyle\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{\tau,x};{\bf
u}\rangle\cdot\bm{\varphi}(\tau,\cdot)\,{\rm
d}x-\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{0,x};{\bf
u}\rangle\cdot\bm{\varphi}(0,\cdot)\,{\rm d}x$ (2.14)
$\displaystyle\qquad=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\langle\mathcal{V}^{\omega}_{t,x};{{\bf
u}\otimes{\bf u}}\rangle:\nabla_{x}\bm{\varphi}\,{\rm d}x\,{\rm
d}t+\int_{\mathbb{T}^{3}}\bm{\varphi}\,\int_{0}^{\tau}d\mathcal{M}^{1}_{E}(t)\,{\rm
d}x+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\nabla_{x}\bm{\varphi}:d\mu_{C},$

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

  7. (g)

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

$\displaystyle\begin{aligned}
\mathrm{E}(t+)\,\leq\,&\mathrm{E}(s-)+\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|^{2}}\right\rangle\,\,{\rm
d}x\,{\rm d}\tau\\\
&\quad-\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\Big{(}\mathcal{Q}_{H}\left\langle\mathcal{V}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|}\right\rangle\Big{)}^{2}\,{\rm
d}x\,{\rm
d}\tau+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}d\mu_{D}+\int_{s}^{t}d\mathcal{M}^{2}_{E},\end{aligned}$
(2.15)

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

$\mathrm{E}(t-):=\lim_{r\to
0^{+}}\frac{1}{r}\int_{t-r}^{t}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}{x}+\mathcal{D}(s)\bigg{)}\,{\rm d}s$
$\mathrm{E}(t+):=\lim_{r\to
0^{+}}\frac{1}{r}\int_{t}^{t+r}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}{x}+\mathcal{D}(s)\bigg{)}\,{\rm d}s$

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

$\mathrm{E}(0-)=\int_{\mathbb{T}^{3}}\frac{1}{2}|\textbf{u}_{0}|^{2}\,\,{\rm
d}{x}.$

  8. (h)

there exists a constant $C>0$ such that

$\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\mu_{C}|+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}d|\mu_{D}|\leq
C\int_{0}^{\tau}\mathcal{D}(t)dt,$ (2.16)

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

  9. (i)

For any given stochastic process $h(t)$, adapted to $(\mathbb{F}_{t})_{t\geq
0}$, given by

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

satisfying

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

with

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

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

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

Here $g_{i}$’s are orthonormal basis for $H^{-1}_{\rm div}(\mathbb{T}^{3})$
and bracket $\langle\cdot,\cdot\rangle$ denotes inner product in the same
space.

######  Remark 2.16.

Notice that, a standard Lebesgue point argument applied to (2.15) reveals that
the energy inequality holds for a.e. $0\leq s<t$ in $(0,T)$:

$\displaystyle\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{t,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm
d}{x}+\mathcal{D}(t)\,\leq\,\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm
d}{x}+\mathcal{D}(s)+\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};{|{\textbf{G}_{k}(u)}|^{2}}\right\rangle\,\,{\rm
d}x\,{\rm d}\tau$
$\displaystyle\quad-\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\Big{(}\mathcal{Q}_{H}\left\langle\mathcal{V}^{\omega}_{s,x};{|{\textbf{G}_{k}(u)}|}\right\rangle\Big{)}^{2}\,{\rm
d}x\,{\rm
d}\tau+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}d\mu_{D}+\int_{s}^{t}d\mathcal{M}^{2}_{E},\,\,\mathbb{P}-a.s.$
(2.17)

However, as it is evident from Section 4, we require energy inequality to hold
for _all_ $s,t\in(0,T)$ to demonstrate weak-strong uniqueness principle.

###  2.6 Statements of main results

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

######  Theorem 2.17 (Existence of Measure-Valued Solution).

Assume ${\bf G}_{k}$ satisfies (2.3), (2.4), and ${\bf u}_{\varepsilon}$ be a
family of finite energy weak martingale solutions to the stochastic
incompressible Navier–Stokes system (1.2). Let the corresponding initial data
${\bf u}_{0}$, the initial law $\Lambda$, given on the space $L_{\rm
div}^{2}(\mathbb{T}^{3})$, be independent of $\varepsilon$, satisfying the
following moment estimate

$\displaystyle\int_{L_{\rm div}^{2}}\|\mathbf{q}\|^{p}_{L_{\rm
div}^{2}(\mathbb{T}^{3})}\,\mathrm{d}\Lambda(\mathbf{q})<\infty,$ (2.18)

holds for all $1\leq p<\infty$. Then the family ${\\{{\bf
u}_{\varepsilon}\\}}_{\varepsilon>0}$ generates, as $\varepsilon\mapsto 0$, a
Young measure
${\\{\mathcal{V}^{\omega}_{t,x}\\}}_{t\in[0,T];x\in\mathbb{T}^{3}}$ which is a
dissipative measure-valued martingale solution to the stochastic
incompressible Euler system (1.1), in the sense of Definition 2.15, with
initial data $\mathcal{V}^{\omega}_{0,x}=\delta_{{\bf u}_{0}(x)}$ almost
surely.

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

######  Theorem 2.18 (Weak-Strong Uniqueness For Nonlinear Noise).

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). 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), driven by the same cylindrical Wiener process $W$, given by
$(\bar{{\bf u}},(\tau_{L})_{L\in\mathbb{N}},\mathfrak{t})$ with the initial
data $\bar{\bf u}(0)$ satisfies

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

Then for $L\in\mathbb{N}$, a.e. $t\in[0,T]$, $\mathcal{D}(t\wedge\tau_{L})=0$,
$\mathbb{P}$-a.s., and for a.e. $t\in[0,T]$, $\mathbb{P}$-a.s.

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

Next, we move our attention to the stochastic incompressible Navier–Stokes
equations given by (1.2) (with $\varepsilon=1$). For the deterministic
counterpart of (1.2), Prodi [31] and Serrin [33] established weak-strong
uniqueness principle under additional regularity of a solution $\mathbf{U}\in
L^{r}([0,T],L^{s}(\mathbb{T}^{3}))$, where $r$ and $s$ satisfies the relation
$2/r+3/s=1$, with $s\in(3,\infty)$. In the stochastic setup, uniqueness of
finite energy weak martingale solutions for (1.2) seems to be out of reach.
However, one can obtain a conditional uniqueness result like Prodi and Serrin.
In fact, our objective is to give a sufficient condition under which
martingale solutions of Naiver Stokes equation (1.2) is unique in class of
finite energy weak martingale solutions. Note that, in comparison to
deterministic analysis, we need additional continuity assumption on stochastic
solutions to deal with the stopping time.

######  Theorem 2.19 (Weak-Strong Uniqueness For Naiver Stokes).

Let $\bf u,U$ be two finite energy weak martingale solutions to the system
(1.2), defined on the same stochastic basis
$\big{(}\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},W,\mathbb{P}\big{)}$ with
same intial data. Let the solution $\bf U$ additionally satisfies

$\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|{\bf
U}\|_{L^{s}(\mathbb{T}^{3})}\bigg{]}<\infty,\,\,\text{for
some}\,\,3\textless\,s\,\textless\,\infty.$ (2.19)

Then $\mathbb{P}$-almost surely, for all $t\in[0,T]$

${{\bf u}}({t,x})={{\bf U}(t,x)},\,\mbox{for a.e. }x\in\mathbb{T}^{3}.$

##  3 Proof of Theorem 2.17

The proof of existence is essentially based on compactness method. For the
compactness argument in space and time variables we make use of Young measure
theory, while for the compactness argument in probability variable we rely on
Skorokhod representation theorem. However, as alluded to before, our path
spaces are not Polish spaces (which is required for classical Skorokhod
theorem), therefore we rely on Skorokhod-Jakubowski theorem ([30, Corollary
2])which is taylor made to deal with so called quasi-Polish spaces. As usual,
we obtain the convergence of the approximate sequence on another probability
space and the existence of dissipative measure-valued martingale solution
follows, thanks to Young measure theory. Observe that the existence of a
pathwise solution (typically obtained by Gyöngy-Krylov’s characterization of
convergence in probability) seems not possible due to the lack of pathwise
uniqueness for the underlying system.

###  3.1 A priori Bounds

Recall that, the existence of finite energy weak martingale solution of
stochastic incompressible Navier–Stokes system (1.2)

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

is well established, thanks to the Theorem 2.13. Observe that the filtration
$(\mathbb{F}_{{\varepsilon},t})_{t\geq 0}$ depends on $\varepsilon$, and lack
of pathwise uniqueness for (1.2) does not allow us to choose the filtration
independent of $\varepsilon$. Having said this, however, note that the
Brownian motion and the probability space can be chosen indepedent of
$\varepsilon$.

To obatin a-priori estimate for the approximate solution, we make use of the
energy inequality (2.13) to obatin for all $t\in[0,T]$, and
$1\,\leq\,p\,\textless\,\infty$

$\mathbb{E}\Big{[}\|{\bf
u}_{\varepsilon}(t)\|_{L^{2}(\mathbb{T}^{3})}^{2}\Big{]}^{p}\leq\,\mathbb{E}\Big{[}\|{\bf
u}_{0}\|_{L^{2}(\mathbb{T}^{3})}^{2}\Big{]}^{p}+C\int_{0}^{t}\Bigg{(}1+\mathbb{E}\Big{[}\|{\bf
u}_{\varepsilon}(s)\|_{L^{2}(\mathbb{T}^{3})}^{2}\Big{]}^{p}\Bigg{)}ds.$

Therefore, a simple application Gronwall lemma yield

$\mathbb{E}\Big{[}\|{\bf
u}_{\varepsilon}\|_{L^{2}(\mathbb{T}^{3})}^{2}\Big{]}^{p}\leq\,C\,\Bigg{(}1+\mathbb{E}\Big{[}\|{\bf
u}_{0}\|_{L^{2}(\mathbb{T}^{3})}^{2}\Big{]}^{p}\Bigg{)}$

Again by energy inequality (2.13), we have for $1\,\leq\,p\,\textless\,\infty$

$\displaystyle\mathbb{E}\bigg{[}$ $\displaystyle\sup_{t\in[0,T]}\|{\bf
u}_{\varepsilon}\|_{L^{2}(\mathbb{T}^{3})}^{2}+\varepsilon\int_{0}^{T}\|\nabla_{x}{\bf
u}_{\varepsilon}\|_{L^{2}(\mathbb{T}^{3})}^{2}\,{\rm d}t\bigg{]}^{p}\,$
$\displaystyle\qquad\qquad\leq\,C\int_{L_{\text{div}}^{2}(\mathbb{T}^{3})}\|\mathbf{q}\|^{2p}_{L_{\text{div}}^{2}(\mathbb{T}^{3})}\,\mathrm{d}\Lambda(\mathbf{q})+C\,\mathbb{E}\bigg{[}\sup_{t\in[0,T]}{\int_{\mathbb{T}^{3}}}\int_{0}^{t}{\bf
u}_{\varepsilon}.\mathcal{P}_{H}\mathcal{G}({\bf u}_{\varepsilon})\,{\rm
d}W\,{\rm d}x\bigg{]}^{p}.$

To handle the right most term of the above inequality, we make use of the
classical Burkholder-Davis-Gundi (BDG) inequality to obtain

$\displaystyle\mathbb{E}$
$\displaystyle\bigg{[}\sup_{t\in[0,T]}{\int_{\mathbb{T}^{3}}}\int_{0}^{t}{\bf
u}_{\varepsilon}.\mathcal{P}_{H}\mathcal{G}({\bf u}_{\varepsilon})\,{\rm
d}W\,{\rm
d}x\bigg{]}^{p}\,\leq\,C\,\mathbb{E}\bigg{[}\int_{0}^{T}\sum_{k\,\geq\,1}\bigg{(}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}\textbf{G}_{k}({\bf
u}_{\varepsilon}){\bf u}_{\varepsilon}\,{\rm d}x\bigg{)}^{2}\bigg{]}^{p/2}$
$\displaystyle\leq\,C\,\mathbb{E}\bigg{[}\int_{0}^{T}\|{\bf
u}_{\varepsilon}(t)\|_{L^{2}(\mathbb{T}^{3})}^{2}\,\sum_{k\,\geq\,1}\|{\textbf{G}}_{k}({\bf
u}_{\varepsilon}(t))\|_{L^{2}(\mathbb{T}^{3})}^{2}\,{\rm d}t\bigg{]}^{p/2}$
$\displaystyle\leq\,C\,\mathbb{E}\bigg{[}\int_{0}^{T}\|{\bf
u}_{\varepsilon}(t)\|_{L^{2}(\mathbb{T}^{3})}^{2}\,(1+\|{\bf
u}_{\varepsilon}(t)\|_{L^{2}(\mathbb{T}^{3})}^{2})\,{\rm d}t\bigg{]}^{p/2}$
$\displaystyle\leq\,C\mathbb{E}\bigg{[}\int_{0}^{T}\|{\bf
u}_{\varepsilon}(t)\|_{L^{2}(\mathbb{T}^{3})}^{2}\,{\rm
d}t\bigg{]}^{p}+\mathbb{E}\bigg{[}\int_{0}^{T}(1+\|{\bf
u}_{\varepsilon}(t)\|_{L^{2}(\mathbb{T}^{3})}^{2})\,{\rm
d}t\bigg{]}^{p}\leq\,C\Bigg{(}1+\int_{L_{\text{div}}^{2}(\mathbb{T}^{3})}\|\mathbf{q}\|^{2p}_{L_{\text{div}}^{2}(\mathbb{T}^{3})}\,\mathrm{d}\Lambda(\mathbf{q})\Bigg{)}.$

This implies that, for any $1\leq p<\infty$, we have

$\displaystyle\mathbb{E}\bigg{[}\sup_{0\leq t\leq
T}\int_{\mathbb{T}^{3}}{|{\bf
u}_{\varepsilon}|^{2}}\,\mathrm{d}x+\varepsilon\int_{0}^{T}\int_{\mathbb{T}^{3}}|\nabla_{x}\mathbf{u}_{\varepsilon}|^{2}\mathrm{d}x\,\mathrm{d}s\bigg{]}^{p}\leq\,C\Bigg{(}1+\int_{L_{\text{div}}^{2}(\mathbb{T}^{3})}\|\mathbf{q}\|^{2p}_{L_{\text{div}}^{2}(\mathbb{T}^{3})}\,\mathrm{d}\Lambda(\mathbf{q})\Bigg{)}\leq\,C(p,\Lambda,T)$

Above relation leads to the following uniform bound

$\displaystyle\mathbf{u}_{\varepsilon}$ $\displaystyle\in
L^{p}(\Omega;L^{\infty}(0,T;L_{\text{div}}^{2}(\mathbb{T}^{3}))).$ (3.1)

###  3.2 Tightness and almost sure representations

For our purpose, to secure almost sure convergence in the probability variable
($\omega$-variable) we make use of Skorokhod-Jakubowski version [30, 24] of
the classical Skorokhod representation theorem. It is well-known that such a
result can be obtained by establishing tightness of probability measures
related to the random variables in quasi-Polish spaces. In what follows, our
first aim is to establish the tightness of the probabilty measures (laws)
generated by the approximate solutions. To do so, we first introduce the
following path space $\mathcal{Y}$ for these measures:

$\displaystyle\mathcal{Y}_{\mathbf{u}}$
$\displaystyle=C_{w}([0,T];L_{\text{div}}^{2}(\mathbb{T}^{3})),$
$\displaystyle\mathcal{Y}_{W}$ $\displaystyle=C([0,T];\mathfrak{U}_{0}),$
$\displaystyle\mathcal{Y}_{C}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)},$
$\displaystyle\mathcal{Y}_{E}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)},$
$\displaystyle\mathcal{Y}_{D}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)}$
$\displaystyle\mathcal{Y}_{\mathcal{V}}$
$\displaystyle=\big{(}L^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{3})),w^{*}\big{)},$
$\displaystyle\mathcal{Y}_{X}$
$\displaystyle=C([0,T];H_{\text{div}}^{-1}(\mathbb{T}^{3})),$
$\displaystyle\mathcal{Y}_{Y}$ $\displaystyle=C([0,T];\mathbb{R}),$
$\displaystyle\mathcal{Y}_{F}$
$\displaystyle=\big{(}L^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),w^{*}\big{)},$

Let us denote by $\mu_{\mathbf{u}_{\varepsilon}}$, and $\mu_{W_{\varepsilon}}$
respectively, the law of $\mathbf{u}_{\varepsilon}$, and $W_{\varepsilon}$ on
the corresponding path space. Moreover, for the martingale terms, let
$\mu_{X_{\varepsilon}}$, and $\mu_{Y_{\varepsilon}}$ denote the law
$X_{\varepsilon}:=\int_{0}^{t}{\mathcal{P}_{H}\mathcal{G}}({\bf
u}_{\varepsilon})\,{\rm d}W$, and
$Y_{\varepsilon}:=\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}\cdot{\mathcal{P}_{H}\mathcal{G}_{k}}({\bf
u}_{\varepsilon})\,{\rm d}x\,{\rm d}W$ on the corresponding path spaces
respectively. Furthermore, let $\mu_{C_{\varepsilon}}$,
$\mu_{D_{\varepsilon}}$, $\mu_{E_{\varepsilon}}$, and
$\mu_{{\mathcal{V}}_{\varepsilon}}$ denote the law of

$\displaystyle C_{\varepsilon}:={{\bf u}_{\varepsilon}\otimes{\bf
u}_{\varepsilon}},\quad D_{\varepsilon}:=\sum_{k\geq 1}|{\bf G}_{k}({\bf
u}_{\varepsilon})|^{2},\quad E_{\varepsilon}:={\frac{1}{2}|{\bf
u}_{\varepsilon}|^{2}},\quad\quad{\mathcal{V}}_{\varepsilon}:=\delta_{{\bf
u}_{\varepsilon}},\quad
F_{\varepsilon}:=\frac{1}{2}\sum_{k\,\geq\,1}|\mathcal{Q}_{h}{{\bf
G}_{k}(u_{\varepsilon})}|^{2},$

respectively, on the corresponding path spaces. Finally, we denote by
$\mu^{\varepsilon}$, the joint law of all the variables on $\mathcal{Y}$. As
stated before, our aim is now to establish tightness of
$\\{\mu^{\varepsilon};\,\varepsilon\in(0,1)\\}$. To this end, first note that
tightness of $\mu_{W_{\varepsilon}}$ is straightforward. Therefore, we focus
on proving tightness of other variables.

######  Proposition 3.1.

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

###### Proof.

For convenience, we rewrite the equation (2.11) as

$\displaystyle\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}(t)\cdot\bm{\varphi}\,{\rm d}x=\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}(0)\cdot\bm{\varphi}\,{\rm
d}x+\int_{0}^{t}\int_{\mathbb{T}^{3}}I_{\varepsilon}(s):\nabla\bm{\varphi}\,{\rm
d}x\,{\rm d}s+\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}\mathcal{G}({\bf
u}_{\varepsilon})\cdot\bm{\varphi}\,{\rm d}x\,{\rm d}W$

for all $t\in[0,T]$, for all $\bm{\varphi}\in
C_{\text{div}}^{\infty}(\mathbb{T}^{3})$. where

$I_{\varepsilon}:=-\varepsilon\nabla{\bf u}_{\varepsilon}+{\bf
u}_{\varepsilon}\otimes{\bf u}_{\varepsilon}$

From the a priori estimate in (3.1), we obtain

$I_{\varepsilon}\in L^{1}(\Omega;L^{2}(0,T;L^{1}(\mathbb{T}^{3})))\subset
L^{1}(\Omega;L^{2}(0,T;W^{-2,2}(\mathbb{T}^{3})))$

uniformly in $\varepsilon$. Let us consider the functional

$\langle\mathcal{I}_{\varepsilon}({t),\bm{\varphi}}\rangle:=\int_{0}^{t}\int_{\mathbb{T}^{3}}I_{\varepsilon}(s):\nabla\bm{\varphi}\,{\rm
d}x\,{\rm d}s,$

which is related to the deterministic part of equation. Then we deduce from
above the following estimate

$\mathbb{E}\bigg{[}\|\mathcal{I}_{\varepsilon}\|_{W^{1,2}(0,T;W_{\text{div}}^{-3,2}(\mathbb{T}^{3}))}\bigg{]}\leq\,C(T).$

For the stochastic term, we have, for $a\,\textgreater\,2$

$\displaystyle\mathbb{E}$
$\displaystyle\Bigg{[}\bigg{\|}\int_{t}^{s}\mathcal{P}_{H}\mathcal{G}({\bf
u}_{\varepsilon})dW\bigg{\|}_{L^{2}(\mathbb{T}^{3})}^{a}\Bigg{]}$
$\displaystyle\leq\,C\,\mathbb{E}\Bigg{[}\bigg{(}\int_{t}^{s}\|\mathcal{P}_{H}\mathcal{G}({\bf
u}_{\varepsilon})\|_{L_{2}(\mathfrak{U},L^{2}(\mathbb{T}^{3}))}^{2}d\sigma\bigg{)}^{a/2}\Bigg{]}\leq\,C\,\mathbb{E}\Bigg{[}\bigg{(}\int_{t}^{s}(1+\|{\bf
u}_{\varepsilon}(\sigma)\|_{L^{2}(\mathbb{T}^{3})}^{2})d\sigma\bigg{)}^{a/2}\Bigg{]}$
$\displaystyle\leq\,C\,\bigg{(}|t-s|^{a/2}\bigg{(}1+\mathbb{E}\Big{[}\sup_{t\in[0,T]}\|{\bf
u}_{\varepsilon}(t)\|_{L^{2}(\mathbb{T}^{3})}\Big{]}^{a/2}\bigg{)}\bigg{)}\leq\,C\,|t-s|^{a/2}\Big{(}1+\mathbb{E}\Big{[}\|{\bf
u}_{\varepsilon}(0)\|_{L^{2}(\mathbb{T}^{3})}^{a}\Big{]}\Big{)}.$

As consequence of Kolmogorov continuity theorem (cf. Lemma 2.5), we have

$\mathbb{E}\Bigg{[}\bigg{\|}\int_{0}^{\cdot}\mathcal{P}_{H}\mathcal{G}({\bf
u}_{\varepsilon})dW\bigg{\|}_{C^{\alpha}([0,T];L_{\text{div}}^{2}(\mathbb{T}^{3}))}^{a}\Bigg{]}\leq\,C$

for all $\alpha\in(\frac{1}{a},\frac{1}{2})$, and $a\,\textgreater\,2$.
Combining the previous estimates and using the embeddings $W^{1,2}(0,T)\subset
C^{1/2}[0,T]$, and $L_{\text{div}}^{2}(\mathbb{T}^{3})\subset
W_{\text{div}}^{-3,2}(\mathbb{T}^{3})$, we conclude

$\mathbb{E}\bigg{[}\|{\bf
u}_{\varepsilon}\|_{C^{\alpha}([0,T];W_{\text{div}}^{-3,2}(\mathbb{T}^{3}))}\bigg{]}\leq\,C(T),$

for some $\alpha\,\textless\,\frac{1}{2}$. Next, we recall the following
compact embedding [6, Chapter 1]

$C^{\alpha}([0,T];W_{\text{div}}^{-3,2})\cap
L^{\infty}(0,T;L_{\text{div}}^{2}(\mathbb{T}^{3}))\subset\subset
C_{w}([0,T];L_{\text{div}}^{2}(\mathbb{T}^{3})),$

to conclude that $\mu_{{\bf u}_{\varepsilon}}$ is tight. ∎

######  Proposition 3.2.

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

###### Proof.

By making use of the _a priori_ bound (3.1), and the fact that all bounded
sets in $L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3}))$ are relatively
compact with respect to the weak-$*$ topology, we obtain the desired result. ∎

######  Proposition 3.3.

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

###### Proof.

This follows from the compactness criterion in
$\big{(}L_{w*}^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{3})),w^{*}\big{)}$.
To see that, define the set

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

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

$\displaystyle\mathcal{L}[{\mathcal{V}}_{\varepsilon}](B^{c}_{M})$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{T}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}\Big{(}|\xi_{1}|^{2}\,d{\mathcal{V}}_{t,x}(\xi)\,{\rm
d}x\,{\rm d}t>M\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{T}\int_{\mathbb{T}^{3}}|{\bf
u}_{\varepsilon}|^{2}\,{\rm d}x\,{\rm
d}t>M\Bigg{)}\leq\frac{1}{M}\mathbb{E}\Big{[}\|{\bf
u}_{\varepsilon}\|_{L^{2}(\mathbb{T}^{3})}^{2}\Big{]}\leq\frac{C}{M}.$

The finishes the proof. ∎

######  Proposition 3.4.

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

###### Proof.

To prove the result, it is enough to observe that the random variable
$X_{\varepsilon}=\int_{0}^{t}\,\mathcal{P}_{H}\mathcal{G}(\mathbf{u}_{\varepsilon})\,\mathrm{d}W(s)\in
L^{p}\big{(}\Omega;W^{\alpha,q}(0,T;L_{\rm div}^{2}(\mathbb{T}^{3}))\big{)}$,
for $q\,\geq\,2$ (see [20]). Therefore, a simple application of the compact
embedding result given in Lemma 2.1 yields required tightness. ∎

######  Proposition 3.5.

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

###### Proof.

It is easy to see that, $Y_{\varepsilon}(t)=\sum_{k\geq
1}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}\cdot\mathcal{P}_{H}{{\bf G}_{k}}({\bf u}_{\varepsilon})\,{\rm
d}x\,{\rm d}W$ is a square integrable martingale, for every
$\varepsilon\in(0,1)$. Notice that for $a>2$

$\displaystyle\mathbb{E}\Big{[}\Big{|}\sum_{k\geq
1}\int_{s}^{t}\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}\cdot\mathcal{P}_{H}{{\bf G}_{k}}({\bf
u}_{\varepsilon})\Big{|}^{a}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\int_{s}^{t}\sum_{k=1}^{\infty}\Big{|}\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}\cdot{{\bf G}_{k}}({\bf
u}_{\varepsilon})\Big{|}^{2}\Big{]}^{a/2}$
$\displaystyle\leq|t-s|^{a/2}\,\Big{(}1+\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\|{\bf u}_{\varepsilon}\|^{a}_{L^{2}(\mathbb{T}^{3})}\Big{]}\Big{)}\leq
C|t-s|^{a/2}.$

Therefore, we can apply the classical Kolmogorov continuity theorem (cf. Lemma
2.5) to conclude that, for some $\beta>0$

$\sum_{k\geq 1}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf
u}_{\varepsilon}\cdot{\mathcal{P}_{H}{\bf G}_{k}}({\bf u}_{\varepsilon})\,{\rm
d}x\,{\rm d}W\in L^{a}(\Omega;C^{\beta}(0,T;\mathbb{R})).$

Therefore, using the well-known compact embedding of $C^{\beta}$ into $C^{0}$,
tightness of law follows. ∎

Making use of results obtained from Proposition 3.1, Proposition 3.2,
Proposition 3.3, Proposition 3.4, and Proposition 3.5, we conclude that

######  Corollary 3.6.

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

Having secured all necessary tightness results, we can now apply Jakubowski-
Skorokhod representation theorem (see also Motyl [30]) to extract almost sure
convergence on a new probability space. To that context, we infer the
following result:

######  Proposition 3.7.

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

  1. (1)

the law of $(\tilde{{\bf
u}}_{\varepsilon},\tilde{W}_{\varepsilon},\tilde{C}_{\varepsilon},\tilde{D}_{\varepsilon},\tilde{E}_{\varepsilon},\tilde{X}_{\varepsilon},\tilde{Y}_{\varepsilon},\tilde{F}_{\varepsilon},\tilde{\nu}_{\varepsilon})$
is given by $\mu^{\varepsilon}$, $\varepsilon\in(0,1)$,

  2. (2)

the law of $(\tilde{{\bf
u}},\tilde{W},\tilde{C},\tilde{D},\tilde{E},\tilde{X},\tilde{Y},\tilde{F},\tilde{\nu})$,
denoted by $\mu$, is a Radon measure,

  3. (3)

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

$\displaystyle\tilde{{\bf u}}_{\varepsilon}\rightarrow\bar{{\bf
u}}\,\,\text{in}\,\,C_{w}([0,T];L_{\rm div}^{2}(\mathbb{T}^{3})),\quad$
$\displaystyle\tilde{W}_{\varepsilon}\rightarrow\tilde{W}\,\,\text{in}\,\,C([0,T];\mathcal{U}_{0})),$
$\displaystyle\tilde{C}_{\varepsilon}\rightarrow\tilde{C}\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),\qquad$
$\displaystyle\tilde{D}_{\varepsilon}\rightarrow\tilde{D}\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{Y}_{\varepsilon}\rightarrow\tilde{Y}\,\,\text{in}\,\,C([0,T];\mathbb{R}),\qquad$
$\displaystyle\tilde{E}_{\varepsilon}\rightarrow\tilde{E}\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{\nu}_{\varepsilon}\rightarrow\tilde{\nu}\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}((0,T)\times\mathbb{T}^{3};\mathcal{P}(\mathbb{R}^{3})),\qquad$
$\displaystyle\tilde{X}_{\varepsilon}\rightarrow\tilde{X}\,\,\text{in}\,\,C([0,T];W_{\rm
div}^{-1,2}(\mathbb{T}^{3})),$
$\displaystyle\tilde{F}_{\varepsilon}\rightarrow\tilde{F}\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$

  4. (4)

For any $\varepsilon$, $\tilde{W}_{\varepsilon}=\tilde{W}$,
$\tilde{\mathbb{P}}$-a.s.

  5. (5)

For any Carathéodory function $J=J(t,x,{\bf u})$, where
$(t,x)\in(0,T)\times\mathbb{T}^{3}$ and ${\bf u}\in\mathbb{R}^{3}$, satisfying
for some $p$ the growth condition $|J(t,x,{\bf u})|\leq 1+|{\bf u}|^{p}$,
uniformly in $(t,x)$. Then we have $\tilde{\mathbb{P}}$-a.s.

$J(\tilde{\bf u}_{\varepsilon})\rightarrow\overline{J(\tilde{\bf
u})}\,\,\text{in}\,\,L^{r}((0,T)\times\mathbb{T}^{3}),\,\,\text{for
all}\,\,1<r\leq\frac{2}{p}.$

###### Proof.

Proof of the items $(1)$, $(2)$, and $(3)$ directly follow from Jakubowski-
Skorokhod representation theorem. For the proof of the item $(4)$, we refer to
Theorem 2.4, and [30]. For the proof of the item $(5)$, we refer to the Lemma
2.2. ∎

####  3.2.1 Passing to the limit

Note that, in view of the equality of joint laws, the energy inequality (2.10)
and the a priori estimate (3.1) for the new random variables hold on the new
probability space. Making use of convergence results given by Proposition 3.7,
we can now pass to the limit in approximate equation (1.2), and the energy
inequality (2.10). First we show that the approximations $\tilde{\bf
u}_{\varepsilon}$ solve the equation given by (1.2) on the new probability
space $(\tilde{\Omega},\tilde{\mathbb{F}},\tilde{\mathbb{P}})$. For that
purpose, let us denote by $(\tilde{\mathbb{F}}_{t}^{\varepsilon})$ and
$(\tilde{\mathbb{F}}_{t})$, $\tilde{\mathbb{P}}$-augmented canonical
filtrations of the process $(\tilde{\bf
u}_{\varepsilon},\tilde{W}_{\varepsilon})$ and
$(\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\textbf{u}}\rangle,\tilde{W},\tilde{X},\tilde{Y})$,
respectively. This means

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

where $\mathbf{s}_{t}$ is the restriction operator to the interval $[0,t]$
acting on various path spaces.

######  Proposition 3.8.

For every $\varepsilon\in(0,1)$,
$\big{(}(\tilde{\Omega},\tilde{\mathbb{F}},(\tilde{\mathbb{F}}_{\varepsilon,t})_{t\geq
0},\tilde{\mathbb{P}}),\tilde{{\bf u}}_{\varepsilon},\tilde{W}\big{)}$ is a
finite energy weak martingale solution to (1.2) with the initial law
$\Lambda_{\varepsilon}$.

###### Proof.

Proof of the above proposition is standard, and one can furnish the proof
following the same line of argument, as in the monograph by Breit et. al. [7,
Theorem 2.9.1]. For brevity, we skip all the details. ∎

We remark that, in light of the above proposition, the new random variables
satisfy the following equations and the energy inequality on the new
probability space

  * •

for all $\bm{\varphi}\in C_{\text{div}}^{\infty}(\mathbb{T}^{3})$ we have

$\displaystyle\langle\mathbf{\tilde{u}}_{\varepsilon}(t),\bm{\varphi}\rangle$
$\displaystyle=\langle\mathbf{\tilde{u}}_{\varepsilon}(0),\bm{\varphi}\rangle-\int_{0}^{t}\langle\mathbf{\tilde{u}}_{\varepsilon}\otimes\mathbf{\tilde{u}}_{\varepsilon},\nabla_{x}\bm{\varphi}\rangle\,\mathrm{d}s+{\varepsilon}\,\int_{0}^{t}\langle\nabla_{x}\mathbf{\tilde{u}}_{\varepsilon}\,,\nabla_{x}\bm{\varphi}\rangle\,\mathrm{d}s+\int_{0}^{t}\langle\mathcal{G}(\mathbf{\tilde{u}}_{\varepsilon}),\bm{\varphi}\rangle\,\mathrm{d}W$
(3.2)

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

  * •

the energy inequality

$\displaystyle-\int_{0}^{T}\partial_{t}\phi\int_{\mathbb{T}^{3}}\frac{1}{2}|\tilde{{\bf
u}}_{\varepsilon}|^{2}\,{\rm d}x\,{\rm
d}t+{\varepsilon}\,\int_{0}^{T}\phi\int_{\mathbb{T}^{3}}|\nabla_{x}\tilde{{\bf
u}}_{\varepsilon}|^{2}\,{\rm d}x\,{\rm d}t$ (3.3)
$\displaystyle\leq\phi(0)\int_{\mathbb{T}^{n}}\frac{1}{2}{|\mathbf{u}_{\varepsilon}(0)|^{2}}\,{\rm
d}x+\sum_{k=1}^{\infty}\int_{0}^{T}\phi\bigg{(}\int_{\mathbb{T}^{n}}\mathbf{G}_{k}(\tilde{{\bf
u}}_{\varepsilon})\cdot\tilde{{\bf u}}_{\varepsilon}\,{\rm d}{x}\bigg{)}{\rm
d}W_{k}+\frac{1}{2}\sum_{k=1}^{\infty}\int_{0}^{T}\phi\int_{\mathbb{T}^{n}}|\mathcal{P}_{H}\mathbf{G}_{k}(\tilde{{\bf
u}}_{\varepsilon})|^{2}\,{\rm d}t$

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

Now we are in a position to pass to the limit in $\varepsilon$ in (3.2) and
(3.3). To see this, note that we have a priori estimate (3.1) for the new
random variable. Therefore, an application of Lemma 2.2 helps us to conclude
that $\tilde{\mathbb{P}}$-a.s.,

$\displaystyle\tilde{\textbf{u}}_{\varepsilon}\rightharpoonup\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\textbf{u}}\rangle,\,\,\text{weakly
in}\,\,L^{2}((0,T);L_{\text{div}}^{2}(\mathbb{T}^{3})).$

Moreover, making use of item $(5)$ of Lemma 3.7, we conclude that
$\mathbb{P}$-a.s.

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

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

$\mathcal{Q}_{H}\mathbf{G}_{k}(\tilde{\bf
u}_{\varepsilon})\rightharpoonup\mathcal{Q}_{H}\big{\langle}\tilde{\mathcal{V}}_{t,x}^{\omega};\mathbf{G}_{k}(\tilde{\bf
u})\big{\rangle}\,\,\mbox{weakly
in}\,\,L^{2}((0,T);(L^{2}(\mathbb{T}^{3}))^{\perp}).$

Indeed, Let $\mathbf{v}\in L^{2}((0,T);(L^{2}(\mathbb{T}^{3}))^{\perp})$ then
orthogonal property of projection $\mathcal{Q}_{H}$ implies that

$\lim_{\varepsilon\to 0}\langle\mathcal{Q}_{H}\mathbf{G}_{k}(\tilde{\bf
u}_{\varepsilon}),\mathbf{v}\rangle=\lim_{\varepsilon\to
0}\langle\mathbf{G}_{k}(\tilde{\bf
u}_{\varepsilon}),\mathbf{v}\rangle=\langle\langle\tilde{\mathcal{V}}_{t,x}^{\omega};\mathbf{G}_{k}(\tilde{\bf
u})\big{\rangle},\mathbf{v}\rangle=\langle\mathcal{Q}_{H}\langle\tilde{\mathcal{V}}_{t,x}^{\omega};\mathbf{G}_{k}(\tilde{\bf
u})\big{\rangle},\mathbf{v}\rangle$

As usual, to identify the weak limits related to the nonlinear terms present
in the equations, we first need to introduce corresponding concentration
defect measures

$\displaystyle\tilde{\mu}_{C}$
$\displaystyle=\tilde{C}-\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};{\tilde{\bf
u}\otimes\tilde{\bf u}}\right\rangle\,{\rm d}x\,{\rm
d}t,\,\,\tilde{\mu}_{E}=\tilde{E}-\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\frac{1}{2}{|\tilde{\bf
u}|^{2}}\right\rangle\,{\rm d}x\,{\rm d}t,$ $\displaystyle\tilde{\mu}_{D}$
$\displaystyle=\tilde{D}-\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\sum_{k\geq
1}|{{\bf G}}_{k}(\tilde{\bf u})|^{2}\right\rangle\,{\rm d}x\,{\rm
d}t,\,\,\tilde{\mu}_{F}=\tilde{F}-\Big{|}\mathcal{Q}_{H}\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\sum_{k\geq
1}|{{\bf G}}_{k}(\tilde{\bf u})|^{2}\right\rangle\Big{|}^{2}\,{\rm d}x\,{\rm
d}t.$

In view of the discussion in Subsection 2.1.1, and making use of above
concentration defect measures, we can conclude that $\mathbb{\tilde{P}}$
almost surely

$\displaystyle\tilde{C}_{\varepsilon}\rightharpoonup\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};{\tilde{\bf
u}\otimes\tilde{\bf u}}\right\rangle\,{\rm d}x\,{\rm
d}t+\tilde{\mu}_{C},\,\,\text{ weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{D}_{\varepsilon}\rightharpoonup\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\sum_{k\geq
1}|{{\bf G}}_{k}(\tilde{\bf u})|^{2}\right\rangle\,{\rm d}x\,{\rm
d}t+\tilde{\mu}_{D},\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{E}_{\varepsilon}\rightharpoonup\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\frac{1}{2}{|\tilde{\bf
u}|^{2}}\right\rangle\,{\rm d}x\,{\rm d}t+\tilde{\mu}_{E},\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}^{+}_{b}(\mathbb{T}^{3})),$
$\displaystyle\tilde{F}_{\varepsilon}\rightharpoonup\Big{|}\mathcal{Q}_{H}\left\langle\mathcal{\tilde{V}}^{\omega}_{(\cdot,\cdot)};\sum_{k\geq
1}|{{\bf G}}_{k}(\tilde{\bf u})|^{2}\right\rangle\Big{|}^{2}\,{\rm d}x\,{\rm
d}t+\tilde{\mu}_{F},\,\,\text{weak-$*$
in}\,\,L_{w*}^{\infty}(0,T;\mathcal{M}^{+}_{b}(\mathbb{T}^{3})).$

Note that both defect measures $\tilde{\mu}_{E}$, and $\tilde{\mu}_{F}$ are
positive, thanks to the lower semi-continuity property of norms. Next, we move
on to the martingale terms $\tilde{X}_{\varepsilon}$, appearing in the
momentum equation, and $\tilde{Y}_{\varepsilon}$, appearing in the energy
inequality. Regarding convergence of these terms, we state following
propositions.

######  Proposition 3.9.

For every time $t\in[0,T]$, $\mathbb{P}$-almost surely
$\tilde{Y}_{\varepsilon}(t)\rightarrow\tilde{Y}(t)$ in $\mathbb{R}$, where
$\tilde{Y}(t)$ is a real valued square-integrable martingale with respect to
the filtration $(\tilde{\mathbb{F}}_{t})$.

###### Proof.

First of all, in view of the Proposition 3.7, we conclude that
$\tilde{Y}_{\varepsilon}\rightarrow\tilde{Y}$ $\mathbb{P}$-a.s. in
$C([0,T];\mathbb{R})$. To claim that $\tilde{Y}(t)$ is a martingale, as usual,
it is sufficient to show that

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

for all $t,s\in[0,T]$ with $s\leq t$. In other words, it is enough to prove
that

$\tilde{\mathbb{E}}\Big{[}\mathfrak{L}_{s}(\tilde{\Phi})\big{(}\tilde{Y}(t)-\tilde{Y}(s)\big{)}\Big{]}=0,$

where we denote
$\tilde{\Phi}:=(\langle{\mathcal{\tilde{V}}^{\omega}_{t,x}};\tilde{\textbf{u}}\rangle,\tilde{W},\tilde{X},\tilde{Y})$,
and on the path space $\underline{\mathcal{Y}}:=\mathcal{Y}_{{\bf
u}}\times\mathcal{Y}_{W}\times\mathcal{Y}_{X}\times\mathcal{Y}_{Y}$, we denote
$\mathfrak{L}_{s}$ by any bounded continuous functional which depends on on
the values of $\tilde{\Phi}$ restricted to $[0,s]$. The idea is to use the
fact that $\tilde{Y}_{\varepsilon}(t)$ is a martingale, i.e.,

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

for all bounded continuous functional $\mathfrak{L}_{s}$ on the same path
space, and $\tilde{\Phi}_{\varepsilon}=(\tilde{\bf
u}_{\varepsilon},\tilde{W},\tilde{X}_{\varepsilon},\tilde{Y}_{\varepsilon})$.
At this point, we recall Proposition 3.7 to conclude that
$\tilde{\Phi}_{\varepsilon}\rightarrow\tilde{\Phi}$, $\mathbb{P}$-a.s. in the
(weak) topology of $\underline{\mathcal{Y}}$. This, in particular, implies
that
$\mathfrak{L}_{s}(\tilde{\Phi}_{\varepsilon})\rightarrow\mathfrak{L}_{s}(\tilde{\Phi})$
$\mathbb{P}$-a.s. Now given this information, along with the fact that
$\tilde{Y}_{\varepsilon}(t)\in L^{2}(\tilde{\Omega})$, we may apply classical
Vitali’s convergence theorem to pass to the limit in $\varepsilon$ to conclude
that $\tilde{Y}(t)$ is a martingale. ∎

######  Proposition 3.10.

For each time $t\in[0,T]$,
$\tilde{X}_{\varepsilon}(t)\rightarrow\tilde{X}(t)$, $\mathbb{P}$-almost
surely in the topology of $H_{\rm div}^{-1}(\mathbb{T}^{3})$. Moreover,
$\tilde{X}(t)$ is also a $H_{\rm div}^{-1}(\mathbb{T}^{3})$-valued square
integrable martingale with respect to the filtration
$(\tilde{\mathbb{F}}_{t})$.

###### Proof.

Note that, as before, we would not be able to identify the structure of the
martingale $\tilde{X}(t)$, instead we just prove that $\tilde{X}(t)$ is a
martingale. In what follows, with the help of the Proposition 3.7, we can
conclude that for each $t\in[0,T]$,
$\tilde{X}_{\varepsilon}(t)\rightarrow\tilde{X}(t)$, $\mathbb{P}$-a.s. in the
topology of $H^{-1}_{\text{div}}(\mathbb{T}^{3})$. Fianlly to show that
$\tilde{X}(t)$ is a martingale, it is enough to demonstrate that for all
$i\geq 1$

$\tilde{\mathbb{E}}\Big{[}\mathfrak{L}_{s}(\tilde{\Phi})\big{<}\tilde{X}(t)-\tilde{X}(s),g_{i}\big{>}\Big{]}=0,$

where $g_{i}$’s are given orthonormal basis for the space $H_{\rm
div}^{-1}(\mathbb{T}^{3})$. We follow the usual argument to establish the
result. To that ocntext, we first use the information that

$\tilde{\mathbb{E}}\Big{[}\mathfrak{L}_{s}(\tilde{\Phi}_{\varepsilon})\big{<}\tilde{X}_{\varepsilon}(t)-\tilde{X}_{\varepsilon}(s),g_{i}\big{>}\Big{]}=0,$

for all $i\geq 1$. Then, like before, we can pass to the limit in the
parameter $\varepsilon$ to show that $\tilde{M}(t)$ is a martingale. Indeed,
this argument requires uniform integrabilty in $\omega$ variable, and can be
achieved using BDG inequality:

$\displaystyle\tilde{\mathbb{E}}\Big{[}\big{|}\big{<}\tilde{X}_{\varepsilon}(t),g_{i}\big{>}\big{|}^{p}\Big{]}$
$\displaystyle=\tilde{\mathbb{E}}\bigg{[}\bigg{|}\Big{<}\int_{0}^{t}\mathcal{P}_{H}{\mathcal{G}}(\tilde{\textbf{u}}_{\varepsilon})\,{\rm
d}\tilde{W},g_{i}\Big{>}\bigg{|}^{p}\bigg{]}\leq
C\,\tilde{\mathbb{E}}\Bigg{[}\sup_{0\leq t\leq
T}\bigg{\|}\int_{0}^{t}\mathcal{P}_{H}{\mathcal{G}}(\tilde{\textbf{u}}_{\varepsilon})\,{\rm
d}\tilde{W}\bigg{\|}^{p}_{H_{\rm div}^{-1}(\mathbb{T}^{3})}\Bigg{]}$
$\displaystyle\leq
C\,\tilde{\mathbb{E}}\Bigg{[}\bigg{(}\int_{0}^{T}\|{\mathcal{G}}(\tilde{\textbf{u}}_{\varepsilon})\|^{2}_{L_{2}(\mathcal{U},L^{2}(\mathbb{T}^{3}))}\,{\rm
d}s\bigg{)}^{p/2}\Bigg{]}\leq C.$

This finishes the proof. ∎

In view of the above discussions, we can pass to the limit in (3.2) to
conclude that

$\displaystyle\int_{\mathbb{T}^{3}}\langle\tilde{\mathcal{V}}^{\omega}_{\tau,x};{\bf
u}\rangle\cdot\bm{\varphi}(\tau,\cdot)\,{\rm
d}x-\int_{\mathbb{T}^{3}}\langle\tilde{\mathcal{V}}^{\omega}_{0,x};{\bf
u}\rangle\cdot\bm{\varphi}(0,\cdot)\,{\rm d}x$
$\displaystyle\qquad=\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\langle\tilde{\mathcal{V}}^{\omega}_{t,x};{{\bf
u}\otimes{\bf u}}\rangle:\nabla_{x}\bm{\varphi}\,{\rm d}x\,{\rm
d}t+\int_{\mathbb{T}^{3}}\bm{\varphi}\,\int_{0}^{\tau}d\tilde{X}(t)\,{\rm
d}x+\int_{0}^{\tau}\int_{\mathbb{T}^{3}}\nabla_{x}\bm{\varphi}:d\tilde{\mu}_{C},$

holds $\tilde{\mathbb{P}}$-a.s., for all $\tau\in[0,T)$, and for all
$\bm{\varphi}\in C_{\text{div}}^{\infty}(\mathbb{T}^{3};\mathbb{R}^{3})$. This
implies that (2.14) holds. Next, we focus on proving the energy inequality
(2.15). To that context, making use of identifications of weak limits of
various terms involved in the energy inequality, we can pass to the limit in
$\varepsilon$ in (3.3). This yield

$\displaystyle-\int_{0}^{T}\partial_{t}\phi\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}{x}+\mathcal{\tilde{D}}(\tau)\bigg{)}{\rm
d}\tau\leq\,\phi(0)\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{0,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}x$
$\displaystyle\quad+\frac{1}{2}\sum_{k\geq\,1}\int_{0}^{T}\phi(\tau)\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|^{2}}\right\rangle\,{\rm
d}x\,{\rm
d}\tau-\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\phi(\tau)\int_{\mathbb{T}^{3}}\Big{(}\mathcal{Q}_{H}\left\langle\tilde{\mathcal{V}}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|}\right\rangle\Big{)}^{2}\,{\rm
d}x\,{\rm d}\tau$
$\displaystyle\qquad+\frac{1}{2}\int_{0}^{T}\phi(\tau)\int_{\mathbb{T}^{3}}{\rm
d}\tilde{\mu}_{D}\,{\rm
d}\tau-\frac{1}{2}\int_{0}^{T}\phi(\tau)\int_{\mathbb{T}^{3}}{\rm
d}\tilde{\mu}_{F}\,{\rm d}\tau+\int_{0}^{T}\phi(\tau)\,{\rm d}\tilde{Y}(\tau)$
(3.4)

holds $\tilde{\mathbb{P}}$-a.s., for all $\phi\in
C_{c}^{\infty}([0,T)),\,\phi\,\geq\,0$. Note that here
$\mathcal{\tilde{D}}(\tau):=\tilde{\mu}_{E}(\tau)(\mathbb{T}^{3})$. To
proceed, we first 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 us denote by
$\phi_{r}$, a Lipschitz fucntion which is linear on $[s-r,s]$ and $[t,t+r]$
such that

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

Then, a standard regularization argument reveals that $\phi_{r}$ can be used
as an admissible test fuction in (3.3). Therefore, replacing the test function
$\phi$ by $\phi_{r}$ in (3.2.1), we get $\mathbb{P}$-a.s., for all
$0\,\textless\,s\,\textless\,t\,\textless\,T$

$\displaystyle\frac{1}{r}\int_{t}^{t+r}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}{x}+\mathcal{\tilde{D}}(\tau)\bigg{)}{\rm
d}\tau\leq\,\frac{1}{r}\int_{s-r}^{s}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}{x}+\mathcal{\tilde{D}}(\tau)\bigg{)}{\rm
d}\tau$
$\displaystyle\quad+\frac{1}{2}\sum_{k\geq\,1}\int_{s-r}^{t+r}\phi_{r}(\tau)\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|^{2}}\right\rangle\,{\rm
d}x\,{\rm
d}\tau-\frac{1}{2}\sum_{k\geq\,1}\int_{s-r}^{t+r}\phi_{r}(\tau)\int_{\mathbb{T}^{3}}\Big{(}\mathcal{Q}_{H}\left\langle\tilde{\mathcal{V}}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|}\right\rangle\Big{)}^{2}\,{\rm
d}x\,{\rm d}\tau$
$\displaystyle\qquad+\frac{1}{2}\int_{s-r}^{t+r}\phi_{r}(\tau)\int_{\mathbb{T}^{3}}{\rm
d}\tilde{\mu}_{D}\,{\rm
d}\tau-\frac{1}{2}\int_{s-r}^{t+r}\phi_{r}(\tau)\int_{\mathbb{T}^{3}}{\rm
d}\tilde{\mu}_{F}\,{\rm d}\tau+\int_{s-r}^{t+r}\phi_{r}(\tau)\,{\rm
d}\tilde{Y}(\tau).$ (3.5)

Now using the non-negativity of the defect measure $\tilde{\mu}_{F}$, and
letting $r\to 0^{+}$ in (3.2.1), we obtain $\mathbb{P}$-a.s., for all
$0\,\textless\,s\,\textless\,t\,\textless\,T$

$\displaystyle\lim_{r\to
0^{+}}\frac{1}{r}\int_{t}^{t+r}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}{x}+\mathcal{\tilde{D}}(\tau)\bigg{)}{\rm
d}\tau\,$ $\displaystyle\qquad\leq\,\lim_{r\to
0^{+}}\frac{1}{r}\int_{s-r}^{s}\bigg{(}\int_{\mathbb{T}^{3}}\left\langle\mathcal{\tilde{V}}^{\omega}_{\tau,x};\frac{|{\bf
u}|^{2}}{2}\right\rangle\,{\rm d}{x}+\mathcal{\tilde{D}}(\tau)\bigg{)}{\rm
d}\tau+\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|^{2}}\right\rangle\,{\rm
d}{x}\,{\rm d}\tau$
$\displaystyle\qquad\qquad-\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\Big{(}\mathcal{Q}_{H}\left\langle\tilde{\mathcal{V}}^{\omega}_{\tau,x};{|{\textbf{G}_{k}(u)}|}\right\rangle\Big{)}^{2}\,{\rm
d}{x}\,{\rm d}\tau+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}{\rm
d}\tilde{\mu}_{D}\,{\rm d}\tau+\int_{s}^{t}{\rm d}\tilde{Y}(\tau)$ (3.6)

We remark that for $s=0$ we need to use a slightly different test function to
conclude the result. In this case we take

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

and apply the same argument as before to establish that the energy inequality
(2.15) holds.

Now we are only left with the verifications of (2.16), and item (i) of
Definition 2.15. To proceed, we start with the following lemma.

######  Lemma 3.11.

Given a stochastic process $h$, as in item (i) of Definition 2.15

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

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

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

where $g_{i}$’s are orthonormal basis for $H^{-1}_{\rm div}(\mathbb{T}^{3})$
and bracket $\langle\cdot,\cdot\rangle$ denotes inner product in the same
space.

###### Proof.

Following the definition of cross variation between two Hilbert space valued
martingales, given by Da Prato $\&$ Zabczyk [13, Section 3.4], we have

$\displaystyle\Big{<}\Big{<}h(t),\tilde{X}_{\varepsilon}(t)\Big{>}\Big{>}=\sum_{i,j}\Big{<}\Big{<}\big{\langle}h(t),g_{i}\big{\rangle},\big{\langle}\tilde{X}_{\varepsilon}(t),g_{j}\big{\rangle}\Big{>}\Big{>}g_{i}\otimes
g_{j},$

where using the informations of the processes $h(t)$ and
$\tilde{X}_{\varepsilon}(t)$, we have

$\displaystyle\Big{<}\Big{<}\big{\langle}h(t),g_{i}\big{\rangle},\big{\langle}\tilde{X}_{\varepsilon}(t),g_{j}\big{\rangle}\Big{>}\Big{>}$
$\displaystyle=\Big{<}\Big{<}\sum_{k\geq
1}\int_{0}^{t}\big{\langle}\mathbb{D}^{s}_{t}h(e_{k}),g_{i}\big{\rangle}{\rm
d}\tilde{W}_{k},\sum_{k\geq
1}\int_{0}^{t}\big{\langle}\mathcal{P}_{H}\textbf{G}_{k}(\tilde{{\bf
u}}_{\varepsilon}),g_{j}\big{\rangle}{\rm d}\tilde{W}_{k}\Big{>}\Big{>}$
$\displaystyle=\sum_{k\geq\,1}\int_{0}^{t}\big{\langle}\mathbb{D}^{s}_{t}h(e_{k}),g_{i}\big{\rangle}\,\big{\langle}\mathcal{P}_{H}\textbf{G}_{k}(\tilde{{\bf
u}}_{\varepsilon}),g_{j}\big{\rangle}\,{\rm d}s$

Therefore we get

$\displaystyle\Big{<}\Big{<}h(t),\tilde{X}_{\varepsilon}(t)\Big{>}\Big{>}=\sum_{i,j}\Bigg{(}\sum_{k=1}^{\infty}\int_{0}^{t}\big{<}\mathbb{D}^{s}_{t}h(e_{k}),h_{i}\big{>}\,\big{<}\mathcal{P}_{H}\mathbf{G}_{k}(\tilde{\bf
u}_{\varepsilon}),h_{j}\big{>}\,{\rm d}s\Bigg{)}h_{i}\otimes h_{j}.$

This equivalently implies that

$\displaystyle\tilde{\mathbb{E}}\Big{[}\mathfrak{L}_{s}(\tilde{\Phi}_{\varepsilon})\Big{(}\big{<}h(t),g_{i}\big{>}\big{<}\tilde{X}_{\varepsilon}(t),g_{j}\big{>}-\sum_{k=1}^{\infty}\int_{0}^{t}\big{<}\mathbb{D}^{s}_{t}h(e_{k}),g_{i}\big{>}\,\big{<}\mathcal{P}_{H}\mathbf{G}_{k}(\tilde{\bf
u}_{\varepsilon}),g_{j}\big{>}\,{\rm d}s\Big{)}\Big{]}=0.$

Since we have $\mathbb{P}$-a.s.

$\mathbf{G}_{k}(\tilde{\bf
u}_{\varepsilon})\rightharpoonup\big{\langle}\tilde{\mathcal{V}}_{t,x}^{\omega};\mathbf{G}_{k}(\tilde{\mathbf{u}})\big{\rangle},\,\,\mbox{weakly
in}\,\,L^{2}((0,T);L^{2}(\mathbb{T}^{3})),$

This gives $\mathbb{P}$-a.s

$\mathcal{P}_{H}\mathbf{G}_{k}(\tilde{\bf
u}_{\varepsilon})\rightharpoonup\mathcal{P}_{H}\big{\langle}\tilde{\mathcal{V}}_{t,x}^{\omega};\mathbf{G}_{k}(\tilde{\mathbf{u}})\big{\rangle},\,\,\mbox{weakly
in}\,\,L^{2}((0,T);L^{2}_{\rm div}(\mathbb{T}^{3})).$

Moreover, making use of the a priori estimate (3.1), we have

$\displaystyle\tilde{\mathbb{E}}\Big{[}\int_{0}^{T}\|\mathcal{P}_{H}\mathcal{G}(\tilde{\bf
u}_{\varepsilon})\|^{2}_{L_{2}(\mathcal{U};H_{\rm
div}^{-1}(\mathbb{T}^{3}))}\,dt\Big{]}\leq\tilde{\mathbb{E}}\Big{[}\int_{0}^{T}\int_{\mathbb{T}^{3}}(\tilde{1}+|\tilde{\bf
u}_{\varepsilon}|^{2})\,dx\,dt\Big{]}\leq C.$

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

$\mathcal{P}_{H}\mathbf{G}_{k}(\tilde{\bf
u})\rightharpoonup\mathcal{P}_{H}\big{\langle}\tilde{\mathcal{V}}_{t,x,}^{\omega};\mathbf{G}_{k}(\tilde{\bf
u})\big{\rangle},\,\,\mbox{weakly in}\,\,L^{2}((0,T);H_{\rm
div}^{-1}(\mathbb{T}^{3})).$

Therefore we can pass to the limit in $\varepsilon\rightarrow 0$, thanks to
uniform integrabilty, to conclude

$\displaystyle\tilde{\mathbb{E}}\Big{[}\mathfrak{L}_{s}(\tilde{\Phi})$
$\displaystyle\Big{(}\big{<}h(t),g_{i}\big{>}\big{<}\tilde{X}(t),g_{j}\big{>}-\sum_{k=1}^{\infty}\int_{0}^{t}\big{<}\mathbb{D}^{s}_{t}h(e_{k}),g_{i}\big{>}\,\big{<}\mathcal{P}_{H}\left\langle\mathcal{\tilde{V}}^{\omega}_{s,x};\mathbf{G}_{k}(\tilde{\bf
u})\right\rangle,g_{j}\big{>}\,{\rm d}s\Big{)}\Big{]}=0.$

This finishes the proof the lemma. ∎

Finally, regarding the verifications of (2.16), we have the following lemma:

######  Lemma 3.12.

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

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

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

###### Proof.

Clearly, by Lemma 2.3, we can conclude that $\tilde{\mu}_{E}$ dominates the
defect measure $\tilde{\mu}_{C}$. On the other hand, making use of hypotheses
(2.3), (2.4), we can show the required dominance of $\tilde{\mu}_{E}$ over
$\tilde{\mu}_{D}$. Indeed, note that the function

${\bf u}\mapsto\sum_{k\geq 1}{|{{\bf G}}_{k}({\bf u})|^{2}}\ \mbox{is
continuous},$

and clearly dominated by the total energy

$\sum_{k\geq 1}{|{{\bf G}}_{k}({\bf u})|^{2}}\leq C\left(1+{|{\bf
u}|^{2}}\right)$

This finishes the proof of the lemma. ∎

##  4 Weak-Strong Uniqueness Principle for Euler System

In this section we prove Theorem 2.18 through auxiliary results. Essentially
the proof relies upon successful identification of the cross variation between
two processes, given by a measure-valued Euler solution and a local strong
Euler solution. In what follows, we begin with the following lemma.

######  Lemma 4.1 (Weak Itô Product Formula).

Let $\mathbf{q}$ be a stochastic process on
$\big{(}\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},\mathbb{P}\big{)}$ such
that

$\mathbf{q}\in C_{w}([0,T];L^{2}_{\rm div}(\mathbb{T}^{3}))\cap
L^{\infty}((0,T);L^{2}_{\rm div}(\mathbb{T}^{3})),\,\,\mathbb{P}-\text{a.s.}$
$\mathbb{E}\Big{[}\sup_{t\in[0,T]}\|\mathbf{q}\|_{L^{2}_{\rm
div}(\mathbb{T}^{3})}^{2}\Big{]}\,\textless\,+\infty.$

Moreover, it satisfies $\mathbb{P}$-a.s.

$\displaystyle\int_{\mathbb{T}^{3}}\mathbf{q}(t)\cdot\bm{\varphi}\,dx=\int_{\mathbb{T}^{3}}\textbf{q}(0)\cdot\bm{\varphi}\,dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathbf{q}_{1}:\nabla\bm{\varphi}dxds+\int_{0}^{t}\int_{\mathbb{T}^{3}}\nabla\bm{\varphi}:\,d\mu(x,s)\,ds+\int_{\mathbb{T}^{3}}\bm{\varphi}\cdot\int_{0}^{t}dMdx$
(4.1)

for all $t\in[0,T]$, and test function $\varphi\in
C^{\infty}_{\mathrm{div}}(\mathbb{T}^{3})$. Here $M$ is a continuous square
integrable $H^{-1}_{\rm div}(\mathbb{T}^{3})$-valued martingale and
$\mathbf{q}_{1},\,\,\mu$ are progressible measurable with

$\mathbf{q}_{1}\in L^{2}(\Omega;L^{1}(0,T;L^{2}_{\rm
div}(\mathbb{T}^{3}))),\qquad\mu\in
L^{1}(\Omega;L_{w*}^{\infty}(0,T;\mathcal{M}_{b}(\mathbb{T}^{3}))).$

Let Q be a stochastic process on
$\big{(}\Omega,\mathbb{F},(\mathbb{F}_{t})_{t\geq 0},\mathbb{P}\big{)}$
satisfying

$\textbf{Q}\in
C([0,T];C^{1}(\mathbb{T}^{3})),\,\mathbb{P}-\text{a.s.}\,\,\text{and}\,\,\,\,\mathbb{E}\big{[}\sup_{t\in[0,T]}\|\mathbf{Q}\|_{L_{\rm
div}^{2}(\mathbb{T}^{3})\cap{C}(\mathbb{T}^{3})}^{2}\big{]}\textless\,\infty,$

be such that

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

Here $\mathbf{Q}_{1},\mathbf{Q}_{2}$ are progressible measurable with

$\displaystyle\mathbf{Q}_{1}\in L^{2}(\Omega;L^{1}((0,T);$ $\displaystyle
L^{2}_{\rm div}(\mathbb{T}^{3}))),\,\qquad\,\textbf{Q}_{2}\in
L^{2}(\Omega;L^{2}((0,T);L_{2}(\mathfrak{U};L_{\rm
div}^{2}(\mathbb{T}^{3})))),$
$\displaystyle\sum_{k=1}^{\infty}\int_{0}^{T}\|\mathbf{Q}_{2}(e_{k})\|^{2}_{L^{2}(\mathbb{T}^{3})}\in
L^{1}(\Omega).$

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

$\displaystyle\int_{\mathbb{T}^{3}}\mathbf{q}(t)\cdot\mathbf{Q}(t)\,{\rm
d}{x}$
$\displaystyle=\int_{\mathbb{T}^{3}}\mathbf{q}(0)\cdot\mathbf{Q}(0)\,{\rm
d}{x}+\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathbf{q}_{1}:\nabla\mathbf{Q}\,{\rm
d}{x}\,{\rm d}s+\int_{0}^{t}\int_{\mathbb{T}^{3}}\nabla\mathbf{Q}:\,{\rm
d}\mu\,{\rm d}s$
$\displaystyle\qquad+\int_{\mathbb{T}^{3}}\int_{0}^{t}\mathbf{Q}\cdot{\rm
d}M\,{\rm
d}x+\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathbf{Q}_{1}\cdot\mathbf{q}\,{\rm
d}x\,{\rm
d}s+\int_{\mathbb{T}^{3}}\int_{0}^{t}\mathbf{q}\cdot\mathbf{Q}_{2}\,{\rm
d}W\,{\rm d}x$
$\displaystyle\qquad+\int_{\mathbb{T}^{3}}\Big{<}\Big{<}M(t)\,,\,\mathbf{Q}(t)\Big{>}\Big{>}\,\,{\rm
d}{x}.$ (4.2)

###### Proof.

The proof of this lemma is straightforward. For the sake of completeness, we
briefly mention the proof. Note that, in order to prove the claim, we need to
compute ${\rm d}\int_{\mathbb{T}^{3}}\textbf{q}\cdot{\bf Q}\,{\rm d}x$. Due to
lack of regularity, it is customary to use regularization by convolutions. To
that context, let us denote by $(\bm{\rho}_{\alpha})$, an approximation to the
identity on $\mathbb{T}^{3}$. Let $\bm{\varphi}\in
L^{2}_{\text{div}}(\mathbb{T}^{3})$, then
$\bm{\varphi}_{\alpha}=\bm{\varphi}*\bm{\rho}_{\alpha}\,\in
C_{\text{div}}^{\infty}(\mathbb{T}^{3})$. Then using $\bm{\varphi}_{\alpha}$
as a test function in (4.1), we obtain $\mathbb{P}$-almost surely, for all
$t\in[0,T]$

$\displaystyle\int_{\mathbb{T}^{3}}\mathbf{q}_{\alpha}(t)\cdot\bm{\varphi}\,{\rm
d}x$
$\displaystyle=\int_{\mathbb{T}^{3}}\textbf{q}_{\alpha}(0)\cdot\bm{\varphi}\,{\rm
d}x+\int_{0}^{t}\int_{\mathbb{T}^{3}}(\mathbf{q}_{1})_{\alpha}:\nabla\bm{\varphi}\,{\rm
d}x\,{\rm
d}s+\int_{0}^{t}\int_{\mathbb{T}^{3}}\nabla\bm{\varphi}:\,d\mu_{\alpha}(x,s)\,{\rm
d}s$
$\displaystyle\qquad+\int_{\mathbb{T}^{3}}\bm{\varphi}\cdot\int_{0}^{t}dM_{\alpha}\,{\rm
d}x.$

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

$\displaystyle\mathbf{q}_{\alpha}(t)$
$\displaystyle=\textbf{q}_{\alpha}(0)-\int_{0}^{t}\mathcal{P}_{H}\big{(}{\rm
div}(\mathbf{q}_{1})_{\alpha}\big{)}{\rm
d}s-\int_{0}^{t}\,\mathcal{P}_{H}\big{(}{\rm div}\mu_{\alpha}\big{)}\,{\rm
d}x\,{\rm d}s+\int_{0}^{t}dM_{\alpha}\,{\rm d}x.$

Now, we can apply Itô’s formula to the process
$t\to\int_{\mathbb{T}^{3}}\textbf{q}_{r}\cdot{\bf Q}\,{\rm d}x$, then we
obtain for all $t\in[0,T],\,\,\mathbb{P}$-a.s

$\displaystyle\int_{\mathbb{T}^{3}}\mathbf{q}_{\alpha}(t)\cdot\mathbf{Q}(t)\,{\rm
d}{x}$
$\displaystyle=\int_{\mathbb{T}^{3}}\mathbf{q}_{\alpha}(0)\cdot\mathbf{Q}(0)\,{\rm
d}{x}+\int_{0}^{t}\int_{\mathbb{T}^{3}}(\mathbf{q}_{1})_{\alpha}:\nabla\mathbf{Q}\,{\rm
d}{x}\,{\rm d}s+\int_{0}^{t}\int_{\mathbb{T}^{3}}\nabla\mathbf{Q}:\,{\rm
d}\mu_{\alpha}\,{\rm d}s$
$\displaystyle\qquad+\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathbf{Q}\cdot{\rm
d}M_{\alpha}\,{\rm
d}x+\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathbf{Q}_{1}\cdot\mathbf{q}_{\alpha}\,{\rm
d}x\,{\rm
d}s+\int_{0}^{t}\int_{\mathbb{T}^{3}}\mathbf{q}_{\alpha}\cdot\mathbf{Q}_{2}\,{\rm
d}W\,{\rm d}x$
$\displaystyle\qquad+\int_{\mathbb{T}^{3}}\Big{<}\Big{<}M_{\alpha}(t)\,,\,\mathbf{Q(t)}\Big{>}\Big{>}\,\,{\rm
d}{x}\,.$

By using given hypothese, we can perform the limit $\alpha\to 0$ in above
relation to conclude the proof. ∎

###  4.1 Relative Energy Inequality (Euler System)

It is well-known that the relative energy inequality is very useful for the
comparison of a measure valued solution and a smooth given function. To see
this, let us first introduce the relative energy (entropy) functional in the
context of measure-valued solutions to the stochastic incompressible Euler
system as

$\displaystyle\mathrm{F}_{\mathrm{mv}}^{1}\left({\bf u}\ \Big{|}{\bf
U}\right)(t):=\int_{\mathbb{T}^{3}}\langle{\mathcal{V}^{\omega}_{t,x}};\frac{1}{2}{|{\bf
u}|^{2}}\rangle\,{\rm
d}x+\mathcal{D}(t)-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
u}\big{\rangle}\,\cdot{\bf U}\,{\rm d}x+\frac{1}{2}\int_{\mathbb{T}^{3}}|{\bf
U}|^{2}\,{\rm d}x.$

In view of the energy inequality (2.16), it is clear that the above relative
energy functional is defined for all $t\in[0,T]\setminus\mathcal{N}$, where
the null (i.e., Lebesgue measure zero) set $\mathcal{N}$ may depends on
$\omega\in\Omega$. We also define relative energy functional for all
$t\in\mathcal{N}$ as follows:

$\displaystyle\mathrm{F}_{\mathrm{mv}}^{2}\left({\bf u}\ \Big{|}{\bf
U}\right)(t):=\lim_{r\to
0^{+}}\frac{1}{r}\int_{t}^{t+r}\bigg{[}\int_{\mathbb{T}^{3}}\langle{\mathcal{V}^{\omega}_{s,x}};\frac{1}{2}{|{\bf
u}|^{2}}\rangle\,{\rm
d}x+\mathcal{D}(s)\bigg{]}ds-\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
u}\big{\rangle}\,\cdot{\bf U}\,{\rm d}x+\frac{1}{2}\int_{\mathbb{T}^{3}}|{\bf
U}|^{2}\,{\rm d}x.$

Using above, we define relative energy functional, which is well-defined
defined for all $t\in[0,T]$, as follows:

$\displaystyle\mathrm{F}_{\mathrm{mv}}\left({\bf u}\ \Big{|}{\bf
U}\right)(t):=\begin{cases}\mathrm{F}_{\mathrm{mv}}^{1}({\bf u}\ \big{|}{\bf
U})(t),&\text{if $t\in[0,T]\setminus\mathcal{N};$}\\\
\mathrm{F}_{\mathrm{mv}}^{2}({\bf u}\ \big{|}{\bf U})(t),&\text{if
$t\in\mathcal{N}$.}\end{cases}$ (4.3)

To proceed further, we make use of the relative energy (4.3) to derive the
relative energy inequality given by (LABEL:relativeEntropy).

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

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). Suppose $\mathbf{U}$
be stochastic processes which is adapted to the filtration
$(\mathbb{F}_{t})_{t\geq 0}$ and satisfies

$\displaystyle\mathrm{d}\mathbf{U}$
$\displaystyle=\mathbf{U}_{1}\,\mathrm{d}t+\mathcal{P}_{H}\mathbf{U}_{2}\,\mathrm{d}W,$

with

$\displaystyle\begin{aligned} {\bf U}\in C([0,T];C_{\rm
div}^{1}(\mathbb{T}^{3})),\
\quad\text{$\mathbb{P}$-a.s.},\qquad\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|{\bf
U}\|_{L_{\rm
div}^{2}(\mathbb{T}^{3})}^{2}\bigg{]}\textless\,\infty,\end{aligned}$ (4.4)

Moreover, ${\bf U}$ satisfy

$\displaystyle{\bf U}_{1}\in L^{2}(\Omega;L^{2}(0,T;L_{\rm
div}^{2}(\mathbb{T}^{3}))),\quad{\bf U}_{2}\in
L^{2}(\Omega;L^{2}(0,T;L_{2}(\mathfrak{U};L^{2}(\mathbb{T}^{3})))),$ (4.5)
$\int_{0}^{T}\sum_{k\geq 1}\|\mathcal{P}_{H}{\bf
U}_{2}(e_{k})\|_{L^{2}(\mathbb{T}^{3})}^{2}\in L^{1}(\Omega).$

Then the following _relative energy inequality_ holds:

$\displaystyle\mathrm{F}_{\mathrm{mv}}\left({\bf u}\ \Big{|}{\bf
U}\right)(t)\leq\mathrm{F}_{\mathrm{mv}}\left({\bf u}\ \Big{|}{\bf
U}\right)(0)+\mathcal{M}_{RE}(t)+\int_{0}^{t}\mathfrak{R}_{\mathrm{mv}}\big{(}{\bf
u}\left|\right.\mathbf{U}\big{)}(s)\,\mathrm{d}s$ (4.6)

$\mathbb{P}$-almost surely, for all $t\in[0,T]$ with

$\displaystyle\mathfrak{R}_{\mathrm{mv}}\big{(}{\bf
u}\left|\right.\mathbf{U}\big{)}$
$\displaystyle=\int_{\mathbb{T}^{3}}\left\langle{\mathcal{V}^{\omega}_{t,x}};{{\bf
u}\otimes{\bf u}}\right\rangle:\nabla_{x}{\bf
u}\,\mathrm{d}x+\int_{\mathbb{T}^{3}}\left\langle{\mathcal{V}^{\omega}_{t,x}};\bf
u\right\rangle\cdot{\bf U}_{1}\,{\rm d}x\,{\rm
d}t-\int_{\mathbb{T}^{3}}\nabla_{x}\mathbf{U}:d\mu_{C}+\frac{1}{2}\int_{\mathbb{T}^{3}}d\mu_{D}$
$\displaystyle\qquad+\frac{1}{2}\sum_{k\in\mathbb{N}}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\big{|}{\mathbf{G}_{k}({\bf
u})}-\mathbf{U}_{2}(e_{k})\big{|}^{2}\big{\rangle}\,\mathrm{d}x.$ (4.7)

Here $\mathcal{M}_{RE}(t)$ is a $\mathbb{R}$-valued square integrable
martingale, whose norm depends on the norms of smooth function $\mathbf{U}$ in
the aforementioned spaces.

###### Proof.

We follow the usual strategy and express all the integrals on the right hand
side of (4.3) by making use of the energy inequality (2.15) and the field
equation (2.14). Therefore, we shall make use of Itô’s formula and the energy
inequality (2.15) to compute the right hand side of (4.3).

Step 1: In order to compute ${\rm
d}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
u}\big{\rangle}\,\cdot{\bf U}\,{\rm d}x$, we first recall that
$\mathbf{q}=\langle{\mathcal{V}^{\omega}_{t,x}};{\bf u}\big{\rangle}$
satisfies hypothesis of Lemma 4.1. Therefore we can apply the Lemma 4.1 to
conclude that $\mathbb{P}$-almost surely

$\begin{split}&{\rm
d}\left(\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
u}\big{\rangle}\cdot{\bf U}\,{\rm
d}x\right)=\int_{\mathbb{T}^{3}}\left[\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
u}\big{\rangle}\cdot{\bf U}_{1}+\left\langle{\mathcal{V}^{\omega}_{t,x}};{{\bf
u}\otimes{\bf u}}\right\rangle:\nabla_{x}{\bf u}\right]\,{\rm d}x{\rm d}t\\\
&\qquad+\sum_{k\geq 1}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}{\bf
U}_{2}(e_{k})\cdot\mathcal{P}_{H}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}({\bf u})\big{\rangle}\,{\rm d}x\,{\rm
d}t+\int_{\mathbb{T}^{3}}\nabla_{x}{\bf U}:d\mu_{C}\,dt+{\rm
d}\mathcal{M}_{1},\end{split}$ (4.8)

where the square integrable martingale $\mathcal{M}_{1}(t)$ is given by

$\mathcal{M}_{1}(t)=\int_{\mathbb{T}^{3}}\int_{0}^{t}{\bf
U}\,dM^{1}_{E}\,dx+\int_{0}^{t}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
u}\big{\rangle}\cdot\mathcal{P}_{H}{\bf U}_{2}\,{\rm d}x\,{\rm d}W$

We remark that the item (i) of the Definition 2.15 is used to identify the
cross variation in (4.8). Indeed, notice that

$\displaystyle\int_{\mathbb{T}^{3}}\Big{<}\Big{<}f(t),M^{1}_{E}(t)\Big{>}\Big{>}dx$
$\displaystyle=\int_{\mathbb{T}^{3}}\sum_{i,j}\Bigg{(}\sum_{k=1}^{\infty}\int_{0}^{t}\langle\mathcal{P}_{H}\left\langle\mathcal{V}^{\omega}_{s,x};\mathbf{G}_{k}({\bf
u})\right\rangle,h_{i}\rangle\,\langle\mathbb{D}^{s}f(e_{k}),h_{j}\rangle\,ds\Bigg{)}h_{i}\otimes
h_{j}{\rm d}x.$
$\displaystyle=\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\int_{0}^{t}\mathcal{P}_{H}{\bf
U}_{2}(e_{k})\cdot\mathcal{P}_{H}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}(\varrho,{\bf u})\big{\rangle}\,{\rm d}x{\rm d}t$

Step 2: Next, we see that

$\begin{split}{\rm d}\left(\int_{\mathbb{T}^{3}}\frac{1}{2}|{\bf U}|^{2}\,{\rm
d}x\right)&=\frac{1}{2}\sum_{k\geq 1}\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}{\bf
U}_{2}(e_{k})|^{2}\,{\rm d}x\ {\rm d}t+{\rm d}\mathcal{M}_{2},\end{split}$
(4.9)

where

$\mathcal{M}_{2}(t)=\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf
U}\cdot\mathcal{P}_{H}{\bf U}_{2}\,{\rm d}x\,{\rm d}W.$

Step 3: We have from energy inequality

$\displaystyle\mathrm{E}(t+)\,\leq\,$
$\displaystyle\mathrm{E}(s-)+\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};{|{\mathbb{G}_{k}(u)}|^{2}}\right\rangle\,{\rm
d}{x}{\rm
d}\tau-\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\Big{(}\mathcal{Q}_{H}\big{\langle}\mathcal{V}^{\omega}_{s,x};{|{\mathbb{G}_{k}(u)}|}\big{\rangle}\Big{)}^{2}\,{\rm
d}{x}{\rm d}\tau$
$\displaystyle\qquad+\frac{1}{2}\int_{s}^{t}\int_{\mathbb{T}^{3}}d\mu_{D}+\int_{s}^{\tau}d\mathcal{M}^{2}_{E}.$
(4.10)

We now manipulate the product term in the equality (4.8) using properties of
projections $\mathcal{P}_{H}$ and $\mathcal{Q}_{H}$. Indeed, note that

$\displaystyle\int_{\mathbb{T}^{3}}\mathcal{P}_{H}{\bf
U}_{2}(e_{k})\cdot\mathcal{P}_{H}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}({\bf u})\big{\rangle}\,{\rm d}x$
$\displaystyle\qquad=\int_{\mathbb{T}^{3}}{\bf
U}_{2}(e_{k})\cdot\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf G}_{k}({\bf
u})\big{\rangle}\,{\rm d}x-\int_{\mathbb{T}^{3}}\mathcal{Q}_{H}{\bf
U}_{2}(e_{k})\cdot\mathcal{Q}_{H}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}({\bf u})\big{\rangle}\,{\rm d}x,$

and

$\displaystyle\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}{\bf
U}_{2}(e_{k})|^{2}\,{\rm d}x=\int_{\mathbb{T}^{3}}|{\bf
U}_{2}(e_{k})|^{2}\,{\rm d}x-\int_{\mathbb{T}^{3}}|\mathcal{Q}_{H}{\bf
U}_{2}(e_{k})|^{2}\,{\rm d}x.$

These properties of projections imply that

$\displaystyle\begin{aligned} &\frac{1}{2}\sum_{k\geq
1}\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}{\bf U}_{2}(e_{k})|^{2}\,{\rm
d}x-\sum_{k\geq 1}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}{\bf
U}_{2}(e_{k})\cdot\mathcal{P}_{H}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}({\bf u})\big{\rangle}\,{\rm d}x\\\
&\qquad\qquad+\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\left\langle\mathcal{V}^{\omega}_{s,x};{|{{\bf
G}_{k}(u)}|^{2}}\right\rangle\,{\rm d}{x}\,{\rm
d}\tau-\frac{1}{2}\sum_{k\geq\,1}\int_{s}^{t}\int_{\mathbb{T}^{3}}\Big{(}\mathcal{Q}_{H}\big{\langle}\mathcal{V}^{\omega}_{s,x};{|{{\bf
G}_{k}(u)}|}\big{\rangle}\Big{)}^{2}\,{\rm d}{x}\,{\rm d}\tau\\\
&=\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\Big{\langle}{\mathcal{V}^{\omega}_{t,x}};\big{|}{\mathbf{G}_{k}({\bf
u})}-\mathbf{U}_{2}(e_{k})\big{|}^{2}\Big{\rangle}\,\mathrm{d}x-\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\Big{|}\mathcal{Q}_{H}\mathbf{U}_{2}(e_{k})-\mathcal{Q}_{H}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};{\bf
G}_{k}({\bf u})\big{\rangle}\Big{|}^{2}\,\,{\rm d}{x}\\\
&\leq\,\frac{1}{2}\sum_{k\geq\,1}\int_{\mathbb{T}^{3}}\Big{\langle}{\mathcal{V}^{\omega}_{t,x}};\big{|}{\mathbf{G}_{k}({\bf
u})}-\mathbf{U}_{2}(e_{k})\big{|}^{2}\Big{\rangle}\,\mathrm{d}x.\end{aligned}$
(4.11)

Finally, in view of the above observations given by (4.8)-(4.11), we can now
add the resulting expressions to establish (LABEL:relativeEntropy). Note that
the square integrable martingale $\mathcal{M}_{RE}(t)$ is given by
$\mathcal{M}_{RE}(t):=\mathcal{M}_{1}(t)+\mathcal{M}_{2}(t)+\mathcal{M}_{E}^{2}(t)$.
∎

###  4.2 Proof of Theorem 2.18

In this subsection, we aim at establishing the desired weak (measure-
valued)–strong uniqueness principle given by Theorem 2.18. To do so, we need
to apply the relative energy inequality (2.15) with a specific choice of the
smooth given function $\mathbf{U}=\bar{\mathbf{u}}(\cdot\wedge\tau_{L})$,
where $(\bar{\mathbf{u}},(\tau_{L})_{L\in\mathbb{N}},\tau)$ is the unique
maximal strong pathwise solution to (1.1). For technical reason, note that the
stopping time $\tau_{L}$ announces the blow-up and satisfies

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

Furthermore, it is evident that $\bar{\mathbf{u}}$ satisfies the equation
(4.2), with

$\displaystyle\mathbf{U}_{1}=\mathcal{P}_{H}(\bar{\mathbf{u}}\cdot\nabla_{x}\bar{\mathbf{u}}),\quad\mathbf{U}_{2}=\mathbb{G}(\bar{{\bf
u}}).$

Clearly, in view of Theorem 2.9 and (2.3)–(2.4), the conditions (4.4) and
(4.5) are satisfied for $t\leq\mathfrak{t}_{L}$. Therefore, the inequality
(LABEL:relativeEntropy) holds, and we can also deduce from (4.2) that

$\displaystyle\mathrm{F}_{\mathrm{mv}}\left({\bf
u}\Big{|}\bar{\mathbf{u}}\right)(t\wedge\tau_{L})\leq\mathrm{F}_{\mathrm{mv}}\left({\bf
u}\Big{|}\bar{\mathbf{u}}\right)(0)+M_{RE}(t\wedge\tau_{L})+\int_{0}^{t\wedge\tau_{L}}\mathfrak{R}_{\mathrm{mv}}\big{(}{\bf
u}\left|\right.\bar{\mathbf{u}}\big{)}(s)\,\mathrm{d}s,$ (4.12)

holds for each $L\in\mathbb{N}$, for all $t\in[0,T]$, $\mathbb{P}$-almost
surely. Here after manipulating terms in (4.2), as in [37], we obtain

$\displaystyle\mathfrak{R}_{\mathrm{mv}}\left({\bf
u}\Big{|}\bar{\mathbf{u}}\right)=$
$\displaystyle\,\int_{\mathbb{T}^{3}}\left\langle{\mathcal{V}^{\omega}_{t,x}};\left|{({\bf
u}-\bar{\mathbf{u}})\otimes(\bar{\mathbf{u}}-{\bf
u})}\right|\right\rangle|\nabla_{x}\bar{\mathbf{u}}|\,dx+\frac{1}{2}\sum_{k\in\mathbb{N}}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\big{|}{\mathbf{G}_{k}(\mathbf{u})}-{\mathbf{G}_{k}(\bar{\mathbf{u}})}\big{|}^{2}\big{\rangle}\,\mathrm{d}x$
$\displaystyle\qquad+\int_{\mathbb{T}^{3}}|\nabla_{x}\bar{\mathbf{u}}|\cdot
d|\mu_{C}|+\int_{\mathbb{T}^{3}}d|\mu_{D}|.$

Since $\|\mathbf{\bar{u}}\|_{W^{1,\infty}(\mathbb{T}^{3})}\leq\,c(L)$ for
$t\,\leq\,\tau_{L}$, we can control the terms $|\nabla_{x}\bar{\mathbf{u}}|$
by some constant. It is also clear that

$\left|{({\bf u}-\bar{\mathbf{u}})\otimes(\bar{\mathbf{u}}-{\bf
u})}\right|\leq|{\bf u}-\bar{\mathbf{u}}|^{2},$

and

$\displaystyle\sum_{k\geq
1}\big{|}{\mathbf{G}_{k}(\mathbf{u})}-{\mathbf{G}_{k}(\bar{\mathbf{u}})}\big{|}^{2}\leq
D_{1}|\mathbf{u}-\bar{\mathbf{u}}|^{2}.$

Finally, we also see that

$\displaystyle\frac{1}{2}\sum_{k\in\mathbb{N}}\int_{\mathbb{T}^{3}}\big{\langle}{\mathcal{V}^{\omega}_{t,x}};\big{|}{\mathbf{G}_{k}(\mathbf{u})}-{\mathbf{G}_{k}(\bar{\mathbf{u}})}\big{|}^{2}\big{\rangle}\,\mathrm{d}x\leq
c(L)\,\mathrm{F}^{1}_{\mathrm{mv}}\big{(}{\mathbf{u}}\left|\right.\bar{\mathbf{u}}\big{)}.$

Collecting all the above estimates and using the item $(h)$ of Definition
2.15, we conclude that for all $t\in[0,T]$, $\mathbb{P}$-a.s.

$\displaystyle\int_{0}^{t\wedge\tau_{L}}\mathfrak{R}_{\mathrm{mv}}\big{(}{\bf
u}\left|\right.\bar{\mathbf{u}}\big{)}\,\mathrm{d}s\leq
c(L)\,\int_{0}^{t\wedge\tau_{L}}\Big{(}\mathrm{F}_{\mathrm{mv}}\big{(}{\bf
u}\left|\right.\bar{\mathbf{u}}\big{)}(s)\Big{)}\,\mathrm{d}s.$ (4.13)

We now combine (4.13) and (LABEL:relativeEntropy1), and apply classical
Gronwall’s lemma, to obtain for all $t\in[0,T]$

$\displaystyle\mathbb{E}\,\Big{[}\mathrm{F}_{\mathrm{mv}}\big{(}{\mathbf{u}}\left|\right.\bar{\mathbf{u}}\big{)}(t\wedge\tau_{L})\Big{]}\leq
c(L)\,\mathbb{E}\,\Big{[}\mathrm{F}_{\mathrm{mv}}\big{(}{\mathbf{u}}\left|\right.\bar{\mathbf{u}}\big{)}(0)\Big{]}.$

We recall that

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

which, by assumption, vanishes in expectation. Therefore, we conclude that

$\mathbb{E}\,\Big{[}\mathrm{F}_{\mathrm{mv}}\big{(}{\mathbf{u}}\left|\right.\bar{\mathbf{u}}\big{)}(t\wedge\tau_{L})\Big{]}=0,\,\,\text{for
all}\,\,\,t\in[0,T].$

This also implies that

$\lim_{r\to
0^{+}}\frac{1}{r}\int_{t}^{t+r}\mathbb{E}\,\Big{[}\mathrm{F}_{\mathrm{mv}}\big{(}{\mathbf{u}}\left|\right.\bar{\mathbf{u}}\big{)}(s\wedge\tau_{L})\Big{]}\,{\rm
d}s=0.$

Keeping in mind a priori estimates, a standard Lebesgue point argument in
combination with classical Fubini’s theorem reveals that for a.e. $t\in[0,T]$,

$\mathbb{E}\,\Big{[}\mathrm{F}_{\mathrm{mv}}^{1}\big{(}{\mathbf{u}}\left|\right.\bar{\mathbf{u}}\big{)}(t\wedge\tau_{L})\Big{]}=0.$

But since the defect measure $\mathcal{D}\geq 0$, above equality implies for
a.e. $t\in[0,T]$, $\mathbb{P}$-almost surely

$\mathcal{D}(t\wedge\tau_{L})=0,\,\,\text{and}\,\,\,\bar{{\bf
u}}(x,t\wedge\tau_{L})={\bf u}(x,t\wedge\tau_{L}),\,\,\text{ for
a.e.}\,\,x\in\mathbb{T}^{3}.$

##  5 Weak-Strong Uniqueness for Navier–Stokes System

Let $\bf u$ and $\bf U$ be two finite energy weak martingale solutions to
(1.2), with same inital data $u_{0}$, defined on the same stochastic basis.
The commonly used form of the relative energy functional in the context of
weak solutions to the incompressible Naiver stokes system reads

$\displaystyle\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}\,\Big{|}\,{\bf U}\right):=\int_{\mathbb{T}^{3}}\frac{1}{2}{|{\bf
u}|^{2}}\,{\rm d}x-\int_{\mathbb{T}^{3}}{\bf u}\,\cdot{\bf U}\,{\rm
d}x+\frac{1}{2}\int_{\mathbb{T}^{3}}|{\bf U}|^{2}\,{\rm d}x.$ (5.1)

The proof of (weak-strong) uniqueness for finite energy weak martingale
solutions to (1.2) essentially uses similar arguments, as depicted in Section
4. However, the main difficulty lies in the successful identification of cross
variation of two martingale solutions. Indeed, the regularity of finite energy
weak martinagle solutions is not enough to identify the cross variation
between them, and requires one solution to be more regular. In what follows,
we start with the following lemma, whose proof is a simple consequence of the
Hölder and Sobolev inequalities, see [29, Lemma 2.4].

######  Lemma 5.1.

Let $r,s$ satisfy

$\frac{3}{s}+\frac{2}{r}=1,\,\,\,\,s\in(3,\infty)$

and let ${\bf u,w}\in L^{2}((0,T);H^{1}_{\rm div}(\mathbb{T}^{3}))$, and ${\bf
u}\in L^{r}(0,T;L^{s}(\mathbb{T}^{3}))$. Then

$\displaystyle\bigg{|}\int_{0}^{T}\langle\textbf{v}\cdot\nabla\textbf{w,u}\rangle\bigg{|}\leq\,C\bigg{(}\int_{0}^{T}\|\nabla\textbf{w}\|$
${}_{L^{2}(\mathbb{T}^{3})}^{2}dt\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{0}^{T}\|\nabla\textbf{v}\|_{L^{2}(\mathbb{T}^{3})}^{2}\bigg{)}^{\frac{3}{2s}}\bigg{(}\int_{0}^{T}\|{\bf
u}\|_{L^{s}(\mathbb{T}^{3})}^{r}\|\textbf{v}\|_{L^{2}(\mathbb{T}^{3})}^{2}dt\bigg{)}^{\frac{1}{r}}.$

To make use of the above inequality, let us assume that two finite energy weak
martingale solutions ${\bf u,U}\,\in
L^{\infty}((0,T);L^{2}(\mathbb{T}^{3}))\cap
L^{2}((0,T);H^{1}(\mathbb{T}^{3}))$ be diveregence free. Assume, in addition,
that ${\bf U}\in L^{r}((0,T);L^{s}(\mathbb{T}^{3})).$ Then, in view of the
above Lemma 5.1, for every $\tau\in(0,T)$

$\displaystyle\bigg{|}\int_{0}^{\tau}\int_{\mathbb{T}^{3}}$
$\displaystyle\nabla{\bf(u-U):((u-U)}\otimes{\bf U})\,{\rm d}x\,{\rm
d}t\bigg{|}$ (5.2)
$\displaystyle\leq\,C\bigg{(}\int_{0}^{\tau}\int_{\mathbb{T}^{3}}|\nabla{\bf(u-U)}|^{2}\,{\rm
d}x\bigg{)}^{1-1/r}\bigg{(}\int_{0}^{\tau}\|{\bf
U}\|_{L^{s}(\mathbb{T}^{3}))}^{r}\int_{\mathbb{T}^{3}}{\bf|u-U|^{2}}\,{\rm
d}x\,{\rm d}t\bigg{)}^{1/r}.$

In order to calculate the evolution equation satisfies by the second term of
the relative energy (5.1), we need to apply Itô product rule. To do so, we
first regularize (2.11) (with $\varepsilon=1$), for both solutions ${\bf u}$
and $\bf U$, by taking a spatial convolution with a suitable family of
regularizing kernels. We denote by ${\bf v}_{r}$, the regularization of $\bf
v$. For a test fuction $\bm{\varphi}\in L^{2}_{\text{div}}(\mathbb{T}^{3})$,
we have $\bm{\varphi_{r}}\in C_{\text{div}}^{\infty}(\mathbb{T}^{3})$. For
both weak solutions $\bf u$ and $\bf U$, we may write equations

$\displaystyle\langle\mathbf{u}(t),\bm{\varphi}_{r}\rangle$
$\displaystyle=\langle\mathbf{u}(0),\bm{\varphi_{r}}\rangle+\int_{0}^{t}\langle\mathbf{u}\otimes\mathbf{u}(s),\nabla_{x}\bm{\varphi_{r}}\rangle\,\mathrm{d}s-\int_{0}^{t}\langle\nabla_{x}\mathbf{u}_{(}s)\,\nabla_{x}\bm{\varphi_{r}}\rangle\mathrm{d}s+\int_{0}^{t}\langle\mathcal{G}(\mathbf{u}),\bm{\varphi_{r}}\rangle\,\mathrm{d}W,$
$\displaystyle\langle\mathbf{U}(t),\bm{\varphi}_{r}\rangle$
$\displaystyle=\langle\mathbf{U}(0),\bm{\varphi_{r}}\rangle+\int_{0}^{t}\langle\mathbf{U}\otimes\mathbf{U}(s),\nabla_{x}\bm{\varphi_{r}}\rangle\,\mathrm{d}s-\int_{0}^{t}\langle\nabla_{x}\mathbf{U}(s)\,,\nabla_{x}\bm{\varphi_{r}}\rangle\mathrm{d}s+\int_{0}^{t}\langle\mathcal{G}(\mathbf{U}),\bm{\varphi_{r}}\rangle\,\mathrm{d}W.$

After shiftting regularizing kernel from test fucntion to solutions term, we
obtain

$\displaystyle\mathbf{u}_{r}(t)$
$\displaystyle=\mathbf{u}_{r}(0)-\int_{0}^{t}\mathcal{P}_{H}({\rm
div}(\mathbf{u}\otimes\mathbf{u})_{r})\mathrm{d}s+\int_{0}^{t}\Delta\mathbf{u}_{r}(s)\mathrm{d}s+\int_{0}^{t}\mathcal{P}_{H}(\mathcal{G}(\mathbf{u})_{r})\mathrm{d}W,$
$\displaystyle\mathbf{U}_{r}(t)$
$\displaystyle=\mathbf{U}_{r}(0)-\int_{0}^{t}\mathcal{P}_{H}({\rm
div}(\mathbf{U}\otimes\mathbf{U})_{r})\mathrm{d}s+\int_{0}^{t}\Delta\mathbf{U}_{r}(s)\mathrm{d}s+\int_{0}^{t}\mathcal{P}_{H}(\mathcal{G}(\mathbf{U})_{r})\mathrm{d}W,$

in $(L^{2}_{\mathrm{div}}(\mathbb{T}^{3}))^{\prime}.$ We can now apply
classical Itô’s formula to the process $t\,\to\,\int_{\mathbb{T}^{3}}{\bf
u}_{r}\cdot\bf{U}_{r}\,{\rm d}x$, to obtain $\mathbb{P}$-almost surely

$\begin{split}{\rm d}\left(\int_{\mathbb{T}^{3}}{{\bf
u}_{r}\cdot{\bf{U}}_{r}}\,{\rm d}x\right)&=\int_{\mathbb{T}^{3}}\Big{[}-{{{\bf
u}}_{r}\cdot{\rm div}({\bf U\otimes\bf U})}_{r}+({{{\bf u}\otimes{\bf
u}}})_{r}:\nabla_{x}{{\bf u}}_{r}\Big{]}\,{\rm d}x\,{\rm d}t\\\
&\quad+\sum_{k\geq
1}\int_{\mathbb{T}^{3}}\mathcal{P}_{H}({\mathbb{G}}_{k}({\bf
U})_{r})\cdot\mathcal{P}_{H}({\bf G}_{k}({{\bf u})_{r})}\,{\rm d}x\,{\rm
d}t-2\int_{\mathbb{T}^{3}}\nabla_{x}{\bf U}_{r}:\nabla_{x}{\bf u}_{r}\,{\rm
d}x\,{\rm d}t\\\ &\qquad+\int_{0}^{t}\int_{\mathbb{T}^{3}}({{\bf
u}}_{r}\cdot{\mathbb{G}(\bf U)}_{r}+{\bf
U}_{r}\cdot{\mathbb{G}}({\bf{u}}_{r}))\,{\rm d}x\,{\rm d}W,\\\ \end{split}$
(5.3)

We now wish to let $r\to 0$ in the above relation (5.3). The main difficulty
in passing to the limits in the parameter $r$ stems from the nonlinear terms,
treatment of others terms is classical. Indeed, we may apply the classical BDG
inequality, with the help of a priori estimate and given conditions for noise
coefficients, to handle stochastic terms appeared in (5.3). In what follows,
we show, with the help of Lemma (5.1) and extra regularity of ${\bf U}$, we
can pass to the limit in the nonlinear terms. Observe that

$\displaystyle\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{u}}_{r}\cdot{\rm
div}({\bf{U}}\otimes{\bf{U}})_{r}\,{\rm d}s\,{\rm
d}x-\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{u}}\cdot{\rm
div}({\bf{U}}\otimes{\bf{U}})\,{\rm d}s\,{\rm d}x$
$\displaystyle\qquad=\bigg{(}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{u}}_{r}\cdot{\rm
div}({\bf{U}}\otimes{\bf{U}})_{r}\,{\rm d}s\,{\rm
d}x-\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{u}}_{r}\cdot{\rm
div}({\bf{U}}\otimes{\bf{U}})\,{\rm d}s\,{\rm d}x\bigg{)}$
$\displaystyle\qquad\qquad+\bigg{(}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{u}}_{r}\cdot{\rm
div}({\bf{U}}\otimes{\bf{U}})_{r}\,{\rm d}s\,{\rm
d}x-\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{u}}\cdot{\rm
div}({\bf{U}}\otimes{\bf{U}})\,{\rm d}s\,{\rm
d}x\bigg{)}=:I_{1}^{r}+I_{2}^{r},$

For the second nonlinear term,

$\displaystyle\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{U}}_{r}\cdot{\rm
div}({\bf{u}}\otimes{\bf{u}})_{r}\,{\rm d}s\,{\rm
d}x-\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{U}}\cdot{\rm
div}({\bf{u}}\otimes{\bf{u}})\,{\rm d}s\,{\rm d}x$
$\displaystyle\qquad=\bigg{(}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{U}}_{r}\cdot{\rm
div}({\bf{u}}\otimes{\bf{u}})_{r}\,{\rm d}s\,{\rm
d}x-\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{U}}_{r}\cdot{\rm
div}({\bf{u}}\otimes{\bf{u}})\,{\rm d}s\,{\rm d}x\bigg{)}$
$\displaystyle\qquad\qquad+\bigg{(}\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{U}}_{r}\cdot{\rm
div}({\bf{u}}\otimes{\bf{u}})_{r}\,{\rm d}s\,{\rm
d}x-\int_{0}^{t}\int_{\mathbb{T}^{3}}{\bf{U}}\cdot{\rm
div}({\bf{u}}\otimes{\bf{u}})\,{\rm d}s\,{\rm
d}x\bigg{)}=:J_{1}^{r}+J_{2}^{r},$

First note that

$\displaystyle|I_{1}^{r}|\,$ $\displaystyle\leq\,C\|({\bf{U}}\otimes{\bf
U})_{r}-({\bf U}\otimes{\bf U})\|_{L^{1}([0,T];L^{1}(\mathbb{T}^{3}))},$
$\displaystyle|J_{1}^{r}|$ $\displaystyle\leq\,C\,\|({\bf{u}}\otimes{\bf
u})_{r}-({\bf u}\otimes{\bf u})\|_{L^{1}([0,T];L^{1}(\mathbb{T}^{3}))}.$

Moreover, thanks to Lemma 5.1, we conclude