# Existence and uniqueness of weak solutions for the generalized stochastic
Navier-Stokes-Voigt equations111 1,3Department of Mathematics, Indian
Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee,
Uttarakhand 247667, INDIA.
2FCT - Universidade de Algarve, Faro, Portugal & CMAFcIO - Universidade de
Lisboa, Lisbon, Portugal. e-mail: Manil T. Mohan<EMAIL_ADDRESS><EMAIL_ADDRESS>e-mail: Ankit Kumar:
akumar14<EMAIL_ADDRESS>e-mail: Hermenegildo Borges
de Oliveira<EMAIL_ADDRESS>*Corresponding author. Keywords: Stochastic
generalized Navier-Stokes-Voigt equations, Gaussian noise, martingale
solution, strong solution. Mathematics Subject Classification (2020): Primary
60H15, 35R60; Secondary 35Q35, 76D03, 76A05.

Ankit Kumar1, Hermenegildo Borges de Oliveira2 and Manil T. Mohan3*

###### Abstract.

In this work, we consider the incompressible generalized Navier-Stokes-Voigt
equations in a bounded domain $\mathcal{O}\subset\mathbb{R}^{d}$, $d\geq 2$,
driven by a multiplicative Gaussian noise. The considered momentum equation is
given by:

$\displaystyle\mathrm{d}\mathopen{}\mathclose{{}\left(\bm{u}-\kappa\Delta\bm{u}}\right)=\mathopen{}\mathclose{{}\left[\bm{f}+\operatorname{div}\mathopen{}\mathclose{{}\left(-\pi\mathbf{I}+\nu|\mathbf{D}(\bm{u})|^{p-2}\mathbf{D}(\bm{u})-\bm{u}\otimes\bm{u}}\right)}\right]\mathrm{d}t+\Phi(\bm{u})\mathrm{d}\mathrm{W}(t).$

In the case of $d=2,3$, $\bm{u}$ accounts for the velocity field, $\pi$ is the
pressure, $\bm{f}$ is a body force and the final term stay for the stochastic
forces. Here, $\kappa$ and $\nu$ are given positive constants that account for
the kinematic viscosity and relaxation time, and the power-law index $p$ is
another constant (assumed $p>1$) that characterizes the flow. We use the usual
notation $\mathbf{I}$ for the unit tensor and
$\mathbf{D}(\bm{u}):=\frac{1}{2}\mathopen{}\mathclose{{}\left(\nabla\bm{u}+(\nabla\bm{u})^{\top}}\right)$
for the symmetric part of velocity gradient. For
$p\in\big{(}\frac{2d}{d+2},\infty\big{)}$, we first prove the existence of a
_martingale solution_. Then we show the _pathwise uniqueness of solutions_. We
employ the classical _Yamada-Watanabe theorem_ to ensure the existence of a
unique _probabilistic strong solution_.

##  1\. Introduction

###  1.1. The model

We study flows of incompressible fluids with elastic properties governed by
the incompressible generalized Navier-Stokes-Voigt (NSV) equations in a
bounded domain $\mathcal{O}\subset\mathbb{R}^{d}$, $d\geq 2$, with its
boundary denoted by $\partial\mathcal{O}$, during the time interval $[0,T]$,
with $0<T<\infty$. The momentum equation is assumed here to be perturbed by a
stochastic term, that is, infinite-dimensional multiplicative Gaussian noise
which is the product of $\Phi(\bm{u})$ and a Wiener process
$\mathrm{W}(\cdot)$. The problem under consideration is posed by the following
system of equations:

$\displaystyle\mathrm{d}\mathopen{}\mathclose{{}\left(\bm{u}-\kappa\Delta\bm{u}}\right)$
$\displaystyle=\mathopen{}\mathclose{{}\left[\bm{f}+\operatorname{div}\mathopen{}\mathclose{{}\left(-\pi\mathbf{I}+\nu|\mathbf{D}(\bm{u})|^{p-2}\mathbf{D}(\bm{u})-\bm{u}\otimes\bm{u}}\right)}\right]\mathrm{d}t+\Phi(\bm{u})\mathrm{d}\mathrm{W}(t),\text{
in }\mathcal{O}_{T},$ (1.1) $\displaystyle\operatorname{div}\bm{u}$
$\displaystyle=0,\text{ in }\mathcal{O}_{T},$ (1.2) $\displaystyle\bm{u}$
$\displaystyle=\bm{u}_{0},\text{ in }\mathcal{O}\times\\{0\\},$ (1.3)
$\displaystyle\bm{u}$ $\displaystyle=\bm{0},\text{ on }\Gamma_{T},$ (1.4)

where $\mathcal{O}_{T}:=\mathcal{O}\times(0,T)$ and
$\Gamma_{T}:=\partial\mathcal{O}\times[0,T]$. Here,
$\bm{u}=(u_{1},\dots,u_{d})$ and $\bm{f}=(f_{1},\dots,f_{d})$ are vector-
valued functions, $\pi$ is a scalar-valued function, whereas $\nu$ and
$\kappa$ are positive constants. In real-world applications, space dimensions
of interest are $d=2$ and $d=3$, and in that case $\bm{u}$ accounts for the
velocity field, $\pi$ is the pressure, $\bm{f}$ represents an external forcing
field. Greek letters $\nu$ and $\kappa$ stay for positive constants that, in
the dimensions of physical interest, account for the kinematics viscosity and
for the relaxation time, that is, the time required for a viscoelastic fluid
relax from a deformed state back to its equilibrium configuration. The power-
law index $p$ is a constant that characterizes the flow and is assumed to be
such that $1<p<\infty$. In particular, for $1<p<2,$ the model describes shear-
thinning fluids, when $p=2,$ we recover the model that governs Newtonian
fluids, while for $p>2,$ the model describes shear-thickening fluids. The
capital letters stay for tensor-valued functions, in particular $\mathbf{I}$
is the unit tensor and $\mathbf{D}(\bm{u})$ is the symmetric part of the
velocity gradient, that is,
$\mathbf{D}(\bm{u}):=\frac{1}{2}\mathopen{}\mathclose{{}\left(\nabla\bm{u}+(\nabla\bm{u})^{\top}}\right)$.
The idea behind the presence of the noise $\Phi(\bm{u})$ is an interaction
between the solution $\bm{u}$ and the random perturbation caused by the Wiener
process $\mathrm{W}(\cdot)$, and the product of these two terms represents the
stochastic part in the momentum equation (1.1). This stochastic part can be
interpreted in different ways. It can be understood as a turbulence in the
fluid motion or it can be interpreted as a perturbation from the physical
model. On the other hand, apart from the forcing term $\bm{f},$ there might be
further quantities with influence on the motion which can be characterized by
this stochastic part as well.

###  1.2. Literature review

We start with the physical significance of NSV equations, also named Kelvin-
Voigt (KV) equations, as we have considered the generalized stochastic NSV
equations (1.1)-(1.4). KV equations are used in the applications to model the
response of materials that exhibit all intermediate range of properties
between an elastic solid and a viscous fluid, as, for instance, polymers.
These materials have some memory in the sense that they can come back to some
previous state when the shear stress is removed (see [10]). The deterministic
KV equations were extensively studied in the article [54], whose author coined
the name Kelvin-Voigt for the incompressible Navier-Stokes equations perturbed
by the relaxation term $\kappa\partial_{t}\Delta\bm{u}$ (see [50] also).
However, and as observed in the work [66], neither Kelvin nor Voigt have
suggested any stress-strain relation, or system of governing equations for
viscoelastic fluids (see [36]). Currently, the Navier-Stokes-Voigt name for
the associated system of equations seems to be the most accepted by the
researchers in this field. Mathematically speaking, and was noticed by
Ladyzhenskaya (see [41]), the most important property of NSV equations is that
the term $\kappa\partial_{t}\Delta\bm{u}$ works as a regularization of the 3D
incompressible Navier-Stokes (NS) equations ensuring the unique global
solvability of the corresponding problem. The existence and uniqueness of the
weak and strong solutions of NSV equations have been established in the work
[54] (cf. [42]). A local in time existence and uniqueness result of an inverse
problem for NSV equations is established in [40]. Several other authors have
also studied some variants of the NSV equations, in many settings and under
different assumptions, with respect to the well-posedness and long time
behavior. It is worth to mention that the works [39, 55] established a
connection between NSV and the Oldroyd models, and proved the existence of
classical global solutions for the models. Also, in the works [52, 66, 67],
the authors proved the existence and uniqueness of weak solutions in case of
domain varying with time and with the optimal control. The authors in [4, 5,
6, 8, 9] discussed the existence, uniqueness and some qualitative properties
of the solutions to some variants of NSV equations, more specifically the case
with power-laws and anisotropic diffusion and several other properties. In
[3], the authors considered the generalized NSV equations for non-homogeneous
and incompressible fluid flows with the diffusion and relaxation terms
described by two distinct power-laws and proved the existence of weak
solutions as well as large-time behavior of the solutions. The works [20, 37,
45, 58] established a clear connection between NSV equations and the
turbulence modeling, to be more specific, with Bardina turbulence models. The
same type of connection was firstly studied in the work [43] and followed by
the work [44]. The authors in [1] connects NSV equations with the Prandtl-
Smagorinsky and turbulent kinematic energy models in order to compute the
turbulent viscosity. Several questions on NSV equations as a regularization of
both NS and Euler equations have been tackled in [12, 13, 37]. Also, the
author in the work [25] considered generalized NS equations (NS equations with
power-law) with the presence of damping term in the momentum equation which
governs isothermal flows of incompressible and homogeneous non-Newtonian
fluids and established the existence of weak solutions using regularization
techniques, the monotone theory, compactness arguments and truncation method.
In a recent work [7], the authors considered the classical NSV problem for
incompressible fluids with unknown non-constant density and established the
existence of weak solutions (velocity and density), unique pressure and some
regularity results of the solutions on the smoothness of the given data.

We are interested in the generalized stochastic NSV equations (1.1)-(1.4). As
mentioned above, from several physical point of view it is important to have a
stochastic part in the equation of motion:

  1. (1)

It gives a clear understanding of turbulence in the fluid motion (see [49]).

  2. (2)

It could be viewed as a deviation from the physical model.

  3. (3)

We are seeing that there may be additional values, generally minor ones, that
have an impact on the motion in addition to the force $\bm{f}$.

Several works are available in the literature dealing with the existence of
weak solutions of stochastic NS equations starting from the article [11]. One
can see the work [28] for an overview. However, only a few works are available
dealing with the generalized ($p\not=2$) NS problem. In the article [21], the
authors discussed the bipolar shear thinning fluid. In that work, the presence
of additional bi-Laplacian term $\Delta^{2}\bm{u}$ makes the computations
easy. Since, the additional term gives enough initial regularity which leads
to the proof using monotone operators without any truncation arguments.

In [62, 64], the authors considered the stochastic power-law fluids and
established the existence and uniqueness of weak solutions and strong
convergence of the Galerkin approximation and energy equality, respectively.
There were no interaction between the solution and Brownian motion in [62, 64]
(additive noise). Further, in [14], the author extended the existence theory
of the stochastic power-law fluids and improved the results of [62, 64]. He
proved the existence of weak solutions (in both analytic and probabilistic
sense), using the standard procedure as the pressure term disappear in the
weak formulation and later it reconstructed into the formulation. He borrowed
the idea from [63] (for deterministic setting), which relates each term in the
equation a pressure part. Moreover, he stabilized the auxiliary equations by
adding a large power of $\bm{u}$ and the approach was based on Galekin method.
Later, he extended the deterministic approach of the work [63] to the
stochastic case and combined the techniques from nonlinear partial
differential equations with stochastic calculus for martingales. Since the
author cannot apply test functions in this situation, he instead employed
Itô’s formula to obtain energy estimates and other stochastic tools. Finally,
in order to justify the limit in the nonlinear terms, he used the monotone
operator theory together with the $\mathrm{L}^{\infty}$–truncation.

Let us mention some works available for stochastic NSV equations. In [46], the
authors considered 3D stochastic NSV equations in bounded domains perturbed by
a finite-dimensional Brownian motion and established an LDP for the solution
using weak convergence approach. The existence and long-time behavior of the
solutions (more specifically mean square exponentially stability and the
almost sure exponential stablility of the stationary solutions) to 3D
stochastic NSV equations in bounded domains perturbed by infinite-dimensional
Wiener process obtained in [2]. In a recent work [65], the author studied the
stability of pullback random attractors for 3D stochastic NSV equations with
delays.

###  1.3. Objectives and novelties of the paper

The main aim of this work is to prove the existence of martingale solution and
unique probabilistic strong solution of generalized stochastic NSV equations
perturbed by infinite-dimensional Wiener process. There seems to be a very
scarce knowledge about the model presented in this work, and to the authors
best knowledge the analysis presented here has not yet been investigated in
the available literature.

  * •

As we have mentioned above, there are only a few works available in the
literature dealing with stochastic power-law fluids [14, 62, 64]. In the first
work [14], the author proved the existence theory for stochastic power-law
fluids driven by an infinite-dimensional multiplicative Gaussian noise. The
latter two works [62, 64] discussed the existence and uniqueness of weak
solutions for stochastic power-law fluids driven by a finite-dimensional
Brownian motion.

  * •

In this work, we are following the stochastic approach considered in the work
[14], which in turn was an extension to the stochastic case of the
deterministic problem studied in the work [63]. We are also considering the
perturbation by an infinite-dimensional multiplicative Gaussian noise.

In addition to the stochastic situation, the main conclusion of this work is
that the Voigt term $\kappa\partial_{t}\Delta\bm{u}$ regularizes the power-law
model of the NS equations in such a way that the existence result can be
proven up to the smallest possible value (in the sense that the Gelfand triple
can still be used) of the power-law exponent $p$ ($p>\frac{2d}{d+2}$), (see
Theorem 2.6 below). The NS power-law model, which corresponds to considering
$\kappa=0$ in the momentum equation (1.1), was considered in [14]. But there
the author used the $\mathrm{L}^{\infty}$–truncation method, and obtained the
existence result for values of $p>\frac{2(d+1)}{d+2}$.

###  1.4. Organization

This article is organized as follows: We start with the basic formulation of
the stochastic background, as well as Definitions 2.3 and 2.4 of martingale
solution and probabilistically strong solution, respectively, and then we
state our main Theorem 2.6 in Section 2. In Section 3, we discuss the
decomposition of pressure terms. As in the weak formulation of the problem,
the pressure term disappears (see Definition 2.3), later it can be recovered
by using a version of de Rham’s theorem. The idea of decomposing the pressure
term that we use in the sequel is borrowed from the work [14], but its first
application dates back to [63]. In Section 4, we consider an approximate
system with a stabilization term (a large power of $\bm{u}$), and, by using
Galerkin approximations, we obtain the uniform energy estimate, followed by
the existence of martingale solution for this approximate system, and for
which we still have to use the Skorokhod representation theorem. Later, we
reconstruct the pressure term and using monotone operator theory leads to
existence of martingale solution to the system (1.1)-(1.4). We wind up the
article by establishing the pathwise uniqueness of the solution to the system
(1.1)-(1.4) and hence the existence of a unique probabilistic strong solution
by an application of the classical Yamada-Watanabe theorem.

##  2\. Auxiliary results

###  2.1. Function spaces

Let us introduce the functions spaces used to characterize the solutions of
Fluid Dynamics problems. Let $\mathcal{O}$ be a domain in $\mathbb{R}^{d}$,
that is, an open and connected subset of $\mathbb{R}^{d}$. By
${\mathrm{C}_{0}^{\infty}(\mathcal{O})^{d}},$ we denote the space of all
infinitely differentiable $\mathbb{R}^{d}-$valued functions with compact
support in $\mathcal{O}$. For $1\leq p\leq\infty$, we denote by
${\mathrm{L}^{p}(\mathcal{O})^{d}},$ the Lebesgue space consisting of all
$\mathbb{R}^{d}-$valued measurable (equivalence classes of) functions that are
$p-$summable over $\mathcal{O}$. By $|\mathcal{O}|,$ we denote the
$d$–Lebesgue measure of $\mathcal{O}$. The corresponding Sobolev spaces are
represented by ${\mathrm{W}^{k,p}(\mathcal{O})^{d}}$, where $k\in\mathbb{N}$.
When $p=2$, ${\mathrm{W}^{k,2}(\mathcal{O})^{d}}$ are Hilbert spaces that we
denote by ${\mathrm{H}^{k}(\mathcal{O})^{d}}$. We define

$\displaystyle\bm{\mathcal{V}}$
$\displaystyle:=\big{\\{}\bm{u}\in{\mathrm{C}_{0}^{\infty}(\mathcal{O})^{d}}:\nabla\cdot\bm{u}=0\big{\\}},$
$\displaystyle\mathbf{H}$ $\displaystyle:=\text{ the closure of
}\bm{\mathcal{V}}\text{ in the Lebesgue space
}{\mathrm{L}^{2}(\mathcal{O})^{d}},$
$\displaystyle\mathrm{L}_{\sigma}^{p}(\mathcal{O})^{d}$ $\displaystyle:=\text{
the closure of }\bm{\mathcal{V}}\text{ in the Lebesgue space
}{\mathrm{L}^{p}(\mathcal{O})^{d}},$
$\displaystyle\mathrm{W}_{\sigma}^{p,k}(\mathcal{O})^{d}$
$\displaystyle:=\text{ the closure of }\bm{\mathcal{V}}\text{ in the Sobolev
space }{\mathrm{W}^{k,p}(\mathcal{O})^{d}},$

for $1\leq p<\infty$ and $k\in\mathbb{N}$. In the particular case of $p=2$, we
denote the space $\mathrm{W}_{\sigma}^{p,k}(\mathcal{O})^{d}$ by
$\mathbf{V}^{k}$, and if $k=1$, we denote it by $\mathbf{V}_{p}$. If both
$p=2$ and $k=1$, we denote $\mathrm{W}_{\sigma}^{p,k}(\mathcal{O})^{d}$ solely
by $\mathbf{V}$. Let $(\cdot,\cdot)$ stand for the inner product of the
Hilbert space $\mathrm{L}^{2}(\mathcal{O})^{d}$, and we denote by
$\langle\cdot,\cdot\rangle,$ the induced duality product between the space
${\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}}$ and its dual
${\mathrm{W}^{-1,p^{\prime}}(\mathcal{O})^{d}}$, as well as between
${\mathrm{L}^{p}(\mathcal{O})^{d}}$ and its dual
${\mathrm{L}^{p^{\prime}}(\mathcal{O})^{d}}$, where
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$. In the sequel, the $\mathrm{L}^{p}$,
$\mathrm{W}^{k,p}$ and $\mathrm{W}^{-k,p^{\prime}}$ norms will be denoted in
short by $\|\cdot\|_{p}$, $\|\cdot\|_{k,p}$ and $\|\cdot\|_{-k,p^{\prime}}$.
On $\mathbf{V}$, we consider the equivalent norm
$\|\bm{u}\|_{\mathbf{V}}:=\|\nabla\bm{u}\|_{2}$, $\bm{u}\in\mathbf{V}$.

###  2.2. Stochastic setting

In this subsection, we start by recalling some notions of stochastic processes
necessary to study stochastic partial differential equations. We consider the
scalar-valued case, being the vectorial case a straightforward generalization.
Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space, where $\Omega$
denotes the set of all possible outcomes $\omega$, $\mathscr{F}$ is the set of
events and $\mathbb{P}:\mathscr{F}\longrightarrow[0,1]$ accounts for the
probability measure function. We assume that $(\Omega,\mathscr{F},\mathbb{P})$
is equipped with a filtration $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$, that is,
with a nondecreasing family of sub-sigma fields of $\mathscr{F}$
($\mathscr{F}_{s}\subset\mathscr{F}_{t}$ whenever $s<t$), such that
$\mathscr{F}_{0}$ contains all the null elements (elements $A\in\mathscr{F}$
with $\mathbb{P}(A)=0$) and $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$ is right-
continuous ($\mathscr{F}_{t}=\mathscr{F}_{t+}=\cap_{s>t}\mathscr{F}_{s}$ for
all $t\in[0,T]$). In this case, we denote a filtered probability space by
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\in[0,T]},\mathbb{P})$. Letting
$1\leq p\leq\infty$ and given a Banach space $\mathbf{B}$, we denote by
$\mathrm{L}^{p}(\Omega,\mathscr{F},\mathbb{P};\mathbf{B}),$ the space of all
measurable mappings
$v:(\Omega,\mathscr{F})\longrightarrow(\mathbf{B},\mathscr{B}(\mathbf{B}))$,
where $\mathscr{B}(\mathbf{B})$ denotes the Borel $\sigma-$algebra on
$\mathbf{B}$, such that

$\mathbb{E}\big{[}\|v\|_{\mathbf{B}}^{p}\big{]}<\infty,$

where the expectation $\mathbb{E}$ is taken with respect to
$(\Omega,\mathscr{F},\mathbb{P})$. A $\mathbf{B}-$valued stochastic process
can be interpreted as a mapping
$X:[0,T]\times\Omega\longrightarrow(\mathbf{B},\mathscr{B}(\mathbf{B}))$,
denoted by $X(t,\omega)$. For a fixed $\omega\in\Omega$, the mapping $t\mapsto
X(t,\omega)$ is called the _path or trajectory_ of $X$. We say that a
$\mathbf{B}-$valued stochastic process $X$ is _adapted to a filtration_
$\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$, or just $\mathscr{F}_{t}-$adapted, if the
mapping $X(t):\Omega\longrightarrow(\mathbf{B},\mathscr{B}(\mathbf{B}))$,
defined by $\omega\mapsto X(t,\omega)$, is measurable for all $t\in[0,T]$. The
$\mathbf{B}-$valued stochastic process $X$ is _progressively measurable_ on
the probability space $(\Omega,\mathscr{F},\mathbb{P})$, with respect to the
filtration $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$, that is, is
$\\{\mathscr{F}_{t}\\}_{t\in[0,T]}-$measurable, if the mapping
$(s,\omega)\longmapsto X(s,\omega)$ is measurable on
$\mathscr{B}([0,t])\otimes\mathscr{F}_{t}$ for all $t\in[0,T]$. Note that
every progressively measurable stochastic process is
$\mathscr{F}_{t}-$adapted.

We are now ready to define the notion of integral of a stochastic process with
values in specific Banach spaces. To this end, we first consider a bounded
linear operator $\mathrm{Q}\in{\mathcal{L}(\mathbf{U},\mathbf{U})}$, where
$\mathbf{U}$ is a separable Hilbert space with inner product denoted by
$(\cdot,\cdot)_{\mathbf{U}}$ and associated norm by $\|\cdot\|_{\mathbf{U}}$.
If $\mathrm{Q}$ is supposed to be nonnegative, symmetric and with finite
trace, then there exists an orthonormal basis
$\\{\mathbf{e}_{k}\\}_{k\in\mathbb{N}}$ of $\mathbf{U}$ such that

$\mathrm{Q}\mathbf{e}_{k}=\lambda_{k}\mathbf{e}_{k},\qquad\lambda_{k}\geq
0,\quad\forall\ k\in\mathbb{N}$

and so that $0$ is the only accumulation point of the sequence
$\\{\lambda_{k}\\}_{k\in\mathbb{N}}$. For a fixed
$\mathrm{Q}\in\mathcal{L}(\mathbf{U},\mathbf{U})$ nonnegative, symmetric and
with finite trace, a ${\mathbf{U}}-$valued stochastic process $\mathrm{W}$ on
the probability space $(\Omega,\mathscr{F},\mathbb{P})$ is a standard
$\mathrm{Q}-$Wiener process if and only if

$\mathrm{W}(t)=\sum_{k\in\mathbb{N}}\sqrt{\lambda_{k}}\beta_{k}(t)\mathbf{e}_{k},\qquad
t\in[0,T],$

where $\beta_{k}$, with $k\in\\{n\in\mathbb{N}:\lambda_{n}>0\\}$, are
independent real-valued Brownian motions on $(\Omega,\mathscr{F},\mathbb{P})$.
A $\mathrm{Q}-$Wiener process $\mathrm{W}$ is said to be adapted to a
filtration $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$, if
$\mathrm{W}(t)\in\mathscr{F}_{t}$ for every $t\in[0,T]$ and
$\mathrm{W}(t)-\mathrm{W}(s)$ is independent of $\mathscr{F}_{s},$ for all
$0\leq s\leq t\leq T$. Let us now set
$\mathbf{U}_{0}=\mathrm{Q}^{\frac{1}{2}}\mathbf{U}$, where
$\mathrm{Q}^{\frac{1}{2}}$ is the operator defined by

$\mathrm{Q}^{\frac{1}{2}}\mathbf{e}_{k}=\sqrt{\lambda_{k}}\mathbf{e}_{k}.$

Then $\mathbf{U}_{0}$ is a Hilbert space with inner product

$(\bm{u},\bm{v})_{0}=\big{(}\mathrm{Q}^{-\frac{1}{2}}\bm{u},\mathrm{Q}^{-\frac{1}{2}}\bm{v}),\qquad\bm{u},\bm{v}\in{\mathbf{U}_{0}},$

and associated norm denoted by $\|\cdot\|_{0}$, where
$\mathrm{Q}^{-\frac{1}{2}}$ is the pseudo-inverse operator of
$\mathrm{Q}^{\frac{1}{2}}$. If $\\{\mathbf{e}_{k}\\}_{k\in\mathbb{N}}$ is an
orthonormal basis of ${\mathbf{U}}_{0}$ and $\\{\beta_{k}\\}_{k\in\mathbb{N}}$
is a family of independent real-valued Brownian motions on
$(\Omega,\mathscr{F},\mathbb{P})$, then the series
$\sum\limits_{k\in\mathbb{N}}\sqrt{\lambda_{k}}\beta_{k}(t)\mathbf{e}_{k}$ is
convergent in $\mathrm{L}^{2}(\Omega,\mathscr{F},\mathbb{P};{\mathbf{U}})$,
because the inclusion ${\mathbf{U}}_{0}\subset{\mathbf{U}}$ defines a Hilbert-
Schmidt embedding from ${\mathbf{U}}_{0}$ onto ${\mathbf{U}}$. Recall that by
$\mathcal{L}_{2}({\mathbf{U}}_{0},{\mathbf{U}})$, we denote the set of all
linear operators $\mathrm{Q}:\mathbf{U}_{0}\longrightarrow\mathbf{U}$ such
that for every orthonormal basis $\\{\mathbf{e}_{k}^{0}\\}_{k\in\mathbb{N}}$
of ${\mathbf{U}}_{0},$ one has
$\sum\limits_{k\in\mathbb{N}}\|\mathrm{Q}\mathbf{e}_{k}^{0}\|_{\mathbf{U}}^{2}<\infty.$
In the case that $\mathrm{Q}$ is no longer of finite trace, one looses this
convergence, but it is possible to define the Wiener process. For that, we
consider a further Hilbert space ${\mathbf{U}}_{1}$ and a Hilbert-Schmidt
embedding $J:{\mathbf{U}}_{0}\longrightarrow{\mathbf{U}}_{1}$. Let us set
$\mathrm{Q}_{1}=JJ^{\ast}$, where
$J^{\ast}:{\mathbf{U}}_{1}\longrightarrow{\mathbf{U}}_{0}$ is the adjoint
operator of $J$. Then
$\mathrm{Q}_{1}\in\mathcal{L}({\mathbf{U}}_{1},{\mathbf{U}}_{1})$,
$\mathrm{Q}_{1}$ is nonnegative definite and symmetric with finite trace and
the series $\sum\limits_{k\in\mathbb{N}}\beta_{k}(t)J\mathbf{e}_{k}$ converges
in $\mathrm{L}^{2}(\Omega,\mathscr{F},\mathbb{P};{\mathbf{U}}_{1})$ and
defines a $\mathrm{Q}_{1}-$Wiener process on ${\mathbf{U}}_{1}$,

$\displaystyle\mathrm{W}(t)=\sum_{k\in\mathbb{N}}\beta_{k}(t)J\mathbf{e}_{k},\qquad
t\in[0,T].$ (2.1)

Moreover, we have that
$\mathrm{Q}^{\frac{1}{2}}_{1}({\mathbf{U}}_{1})=J({\mathbf{U}}_{0})$ and
$J:{\mathbf{U}}_{0}\longrightarrow\mathrm{Q}^{\frac{1}{2}}_{1}({\mathbf{U}}_{1})$
is an isometry. Since the Hilbert space ${\mathbf{U}}_{1}$ and the Hilbert-
Schmidt embedding $J$ considered above always exist, (2.1) defines a Wiener
process, even though $\mathrm{Q}$ is not of finite trace. The stochastic
process defined this way is usually referred to as a _cylindrical Wiener
process_ in ${\mathbf{U}}$. In this case, we say that a process $\phi(t)$,
$t\in[0,T]$, is integrable with respect to $\mathrm{W}(t)$, if it takes values
in $\mathcal{L}_{2}(\mathrm{Q}^{\frac{1}{2}}({\mathbf{U}}_{1}),{\mathbf{U}})$,
is predictable and if

$\mathbb{P}\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|\phi(s)\|_{\mathcal{L}_{2}(\mathrm{Q}^{\frac{1}{2}}({\mathbf{U}}_{1}),{\mathbf{U}})}\mathrm{d}s<\infty}\right)=1,$

where
$\mathcal{L}_{2}(\mathrm{Q}^{\frac{1}{2}}({\mathbf{U}}_{1}),{\mathbf{U}})$
denotes the set of all Hilbert-Schmidt operators from
$\mathrm{Q}^{\frac{1}{2}}({\mathbf{U}}_{1})$ to ${\mathbf{U}}$. Hence, we can
define the stochastic integral

$\int_{0}^{t}\phi(s)\mathrm{d}\mathrm{W}(s)=\int_{0}^{t}\phi(s)\circ
J^{-1}\mathrm{d}\mathrm{W}(s)=\sum_{k\in\mathbb{N}}\int_{0}^{t}\phi(s)\mathbf{e}_{k}\mathrm{d}\beta_{k}(s),$

which is independent of the choice of ${\mathbf{U}}_{1}$ and $J$. For more
details on stochastic integration, we are referring to [24, 47, 57].

Next, we define some probabilistic spaces of time-dependent functions that we
deal throughout this work. Given $1\leq q\leq\infty$ and a separable Banach
space $\mathbf{B}$, we consider the Bochner space
$\mathrm{L}^{q}(0,T;\mathbf{B})$ formed by all the $\mathbf{B}-$valued
measurable functions $\bm{u}$ on $[0,T]$ such that

$\displaystyle\|\bm{u}\|_{\mathrm{L}^{q}(0,T;\mathbf{B})}=\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|\bm{u}(t)\|_{\mathbf{B}}^{q}\mathrm{d}t}\right)^{\frac{1}{q}}<\infty.$

In the particular case of $\mathbf{B}=\mathrm{L}^{q}(\mathcal{O})^{d}$, we
shall denote $\mathrm{L}^{q}(0,T;\mathbf{B})$ in short by
$\mathbf{L}^{q}(\mathcal{O}_{T})$. Let $\mathrm{L}_{\rm
w}^{q}(0,T;\mathbf{B})$ denote a space $\mathrm{L}^{q}(0,T;\mathbf{B})$ with
the weak topology $\rm w$.

The space $\mathrm{C}([0,T]{\color[rgb]{1,0,0};}\mathbf{B})$ represents the
Banach space of all continuous functions from $[0,T]$ to $\mathbf{B}$ with the
norm

$\|\bm{u}\|_{\mathrm{C}([0,T];\mathbf{B})}=\sup_{t\in[0,T]}\|\bm{u}(t)\|_{\mathbf{B}}.$

Also, $\mathrm{C}_{\rm{w}}([0,T];\mathbf{B})$ denotes the subspace of
$\mathrm{L}^{\infty}(0,T;\mathbf{B})$ formed by weakly continuous functions
from $[0,T]$ into $\mathbf{B}$.

For $1\leq p<\infty$, $1\leq q\leq\infty$ and a separable Banach space
$\mathbf{B}$, we denote by
$\mathrm{L}^{p}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{q}(0,\\\
T;\mathbf{B})),$ the space of all functions $\bm{u}=\bm{u}(x,t,\omega)$
defined on $\mathcal{O}\times[0,T]\times\Omega$ and with values in
$\mathbf{B}$ such that $\bm{u}$ is measurable with respect to $(t,\omega)$ and
$\bm{u}$ is $\mathscr{F}_{t}-$ measurable for a.e. $t$, and

$\displaystyle\|\bm{u}\|_{\mathrm{L}^{p}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{q}(0,T;\mathbf{B}))}=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{ll}\displaystyle\mathopen{}\mathclose{{}\left\\{\mathbb{E}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|\bm{u}(t)\|_{\mathbf{B}}^{q}\mathrm{d}t}\right)^{\frac{p}{q}}}\right]}\right\\}^{\frac{1}{p}}<\infty,&1\leq
q<\infty,\\\
\displaystyle\mathopen{}\mathclose{{}\left\\{\mathbb{E}\mathopen{}\mathclose{{}\left[\displaystyle\operatorname*{ess\,sup}_{t\in[0,T]}\|\bm{u}(t)\|_{\mathbf{B}}^{p}}\right]}\right\\}^{\frac{1}{p}}<\infty,&q=\infty.\end{array}}\right.$

By
$\mathrm{L}^{p}(\Omega,\mathscr{F},\mathbb{P};\mathrm{C}([0,T];\mathbf{B}))$,
with $1\leq p<\infty$, we denote the space of all continuous and progressively
$\\{\mathscr{F}_{t}\\}_{t\in[0,T]}-$measurable $\mathbf{B}-$valued stochastic
processes $\bm{u}$ such that

$\displaystyle\|\bm{u}\|_{\mathrm{L}^{p}(\Omega,\mathscr{F},\mathbb{P};\mathrm{C}([0,T];\mathbf{B}))}=\mathopen{}\mathclose{{}\left\\{\mathbb{E}\mathopen{}\mathclose{{}\left[\displaystyle\sup_{t\in[0,T]}\|\bm{u}(t)\|_{\mathbf{B}}^{p}}\right]}\right\\}^{\frac{1}{p}}<\infty.$

We recall an important result on stochastic ordinary differential equations in
finite-dimensions, where we denote by $\mathcal{M}^{d\times d}(\mathbb{R}),$
the vector space of all real $d\times d-$matrices.

######  Lemma 2.1.

Let the map
$\mathbf{M}:[0,T]\times\mathbb{R}^{d}\times\Omega\longrightarrow\mathcal{M}^{d\times
d}(\mathbb{R})$, defined by
$(t,x,\omega)\mapsto\mathbf{M}=\mathbf{M}(t,x,\omega)$, and
$\mathbf{b}:[0,T]\times\mathbb{R}^{d}\times\Omega\longrightarrow\mathbb{R}^{d}$,
defined by $(t,x,\omega)\mapsto\mathbf{b}=\mathbf{b}(t,x,\omega)$ be both
continuous in $x\in\mathbb{R}^{d}$ for $t\in[0,T]$ and $\omega\in\Omega$, and
be progressively measurable on the probability space
$(\Omega,\mathscr{F},\mathbb{P})$ with respect to the filtration
$\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$. Assume the following conditions are
verified for all $R\in(0,\infty)$:

  1. (1)

$\displaystyle\int_{0}^{T}\sup_{|x|\leq
R}\big{(}|\mathbf{b}(t,x)|+|\mathbf{M}(t,x)|^{2}\big{)}dt<\infty$;

  2. (2)

$2\big{(}x-y\big{)}\cdot\big{(}\mathbf{b}(t,x)-\mathbf{b}(t,y)\big{)}+\big{|}\mathbf{M}(t,x)-\mathbf{M}(t,y)\big{|}^{2}\leq
K_{t}(R)$, $\forall\ t\in[0,T]$, $\forall\ x,y\in\mathbb{R}^{d}:|x|,|y|\leq
R$;

  3. (3)

$2x\cdot\mathbf{b}(t,x)+\big{|}\mathbf{M}(t,x)\big{|}^{2}\leq
K_{t}(1)\big{(}1+|x|^{2}\big{)}$, $\forall\ t\in[0,T]$, $\forall\
x\in\mathbb{R}^{d}:|x|\leq R$;

where $K_{t}(R)$ is an $\mathbb{R}^{+}-$valued process adapted to the
filtration $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$ and such that

$\alpha_{S}(R):=\int_{0}^{S}K_{t}(R)\mathrm{d}t<\infty,\quad\forall\
S\in[0,T].$

Then for any $\mathscr{F}_{0}-$measurable map
$X_{0}:\Omega\longrightarrow\mathbb{R}^{d},$ there exists a _unique solution_
to the stochastic differential equation

$\mathrm{d}X(t)=\mathbf{b}\big{(}t,X(t)\big{)}\mathrm{d}t+\mathbf{M}\big{(}t,X(t)\big{)}\mathrm{d}\mathrm{W}(t).$

###### Proof.

See [57, Theorem 3.1.1]. ∎

We recall also the following inequalities which are classical in the theory of
$p$-Laplace equations.

######  Lemma 2.2.

For all $\mathbf{M},\ \mathbf{N}\in\mathbb{R}^{d\times d}$, the following
assertions hold true:

$\displaystyle 2\leq p<\infty$
$\displaystyle\quad\Rightarrow\quad\frac{1}{2^{p-1}}|\mathbf{M}-\mathbf{N}|^{p}\leq\big{(}|\mathbf{M}|^{p-2}\mathbf{M}-|\mathbf{N}|^{p-2}\mathbf{N}\big{)}:\big{(}\mathbf{M}-\mathbf{N}\big{)};$
(2.2) $\displaystyle 1<p<2$
$\displaystyle\quad\Rightarrow\quad(p-1)|\mathbf{M}-\mathbf{N}|^{2}\leq\big{(}|\mathbf{M}|^{p-2}\mathbf{M}-|\mathbf{N}|^{p-2}\mathbf{N}\big{)}:(\mathbf{M}-\mathbf{N})\big{(}|\mathbf{M}|^{p}+|\mathbf{N}|^{p}\big{)}^{\frac{2-p}{p}}.$
(2.3)

###### Proof.

The proof combines [31, Lemmas 5.1 and 5.2]. ∎

From (2.2) and (2.3), one easily gets

$\displaystyle\big{(}|\mathbf{D}(\bm{u})|^{p-2}\mathbf{D}(\bm{u})-|\mathbf{D}(\bm{v})|^{p-2}\mathbf{D}(\bm{v})\big{)}:\big{(}\mathbf{D}(\bm{u})-\mathbf{D}(\bm{v})\big{)}\geq
0,\ \forall\ \bm{u},\bm{v}\in{\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}},$ (2.4)

and whenever $1<p<\infty$. As a consequence of (2.4), the operator

$\mathbf{A}(\bm{u}):=|\mathbf{D}(\bm{u})|^{p-2}\mathbf{D}(\bm{u})$ (2.5)

is said to be _monotone_ for any $p$ such that $1<p<\infty$.

###  2.3. Probabilistic weak formulation

We are interested in martingale solutions, that is, weak solutions in the
probabilistic sense, to the stochastic problem (1.1)-(1.4). This means that
when seeking martingale solutions of (1.1)-(1.4), constructing a filtered
probability space
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\in[0,T]},\mathbb{P})$ and a
cylindrical Wiener process $\mathrm{W}$ on it are both part of the problem. We
shall prove that a martingale solution typically exists for given Borel
measures $\Lambda_{0}$ and $\Lambda_{\bm{f}}$ that account for initial and
forcing laws as follows,

$\displaystyle\Lambda_{0}=\mathbb{P}\circ\bm{u}_{0}^{-1},\quad\mbox{i.e., }\
\mathbb{P}\big{(}\bm{u}_{0}\in\mathrm{U}\big{)}=\Lambda_{0}(\mathrm{U}),\qquad\forall\
\mathrm{U}\in\mathscr{B}(\mathbf{V}),$ (2.6)
$\displaystyle\Lambda_{\bm{f}}=\mathbb{P}\circ\bm{f}^{-1},\quad\mbox{i.e., }\
\mathbb{P}\big{(}\bm{f}\in\mathrm{U}\big{)}=\Lambda_{\bm{f}}(\mathrm{U}),\qquad\forall\
\mathrm{U}\in\mathscr{B}(\mathbf{L}^{2}(\mathcal{O}_{T})).$ (2.7)

It should be noted that even if the initial datum $\bm{u}_{0}$ and the forcing
term $\bm{f}$ are given, they might live on different probability spaces, and
therefore $\bm{u}_{0}$ and $\bm{u}(0)$ from one hand, and $\bm{f}_{t}$ and
$\bm{f}(t)$ on the other, can only coincide in law. In particular, the
underlying probability space is not a priori known but becomes part of the
solution. Next, we define martingale solutions for the stochastic problem
(1.1)-(1.4) starting with an initial law defined on $\mathbf{V}$ and a forcing
law defined on $\mathbf{L}^{2}(\mathcal{O}_{T})$.

######  Definition 2.3.

Let $\Lambda_{0}$, $\Lambda_{\bm{f}}$ be Borel probability measures on
$\mathbf{V}$ and $\mathbf{L}^{2}(\mathcal{O}_{T})$, respectively. We say that

$\big{(}(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\in[0,T]},\mathbb{P}),\bm{u},\bm{u}_{0},\bm{f},\mathrm{W}\big{)}$

is a _martingale solution_ to the stochastic problem (1.1)-(1.4), with initial
datum $\Lambda_{0}$ and forcing term $\Lambda_{\bm{f}}$, if:

  1. (1)

$\big{(}\Omega,\mathscr{F},\mathbb{P})$ is a stochastic basis with a complete
right-continuous filtration $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}$;

  2. (2)

$\mathrm{W}$ is a cylindrical $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}-$adapted
Wiener process;

  3. (3)

$\bm{u}$ is a progressively $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}-$measurable
stochastic process with $\mathbb{P}-$a.s. paths
$t\mapsto\bm{u}(t,\omega)\in\mathrm{L}^{\infty}(0,T;\mathbf{V})\cap\mathrm{L}^{p}(0,T;{\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}}),$
with a continuous modification having paths in $\mathrm{C}([0,T];\mathbf{V})$;

  4. (4)

$\bm{u}(0)\ (:=\bm{u}_{0})$ is progressively
$\\{\mathscr{F}_{t}\\}_{t\in[0,T]}-$measurable on the probability space
$(\Omega,\mathscr{F},\mathbb{P})$, with $\mathbb{P}-$a.s. paths
$\bm{u}(0,\omega)\in\mathbf{V}$ and
$\Lambda_{0}=\mathbb{P}\circ\bm{u}_{0}^{-1}$ in the sense of (2.6);

  5. (5)

$\bm{f}$ is an $\\{\mathscr{F}_{t}\\}_{t\in[0,T]}-$adapted stochastic process
$\mathbb{P}-$a.s. paths $\bm{f}(t,\omega)\in\mathbf{L}^{2}(\mathcal{O}_{T})$
and $\Lambda_{\bm{f}}=\mathbb{P}\circ\bm{f}^{-1}$ in the sense of (2.7);

  6. (6)

for every $\bm{\varphi}\in{\mathrm{C}_{0}^{\infty}(\mathcal{O})^{d}}$ with
$\operatorname{div}\bm{\varphi}=0$ and all $t\in[0,T],$ the following identity
holds $\mathbb{P}-$a.s.

$\displaystyle\int_{\mathcal{O}}\bm{u}(t)\cdot\bm{\varphi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}(t):\nabla\bm{\varphi}\,\mathrm{d}\bm{x}-\int_{0}^{t}\int_{\mathcal{O}}{\bm{u}\otimes\bm{u}}:\mathds{\nabla}\bm{\varphi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad+\nu\int_{0}^{t}\int_{\mathcal{O}}{|\mathbf{D}(\bm{u})|^{p-2}\mathbf{D}(\bm{u})}:\mathbf{D}\bm{\varphi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\int_{\mathcal{O}}\bm{u}_{0}\cdot\bm{\varphi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}_{0}:\nabla\bm{\varphi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\bm{f}}\cdot\bm{\varphi}\,\mathrm{d}\bm{x}\mathrm{d}s+\int_{0}^{t}\int_{\mathcal{O}}{\Phi(\bm{u})\mathrm{d}\mathrm{W}(s)}\cdot\bm{\varphi}\,\mathrm{d}\bm{x}.$
(2.8)

We are now interested to know whether it is possible to find a
probabilistically strong solution for the problem (1.1)-(1.4), for a given
initial velocity $\bm{u}_{0}$ and forcing $\bm{f}$ (which are random variables
rather than probability laws) on a given probability space.

######  Definition 2.4 (Probabilistically strong solution).

We are given a stochastic basis
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$, initial
datum $\bm{u}_{0}$ and a forcing term $\bm{f}$. Then, the problem (1.1)-(1.4)
has a pathwise _strong probabilistic solution_ if and only if there exists a
$\bm{u}:[0,T]\times\Omega\to\mathbf{V}$ with $\mathbb{P}-$a.s., paths

$\displaystyle\bm{u}(\cdot,\omega)\in\mathrm{L}^{\infty}(0,T;\mathbf{V})\cap\mathrm{L}^{p}(0,T;{\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}}),$

with a continuous modification having $\mathbb{P}-$a.s. paths in
$\mathrm{C}([0,T];\mathbf{V})$, and (6) holds for all
$\bm{\phi}\in\mathbf{V}.$

######  Definition 2.5 (Pathwise uniqueness).

For $i=1,2$, let $\bm{u}_{i}$ be any solution on the stochastic basis
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ to the
system (1.1)-(1.4) with initial datum $\bm{u}_{0}$ and forcing term
${\bm{f}}$. Then, the solutions of the system (1.1)-(1.4) are _pathwise
unique_ if and only if

$\displaystyle\mathbb{P}\big{\\{}\bm{u}_{1}(t)=\bm{u}_{2}(t),\ \forall\ t\geq
0\big{\\}}=1.$

###  2.4. Assumptions and main result

The existence of martingale solutions to the stochastic problem (1.1)-(1.4)
shall be carried out in the forthcoming pages under suitable assumptions on
the initial datum $\bm{u}_{0}$, forcing term $\bm{f}$ and on the noise
coefficient $\Phi(\bm{u})$, and for a suitable range of the summability
Lebesgue exponent $p$. On the initial and forcing laws, we assume that for
some constant $\gamma=\gamma(p,d)$ (to be determined further on)

$\displaystyle\displaystyle\int_{\mathbf{V}}\|\bm{z}\|_{\mathbf{V}}^{\gamma}\mathrm{d}\Lambda_{0}(\bm{z})<\infty,$
(2.9)
$\displaystyle\displaystyle\int_{\mathbf{L}^{2}(\mathcal{O}_{T})}\|\mathbf{g}\|_{\mathrm{L}^{2}(\mathcal{O}_{T})}^{\gamma}\mathrm{d}\Lambda_{\bm{f}}(\mathbf{g})<\infty.$
(2.10)

Let $\mathbf{U}$ be a Hilbert space with orthonormal basis
$\\{\mathbf{e}_{k}\\}_{k\in\mathbb{N}}$ and let
$\mathcal{L}_{2}(\mathbf{U},\mathrm{L}^{2}(\mathcal{O})^{d})$ be the space of
Hilbert-Schmidt operators form $\mathbf{U}$ to
$\mathrm{L}^{2}(\mathcal{O})^{d}$ (to simplify writing, in the sequel we
denote the operator norm
$\|\cdot\|_{\mathcal{L}_{2}(\mathbf{U},\mathrm{L}^{2}(\mathcal{O})^{d})}$
solely by $\|\cdot\|_{\mathcal{L}_{2}}$). Moreover, we define an auxiliary
space $\mathbf{U}_{0}\supset\mathbf{U}$ as

$\displaystyle\mathbf{U}_{0}:=\bigg{\\{}\bm{v}=\sum_{k\in\mathbb{N}}\alpha_{k}\mathbf{e}_{k}:\sum_{k\in\mathbb{N}}\frac{\alpha_{k}^{2}}{k^{2}}<\infty\bigg{\\}}$

equipped with the norm

$\displaystyle\|\bm{v}\|_{\mathbf{U}_{0}}^{2}:=\sum_{k\in\mathbb{N}}\frac{\alpha_{k}^{2}}{k^{2}},\
\ \bm{v}=\sum_{k\in\mathbb{N}}\alpha_{k}\mathbf{e}_{k}.$

Throughout the sequel, we consider a cylindrical
$\\{\mathscr{F}_{t}\\}-$Wiener process $\mathrm{W}=\\{\mathrm{W}(t)\\}_{t\geq
0}$ which has the form

$\displaystyle\mathrm{W}(t)=\sum_{k\in\mathbb{N}}\mathbf{e}_{k}\beta_{k}(t),$

where $\\{\beta_{k}\\}_{k\in\mathbb{N}}$ are independent real-valued Brownian
motions. The embedding $\mathbf{U}\hookrightarrow\mathbf{U}_{0}$ is Hilbert-
Schmidt and trajectories of $\mathrm{W}$ are $\mathbb{P}$-a.s. continuous with
values in $\mathbf{U}_{0}$ (see Subsection 2.2). In this article, we suppose
that the noise coefficient $\Phi(\bm{u})$ satisfies linear growth and
Lipschitz conditions. More precisely, we assume that for each
$\bm{w}\in{\mathrm{L}^{2}(\mathcal{O})^{d}}$ there is a mapping

$\displaystyle\Phi(\bm{w}):$
$\displaystyle\mathbf{U}\longrightarrow{\mathrm{L}^{2}(\mathcal{O})^{d}}$
$\displaystyle\mathbf{e}_{k}\longmapsto\Phi(\bm{w})\mathbf{e}_{k}=\phi_{k}(\bm{w}),$

where $\\{\mathbf{e}_{k}\\}_{k\in\mathbb{N}}$ is an orthonormal basis of
$\mathbf{U}$, such that $\phi_{k}\in\mathrm{C}(\mathbb{R}^{d})$ and the
following conditions hold for some constants $K,L>0$:

$\displaystyle\displaystyle\sum_{k\in\mathbb{N}}|\phi_{k}(\bm{\xi})|\leq
K(1+|\bm{\xi}|),\ \mbox{ and }\
\displaystyle\sum_{k\in\mathbb{N}}|\phi_{k}(\bm{\xi})-\phi_{k}(\bm{\zeta})|\leq
L|\bm{\xi}-\bm{\zeta}|,\ \ {\bm{\xi},\bm{\zeta}\in\mathbb{R}^{d}}.$ (2.11)

Moreover, we are assuming that the following condition holds for some constant
$C>0$,

$\displaystyle\sup_{k\in\mathbb{N}}k^{2}|\phi_{k}(\bm{\xi})|^{2}\leq
C(1+|\bm{\xi}|^{2}),\qquad{\bm{\xi}\in\mathbb{R}^{d}}.$ (2.12)

######  Theorem 2.6.

Let $\mathcal{O}\subset\mathbb{R}^{d}$ be a bounded domain with a smooth
boundary $\partial\mathcal{O}$ of class $\mathrm{C}^{2}$, and assume that
conditions (2.9) and (2.10) hold for

$\gamma\geq\max\bigg{\\{}\frac{pd}{d-2},2+\frac{2p}{d-2},\frac{2d}{d-2}\bigg{\\}}\
(d\neq 2)\ \text{ and }\ \gamma\geq 2\ (d=2),$ (2.13)

and (2.11), (2.12) are fulfilled. If $2\leq d\leq 4$ and

$\displaystyle p>\frac{2d}{d+2},$ (2.14)

then there exists, at least, a _martingale solution_

$\big{(}(\overline{\Omega},\overline{\mathscr{F}},\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]},\overline{\mathbb{P}}),\overline{\bm{u}},\overline{\bm{u}}_{0},\overline{\bm{f}},\overline{\mathrm{W}}\big{)}$

in the sense of Definition 2.3 to the stochastic problem (1.1)-(1.4).

######  Theorem 2.7.

Under the assumptions of Theorem 2.6, there exists a unique _probabilistically
strong solution_ of the system (1.1)-(1.4) in the sense of Definition 2.4.

The proof of the above theorems are presented in the subsequent sections.

##  3\. Pressure decomposition

In this section, our focus will be on the pressure term coming in our model
and we decompose it in such a way that each part of the pressure term
corresponds to one term in the equation. The following theorem generalizes the
idea of [63, Theorem 2.6] to the stochastic case and a similar result has been
established in [14, Section 3] for the stochastic case.

######  Theorem 3.1.

Let us consider a stochastic basis
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$,
$\bm{u}\in\mathrm{L}^{2}({\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{\infty}(0,T;\mathbf{V}))$,
${\mathbfcal{H}\in\mathbf{L}^{r}(\mathcal{O}_{T})}$ for some $1<r\leq 2$, both
adapted to $\\{\mathscr{F}_{t}\\}_{t\geq 0}$. Moreover, if the initial data
$\bm{u}_{0}\in\mathrm{L}^{2}({\Omega,\mathscr{F},\mathbb{P}};\mathbf{V})$ and
$\Phi\in\mathrm{L}^{2}({\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{\infty}(0,T;{\mathcal{L}_{2}(\mathbf{U},\mathrm{L}^{2}(\mathcal{O})^{d})}))$
progressively measurable such that

$\displaystyle\int_{\mathcal{O}}\bm{u}(t)\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}(t):\nabla\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\mathbfcal{H}}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\int_{\mathcal{O}}\bm{u}_{0}\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}_{0}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\Phi(\bm{u})}\mathrm{d}\mathrm{W}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x},$
(3.1)

for all $\bm{\phi}\in\bm{\mathcal{V}}$ and all $t\in[0,T]$. Then there are
functions $\pi_{\mathbfcal{H}}$, $\pi_{\Phi}$ and $\pi_{h}$ adapted to
$\\{\mathscr{F}_{t}\\}_{t\geq 0}$ such that

  1. (1)

We have $\Delta\pi_{h}=0$ and there holds for $m:=\min\\{2,r\\}$

$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\|\pi_{\mathbfcal{H}}(t)\|^{r}_{r}\mathrm{d}t\bigg{]}\leq
C\mathbb{E}\bigg{[}\int_{0}^{T}\|{\mathbfcal{H}}(t)\|^{r}_{r}\mathrm{d}t\bigg{]},$
$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\pi_{\Phi}(t)\|^{2}_{2}\bigg{]}\leq
C\mathbb{E}\bigg{[}\sup_{t\in[0,T]}{\|\Phi(\bm{u}(t))\|^{2}_{\mathcal{L}_{2}}}\bigg{]},$
$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\pi_{h}(t)\|^{m}_{m}\bigg{]}\leq
C\mathbb{E}\bigg{[}1+\sup_{t\in[0,T]}\Big{\\{}\|\bm{u}(t)\|^{2}_{2}+\kappa\|\nabla\bm{u}(t)\|^{2}_{2}\Big{\\}}$
$\displaystyle+\|\bm{u}_{0}\|^{2}_{2}+\kappa\|\nabla\bm{u}_{0}\|^{2}_{2}+{\sup_{t\in[0,T]}\|\Phi(\bm{u}(t))\|^{2}_{\mathcal{L}_{2}}}+\int_{0}^{T}\|{\mathbfcal{H}}(t)\|^{r}_{r}\mathrm{d}t\bigg{]}.$

  2. (2)

There holds

$\displaystyle\int_{\mathcal{O}}\big{(}\bm{u}(t){+}\nabla\pi_{h}(t)\big{)}\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}(t):\nabla\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\mathbfcal{H}}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad{-}\int_{0}^{t}\int_{\mathcal{O}}\pi_{\mathbfcal{H}}\operatorname{div}\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\int_{\mathcal{O}}\bm{u}_{0}\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}_{0}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}{+}\int_{\mathcal{O}}\pi_{\Phi}(t)\operatorname{div}\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\Phi(\bm{u})}\mathrm{d}\mathrm{W}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x},$

for all $\bm{\phi}\in\mathrm{C}_{0}^{\infty}(\mathcal{O})^{d}$. Moreover
$\pi_{h}(0)=\pi_{\mathbfcal{H}}(0)=\pi_{\Phi}(0)=0,\;\mathbb{P}-$a.s.

######  Remark 3.2.

If we place all the decomposed terms of pressure together, we have

$\displaystyle\pi(t)=\pi_{h}(t)+\pi_{\Phi}(t)+\int_{0}^{t}\pi_{\mathbfcal{H}}(s)\mathrm{d}s,$

then there holds
$\pi\in\mathrm{L}^{m}(\Omega,{\mathscr{F},\mathbb{P}};\mathrm{L}^{\infty}(0,T;\mathrm{L}^{m}(\mathcal{O})))$.

Before proceeding with the proof of Theorem 3.1, let us recall some important
auxiliary results. The following lemma is a variant of the well-known de
Rham’s theorem.

######  Lemma 3.3.

Assume $1<m<\infty$.

  1. (1)

For each $\bm{v}^{\ast}\in{\mathrm{W}^{-1,m}(\mathcal{O})^{d}}$ such that

$\big{<}\bm{v}^{\ast},\bm{v}\big{>}=0,\quad\forall\
\bm{v}\in\mathbf{V}_{m^{\prime}},$

there exists a unique $\pi\in\mathrm{L}^{m}(\mathcal{O})$, with
$\int_{\mathcal{O}}\pi\,\,\mathrm{d}\bm{x}=0$, such that

$\mathopen{}\mathclose{{}\left\langle\bm{v}^{\ast},\bm{v}}\right\rangle=\int_{\mathcal{O}}\pi\operatorname{div}\bm{v}\,\,\mathrm{d}\bm{x},\quad\forall\,\bm{v}\in\mathrm{W}_{0}^{1,m^{\prime}}(\mathcal{O})^{d}.$

Moreover, there exists a positive constant $C$ such that

$\|\pi\|_{m}\leq C\|\bm{v}^{\ast}\|_{-1,m}.$

  2. (2)

For each $\xi\in\mathrm{L}^{m}(\Omega)$ with
$\int_{\Omega}\xi\,\,\mathrm{d}\bm{x}=0$, there exists, at least, one solution
$\bm{w}\in\mathrm{W}^{1,m}_{0}(\mathcal{O})^{d}$ to the problem

$\operatorname{div}\bm{w}=\xi\quad\mbox{in}\quad\mathcal{O},$

and such that for some positive constant $C$,

$\|\nabla\bm{w}\|_{m}\leq C\|\xi\|_{m}.$

###### Proof.

The proof is due to Bogovskiǐ [18] (see also [30, Theorems III.3.1 and
III.5.3] and [61, Proposition I.1.1]). ∎

###### Proof.

Step 1: Define a bilinear form $\mathcal{S}$ with the domain
$\mathcal{D}(\mathcal{S})\subseteq\mathrm{L}^{2}(\mathcal{O})^{d}$ by setting
$\mathcal{D}(\mathcal{S})=\mathrm{W}_{0}^{1,2}(\mathcal{O})^{d}$ and

$\mathcal{S}(\bm{u},\bm{v})=\int_{\mathcal{O}}\bm{u}\cdot\bm{v}\,\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}:\nabla\bm{v}\,\,\mathrm{d}\bm{x}.$

Since $\mathrm{W}_{0}^{1,2}(\mathcal{O})^{d}$ is complete with respect to the
norm

$\mathcal{S}(\bm{u},\bm{u})^{\frac{1}{2}}=\mathopen{}\mathclose{{}\left(\|\bm{u}\|_{2}^{2}+\kappa\|\nabla\bm{u}\|^{2}_{2}}\right)^{\frac{1}{2}},$

the form $\mathcal{S}$ is closed. As $\kappa>0$, it is easy to see that
$\mathcal{S}$ is positive and symmetric. Therefore by [59, Lemma II.3.2.1],
there exists a uniquely determined positive self-adjoint operator
$\mathcal{A}:\mathcal{D}(\mathcal{A})\longrightarrow\mathrm{L}^{2}(\mathcal{O})^{d}$,
with dense domain
$\mathcal{D}(\mathcal{A})\subseteq\mathrm{W}_{0}^{1,2}(\mathcal{O})^{d}$, such
that

$\int_{\mathcal{O}}\mathcal{A}(\bm{u})\cdot\bm{v}\,\,\mathrm{d}\bm{x}=\mathcal{S}(\bm{u},\bm{v})=\int_{\mathcal{O}}\bm{u}\cdot\bm{v}\,\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}:\nabla\bm{v}\,\,\mathrm{d}\bm{x}.$

Setting $\bm{v}\in\mathrm{C}_{0}^{\infty}(\mathcal{O})^{d}$, we can see that
$\mathcal{A}(\bm{u})=(\mathbf{I}-\kappa\Delta)(\bm{u})=(\mathbf{I}-\kappa\operatorname{div}\nabla)(\bm{u})$
holds in the distribution sense, and so we can set
$\mathcal{A}=\mathbf{I}-\kappa\Delta$, where $\mathbf{I}$ denotes the identity
operator. In this way, the operator
$\mathbf{I}-\kappa\Delta:\mathcal{D}(\mathbf{I}-\kappa\Delta)\longrightarrow\mathrm{L}^{2}(\mathcal{O})^{d}$
is defined by

$\int_{\mathcal{O}}(\mathbf{I}-\kappa\Delta)(\bm{u})\cdot\bm{v}\,\,\mathrm{d}\bm{x}=\mathcal{S}(\bm{u},\bm{v})=\int_{\mathcal{O}}\bm{u}\cdot\bm{v}\,\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}:\nabla\bm{v}\,\,\mathrm{d}\bm{x},$
(3.2)

for $\bm{u}\in\mathcal{D}(\mathbf{I}-\kappa\Delta)$,
$\bm{v}\in\mathrm{W}_{0}^{1,2}(\mathcal{O})^{d}$, and

$\mathcal{D}(\mathbf{I}-\kappa\Delta)=\mathopen{}\mathclose{{}\left\\{\bm{u}\in\mathrm{W}_{0}^{1,2}(\mathcal{O})^{d}:\bm{v}\longmapsto\mathcal{S}(\bm{u},\bm{v})\
\mbox{is continuous in the $\mathrm{L}^{2}$-norm}}\right\\}.$

Given the above, and observing that $m\leq\min\\{r,2\\}$, we can write (3.1)
in the following form:

$\displaystyle\int_{\mathcal{O}}(\mathbf{I}-\kappa\Delta)(\bm{u}(t))\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\mathbfcal{H}}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\int_{\mathcal{O}}(\mathbf{I}-\kappa\Delta)(\bm{u}_{0})\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u})\mathrm{d}\mathrm{W}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x},$
(3.3)

for all $\bm{\phi}\in\mathrm{W}_{0}^{1,m^{\prime}}(\mathcal{O})^{d}$ with
$\operatorname{div}\bm{\phi}=0$ in $\mathcal{O}$, and all $t\in[0,T]$.

Step 2: Using (3.3) and Lemma 3.3-(1), we can argue as in the proof of [63,
Theorem 2.6] to prove the existence of a unique pressure
$\pi\in\mathrm{C}_{\rm{w}}([0,T];\mathrm{L}^{m}(\mathcal{O}))$, with

$\int_{\mathcal{O}}\pi(t)\,\mathrm{d}\bm{x}=0,\qquad\forall\ t\in[0,T],$ (3.4)

such that

$\displaystyle\int_{\mathcal{O}}\bm{u}(t)\cdot{(\mathbf{I}-\kappa\Delta)(\bm{\phi})}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\mathbfcal{H}}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\int_{\mathcal{O}}\bm{u}_{0}\cdot{(\mathbf{I}-\kappa\Delta)(\bm{\phi})}\,\mathrm{d}\bm{x}{+}\int_{\mathcal{O}}\pi(t)\operatorname{div}\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}{\Phi(\bm{u})}\mathrm{d}\mathrm{W}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x},$
(3.5)

for all $\bm{\phi}\in\mathrm{W}_{0}^{1,m^{\prime}}(\mathcal{O})^{d}$ (note
that $m=\min\\{2,r\\}$), and all $t\in[0,T]$.

Now, we will prove that

$\displaystyle\pi\in\mathrm{L}^{m}(\Omega,{\mathscr{F},\mathbb{P}};\mathrm{L}^{\infty}(0,T;\mathrm{L}^{m}(\mathcal{O}))).$
(3.6)

From (3), we can say that $\pi$ is a progressively measurable process, since
all other terms in this equation are. By Lemma 3.3-(2), with $m^{\prime}$ in
the place of $m$, given $\psi\in\mathrm{L}^{m^{\prime}}(\mathcal{O}),$ there
exists $\bm{\phi}\in\mathrm{W}_{0}^{1,m^{\prime}}(\mathcal{O})^{d}$ such that

$\operatorname{div}\bm{\phi}=\psi-\psi_{\mathcal{O}},\qquad\psi_{\mathcal{O}}:=\frac{1}{|\mathcal{O}|}\int_{\mathcal{O}}\psi\,\mathrm{d}\bm{x}.$
(3.7)

Combining (3.4) and (3.7) with (3), one has

$\displaystyle\int_{\mathcal{O}}\pi(t)\psi\,\mathrm{d}\bm{x}$
$\displaystyle=\int_{\mathcal{O}}\pi(t)(\psi-\psi_{\mathcal{O}})\,\mathrm{d}\bm{x}$
$\displaystyle=\int_{\mathcal{O}}(\bm{u}(t)-\bm{u}_{0})\cdot{(\mathbf{I}-\kappa\Delta)\mathcal{B}(\psi)}\,\mathrm{d}\bm{x}{+}\int_{0}^{t}\int_{\mathcal{O}}{\mathbfcal{H}}:\nabla{\mathcal{B}(\psi)}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad{-}\int_{0}^{t}\int_{\mathcal{O}}{\Phi(\bm{u})}\mathrm{d}\mathrm{W}(s)\cdot{\mathcal{B}(\psi)}\,\mathrm{d}\bm{x},$

for all $\psi\in\mathrm{L}^{m^{\prime}}(\mathcal{O})$, and where
$\mathcal{B}:\mathrm{L}^{m^{\prime}}(\mathcal{O})\longrightarrow\mathrm{W}_{0}^{1,m^{\prime}}(\mathcal{O})^{d}$
is the Bogovskiĭ operator (see [56, Appendix 4.2]). Denoting by
$\mathcal{B}^{\ast}:\mathrm{W}_{0}^{1,m^{\prime}}(\mathcal{O})^{d}\longrightarrow\mathrm{L}^{m^{\prime}}(\mathcal{O}),$
the Bogovskiĭ adjoint operator, this implies

$\displaystyle\pi(t)=\big{(}(\mathbf{I}-\kappa\Delta)\mathcal{B}\big{)}^{\ast}(\bm{u}(t)-\bm{u}_{0})+\int_{0}^{t}(\nabla\mathcal{B})^{*}{\mathbfcal{H}}\mathrm{d}s-\int_{0}^{t}\mathcal{B}^{*}\Phi(\bm{u})\mathrm{d}\mathrm{W}(s),$
(3.8)

with respect to the $\mathrm{L}^{2}(\mathcal{O})$–inner product. From (3.8),
one immediately has $\pi(0)=0$. Averaging (3.8) in the
$\mathrm{L}^{m}(\mathcal{O})$–norm, we get

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\pi(t)\|^{m}_{m}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\bigg{\|}\big{(}(\mathbf{I}-\kappa\Delta)\mathcal{B}\big{)}^{\ast}(\bm{u}(t)-\bm{u}_{0})-\int_{0}^{t}(\nabla\mathcal{B})^{*}{\mathbfcal{H}}(\bm{u}(s))\,\mathrm{d}s+\int_{0}^{t}\mathcal{B}^{*}\Phi(\bm{u}(s))\mathrm{d}\mathrm{W}(s)\bigg{\|}_{m}^{m}\bigg{]}$
$\displaystyle\leq\par
C\bigg{(}1+\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\big{(}(\mathbf{I}-\kappa\Delta)\mathcal{B}\big{)}^{\ast}(\bm{u}(t)-\bm{u}_{0})\|_{2}^{2}\bigg{]}+\mathbb{E}\bigg{[}\int_{0}^{T}\|(\nabla\mathcal{B})^{*}{\mathbfcal{H}}(\bm{u}(t))\|_{m}^{m}\mathrm{d}t\bigg{]}$
$\displaystyle\qquad+\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\bigg{\|}\int_{0}^{t}\mathcal{B}^{*}\Phi(\bm{u}(s))\mathrm{d}\mathrm{W}(s)\bigg{\|}_{2}^{2}\bigg{]}\bigg{)}=:C\bigg{(}1+\sum_{i=1}^{3}I_{i}\bigg{)},$
(3.9)

where we have used the Minkowski, Hölder and Young inequalities, together with
the fact that $m\leq 2$. Using the continuity of the operator
$\mathcal{B}^{\ast}$ from $\mathrm{L}^{2}(\mathcal{O})^{d}$ to
$\mathrm{L}^{2}(\mathcal{O})$, observing first that
$(\mathbf{I}-\kappa\Delta)$ is self-adjoint and therefore
$\big{(}(\mathbf{I}-\kappa\Delta)\mathcal{B}\big{)}^{\ast}=\mathcal{B}^{\ast}(\mathbf{I}-\kappa\Delta)$,
and still using (3.2) and Minkowski’s inequality, we have

$\displaystyle I_{1}$ $\displaystyle\leq
C\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|(\mathbf{I}-\kappa\Delta)(\bm{u}(t)-\bm{u}_{0})\|_{2}^{2}\bigg{]}$
$\displaystyle=C\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\bm{u}(t)-\bm{u}_{0}\|_{2}^{2}+\kappa\|\nabla(\bm{u}(t)-\bm{u}_{0})\|^{2}_{2}\bigg{]}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\Big{\\{}\|\bm{u}(t)\|_{2}^{2}+\kappa\sup_{t\in[0,T]}\|\nabla\bm{u}(t)\|^{2}_{2}\Big{\\}}+\|\bm{u}_{0}\|_{2}^{2}+\kappa\|\nabla\bm{u}_{0}\|^{2}_{2}\bigg{]}.$

Since $\mathbfcal{H}\in\mathbf{L}^{r}(\mathcal{O}_{T})$, $m=\min\\{2,r\\}$ and
$1<r\leq 2$, and still using the fact that $(\nabla\mathcal{B})^{*}$ is a
continuous operator from $\mathrm{L}^{r}(\mathcal{O})^{d\times d}$ to
$\mathrm{L}^{r}(\mathcal{O})$, a similar calculation yields

$\displaystyle I_{2}\leq
C\mathbb{E}\bigg{[}1+\int_{0}^{T}\|{\mathbfcal{H}}(t)\|_{r}^{r}\mathrm{d}t\bigg{]}.$

The stochastic integral term $I_{3}$ appearing in (3) can be handled with the
help of Burkholder-Davis-Gundy (BDG) inequality, using also here the
continuity of $\mathcal{B}^{\ast}$ from $\mathrm{L}^{2}(\mathcal{O})^{d}$ to
$\mathrm{L}^{2}(\mathcal{O})$,

$\displaystyle I_{3}\leq
C\mathbb{E}\bigg{[}\int_{0}^{T}\|\Phi(\bm{u}(t))\|^{2}_{\mathcal{L}_{2}}\mathrm{d}t\bigg{]}.$

Combining the above estimates in (3), we obtain

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\pi(t)\|_{m}^{m}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}1+\sup_{t\in[0,T]}\Big{\\{}\|\bm{u}(t)\|^{2}_{2}+\kappa\|\nabla\bm{u}(t)\|^{2}_{2}\Big{\\}}+\|\bm{u}_{0}\|_{2}^{2}+\kappa\|\nabla\bm{u}_{0}\|^{2}_{2}$
$\displaystyle\qquad+\int_{0}^{T}\|{\mathbfcal{H}}(t)\|_{r}^{r}\mathrm{d}t+\int_{0}^{T}\|\Phi(\bm{u}(t))\|^{2}_{\mathcal{L}_{2}}\mathrm{d}t\bigg{]}.$
(3.10)

On the other hand, by [63, Corollary 2.5], there exist unique functions

$\displaystyle\pi_{0}\in\mathrm{C}_{\rm{w}}([0,T];\Delta\mathrm{W}_{0}^{2,m}(\mathcal{O})),\qquad\Delta\mathrm{W}_{0}^{2,m}(\mathcal{O}):=\mathopen{}\mathclose{{}\left\\{\Delta
v:v\in\mathrm{W}_{0}^{2,m}(\mathcal{O})}\right\\},$
$\displaystyle\pi_{h}\in\mathrm{C}_{\rm{w}}([0,T];\mathrm{L}^{m}_{\Delta}(\mathcal{O})),\qquad\mathrm{L}^{m}_{\Delta}(\mathcal{O}):=\mathopen{}\mathclose{{}\left\\{v\in\mathrm{L}^{m}(\mathcal{O}):\Delta
v=0,\ \ \int_{\mathcal{O}}v\,\mathrm{d}\bm{x}=0}\right\\},$

such that

$\displaystyle\pi:=\pi_{0}+\pi_{h}\qquad\mbox{in}\ \ \mathcal{O}_{T},$
$\displaystyle\sup_{t\in[0,T]}\|\pi_{0}(t)\|_{m}+\sup_{t\in[0,T]}\|\pi_{h}(t)\|_{m}\leq
C\sup_{t\in[0,T]}\|\pi(t)\|_{m},$ (3.11)

for some positive constant $C$. Therefore, we can decompose the pressure term
pointwise on $\mathcal{O}_{T}$ as follows:

$\displaystyle\pi:=\pi_{0}+\pi_{h},\qquad\pi_{0}:=\Delta\Delta_{\mathcal{O}}^{-2}\Delta\pi,\qquad\pi_{h}:=\pi-\pi_{0},$

where $\Delta_{\mathcal{O}}^{-2}$ denotes the solution operator to the bi-
Laplace equation with respect to zero boundary values for the function and
gradient. As a consequence of (3.11), inequality (3) holds true for $\pi_{0}$
and $\pi_{h}$ as well. Into (3), replacing $\pi$ by $\pi_{0}+\pi_{h}$,
inserting $\bm{\phi}=\nabla\psi$,
$\psi\in\mathrm{W}_{0}^{2,m^{\prime}}(\mathcal{O})$, and observing that
$\operatorname{div}\bm{u}=0$, $\operatorname{div}\bm{u}_{0}=0$ and
$\Delta\pi_{h}=0$, one has

$\displaystyle\int_{\mathcal{O}}\pi_{0}(t)\Delta\psi\,\mathrm{d}\bm{x}=\int_{0}^{t}\int_{\mathcal{O}}{\mathbfcal{H}}:\nabla^{2}\psi\,\mathrm{d}\bm{x}\mathrm{d}s-\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u})\mathrm{d}\mathrm{W}(s)\cdot\nabla\psi\,\mathrm{d}\bm{x},$
(3.12)

for all $\psi\in\mathrm{W}_{0}^{2,m^{\prime}}(\mathcal{O})$. From (3.12), it
is clear that $\pi_{0}(0)=0$. And once that $\pi(0)=0$, we also have
$\pi_{h}(0)=0$. As
$\operatorname{div}^{2}{\mathbfcal{H}}=\operatorname{div}(\operatorname{div}{\mathbfcal{H}})\in\mathrm{W}^{-2,m}(\mathcal{O})$,
by [63, Lemma 2.4], there exists a function
$\pi_{\mathbfcal{H}}(t)\in\Delta\mathrm{W}_{0}^{2,m}(\mathcal{O})$ such that

$\displaystyle\int_{\mathcal{O}}\pi_{\mathbfcal{H}}(t)\Delta\psi\,\mathrm{d}\bm{x}=\int_{\mathcal{O}}{\mathbfcal{H}}(t):\nabla^{2}\psi\,\mathrm{d}\bm{x},$

for all $\psi\in\mathrm{C}_{0}^{\infty}(\mathcal{O})$. The measurability of
$\pi_{\mathbfcal{H}}$ follows from the measurability of the right hand side.
On account of the solvability of the bi-Laplace equation, one has

$\displaystyle\|\pi_{\mathbfcal{H}}(t)\|_{m}^{m}\leq
C\|{\mathbfcal{H}}(t)\|_{m}^{m}\qquad\mathbb{P}\otimes\lambda-\text{a.e.}$

for some positive constant $C$, where $\lambda$ is the Lebesgue measure on
$(0,T)$. For more details, see [51], or [63, Lemma 2.1] and [15, Lemma 2.2].
This in turn gives

$\displaystyle\int_{\Omega\times\mathcal{O}_{T}}\|\pi_{\mathbfcal{H}}(t)\|_{m}^{m}\mathrm{d}t\mathrm{d}\mathbb{P}\leq
C\int_{\Omega\times\mathcal{O}_{T}}\|{\mathbfcal{H}}(t)\|_{m}^{m}\mathrm{d}t\mathrm{d}\mathbb{P}.$

Next, we define

$\pi_{\Phi}(t):=\pi_{0}(t)+\int_{0}^{t}\pi_{\mathbfcal{H}}(s)\mathrm{d}s,$

as the unique solution to the following integral equation

$\displaystyle\int_{\mathcal{O}}\pi_{\Phi}(t)\Delta\psi\,\mathrm{d}\bm{x}=\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u}(s))\mathrm{d}\mathrm{W}(s)\cdot\nabla\psi\,\mathrm{d}\bm{x},\qquad\
\psi\in\mathrm{C}_{0}^{\infty}(\mathcal{O}).$ (3.13)

Since $\pi_{\Phi}(t)\in\Delta\mathrm{W}_{0}^{2,m}(\mathcal{O})$, (3.13) can be
written as follows

$\displaystyle\int_{\mathcal{O}}\pi_{\Phi}(t)\psi\,\mathrm{d}\bm{x}=\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u})\mathrm{d}\mathrm{W}(s)\cdot\nabla(\Delta^{-2}_{\mathcal{O}}\Delta\psi)\,\mathrm{d}\bm{x},\qquad\psi\in\mathrm{C}_{0}^{\infty}(\mathcal{O}).$
(3.14)

From here, one immediately has $\pi_{\Phi}(0)=0$. Defining the operator
$\mathscr{D}:=\nabla\Delta_{\mathcal{O}}^{-2}\Delta:\mathrm{W}_{0}^{2,m}(\mathcal{O})\longrightarrow\mathrm{W}_{0}^{1,m}(\mathcal{O})^{d}$,
we can write (3.13) as

$\displaystyle\int_{\mathcal{O}}\pi_{\Phi}(t)\psi\,\mathrm{d}\bm{x}=\int_{0}^{t}\int_{\mathcal{O}}\mathscr{D}^{*}\Phi(\bm{u})\mathrm{d}\mathrm{W}(s)\psi\,\mathrm{d}\bm{x},\qquad\psi\in\mathrm{C}_{0}^{\infty}(\mathcal{O}),$

and hence

$\pi_{\Phi}(t)=\int_{0}^{t}\mathscr{D}^{*}\Phi(\bm{u}(s))\mathrm{d}\mathrm{W}(s),\qquad\mbox{$\mathbb{P}\times\lambda^{d+1}$-a.e.},$

where $\lambda^{d+1}$ is the Lebesgue measure on $\mathcal{O}\times(0,T)$.
Using the BDG inequality, together with the fact that $\mathscr{D}^{*}$ is a
continuous operator from $\mathrm{L}^{2}(\mathcal{O})^{d}$ to
$\mathrm{L}^{2}(\mathcal{O})$, we can show that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\pi_{\Phi}(t)\|_{2}^{2}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\bigg{\|}\int_{0}^{t}\mathscr{D}^{*}\Phi(\bm{u}(s))\mathrm{d}\mathrm{W}(s)\bigg{\|}_{2}^{2}\bigg{]}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\int_{0}^{T}\|\Phi(\bm{u}(t))\|_{\mathcal{L}_{2}}^{2}\mathrm{d}t\bigg{]}.$

Finally, we can see that

$\bar{\pi}_{0}(t):=\pi_{\Phi}(t)+\int_{0}^{t}\pi_{\mathbfcal{H}}(s)\mathrm{d}s$

solves (3.12), and
$\bar{\pi}_{0}(t)\in\Delta\mathrm{W}_{0}^{2,m}(\mathcal{O})$. This implies

$\displaystyle\pi_{0}(t)=\pi_{\Phi}(t)+\int_{0}^{t}\pi_{\mathbfcal{H}}(s)\,\mathrm{d}s,$

and hence we obtained the required result. ∎

For the following result, it is worth keeping in mind that we are assuming a
smooth boundary $\partial\mathcal{O}$, in fact of class $\mathrm{C}^{2}$. See
[22, Section 2.2] and [23, Section 2.2] in the case of non-smooth boundary
$\partial\mathcal{O}$.

######  Corollary 3.4.

Assume the conditions of Theorem 3.1 are satisfied. Then there exists
$\Phi_{\pi}\in\mathrm{L}^{2}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{\infty}(0,T;\mathcal{L}_{2}(\mathbf{U},\mathrm{L}^{2}(\mathcal{O}))))$
progressively measurable such that

$\displaystyle\int_{\mathcal{O}}\pi_{\Phi}(t)\operatorname{div}\bm{\phi}\,\mathrm{d}\bm{x}=\int_{0}^{t}\int_{\mathcal{O}}\Phi_{\pi}\mathrm{d}\mathrm{W}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x},\quad\forall\
\bm{\phi}\in\mathrm{C}_{0}^{\infty}(\mathcal{O}).$ (3.15)

and $\|\Phi_{\pi}\mathbf{e}_{j}\|_{2}\leq
C({\mathcal{O}})\|\Phi\mathbf{e}_{j}\|_{2}$, for all $j$, that is, we have
$\mathbb{P}\otimes\lambda$, a.e.

$\displaystyle\|\Phi_{\pi}\|_{\mathcal{L}_{2}}\leq
C({\mathcal{O}})\|\Phi\|_{\mathcal{L}_{2}}.$

Furthermore, if $\Phi$ satisfies (2.11), then there holds

$\displaystyle\|\Phi_{\pi}(\bm{u}_{1})-\Phi_{\pi}(\bm{u}_{2})\|_{\mathcal{L}_{2}}\leq
C(L,{\mathcal{O}})\|\bm{u}_{1}-\bm{u}_{2}\|_{{2}},$

for all $\bm{u}_{1},\bm{u}_{2}\in\mathrm{L}^{2}(\mathcal{O})^{d}$.

###### Proof.

From (3.14) of the proof of Theorem 3.1, we have for any
$\bm{\phi}\in{\mathrm{C}_{0}^{\infty}(\mathcal{O})^{d}}$

$\displaystyle\int_{\mathcal{O}}\pi_{\Phi}(t)\operatorname{div}\bm{\phi}\,\mathrm{d}\bm{x}$
$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u})\mathrm{d}\mathrm{W}(s)\cdot\nabla(\Delta^{-2}_{\mathcal{O}}\Delta\operatorname{div}\bm{\phi})\,\mathrm{d}\bm{x}$
$\displaystyle=\sum_{j}\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u})\mathbf{e}_{j}\mathrm{d}\beta_{j}(s)\cdot\nabla(\Delta^{-2}_{\mathcal{O}}\Delta\operatorname{div}\bm{\phi})\,\mathrm{d}\bm{x}$
$\displaystyle=\sum_{j}\int_{0}^{t}\int_{\mathcal{O}}\nabla\Delta\Delta^{-2}_{\mathcal{O}}\operatorname{div}\Phi(\bm{u})\mathbf{e}_{j}\mathrm{d}\beta_{j}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x}$
$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\nabla\Delta\Delta^{-2}_{\mathcal{O}}\operatorname{div}\Phi(\bm{u})\mathrm{d}\mathrm{W}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x},$

which proves (3.15) by setting
$\Phi_{\pi}=\nabla\Delta\Delta^{-2}_{\mathcal{O}}\operatorname{div}\Phi$. For
the second part, we use the local regularity theory for the bi-Laplace
equation in the following manner:

$\displaystyle\big{|}(\Phi_{\pi}\mathbf{e}_{k},\bm{\psi})\big{|}=$
$\displaystyle\big{|}(\nabla\Delta\Delta^{-2}_{\mathcal{O}}\operatorname{div}\Phi\mathbf{e}_{k},\bm{\psi})\big{|}\leq\|\nabla\Delta\Delta^{-2}_{\mathcal{O}}\operatorname{div}\Phi\mathbf{e}_{k}\|_{{2}}\|\bm{\psi}\|_{{2}}$
$\displaystyle\leq$ $\displaystyle
C({\mathcal{O}})\|\Delta\Delta^{-2}_{\mathcal{O}}\operatorname{div}\Phi\mathbf{e}_{k}\|_{-1,2}\|\bm{\psi}\|_{{2}}\leq
C({\mathcal{O}})\|\Delta^{-2}_{\mathcal{O}}\operatorname{div}\Phi\mathbf{e}_{k}\|_{3,2}\|\bm{\psi}\|_{{2}}$
$\displaystyle\leq$ $\displaystyle
C({\mathcal{O}})\|\operatorname{div}\Phi\mathbf{e}_{k}\|_{-1,2}\|\bm{\psi}\|_{{2}}\leq
C({\mathcal{O}})\|\Phi\mathbf{e}_{k}\|_{{2}}\|\bm{\psi}\|_{{2}}\leq
C({\mathcal{O}})\|\Phi\mathbf{e}_{k}\|_{{2}}\|\bm{\psi}\|_{{2}},$

for $\bm{\psi}\in\mathrm{L}^{2}(\mathcal{O})^{d}$. Similarly, we can conclude
the final part as follows:

$\displaystyle\|(\Phi_{\pi}(\bm{u}_{1})-\Phi_{\pi}(\bm{u}_{2}))\mathbf{e}_{k}\|_{{2}}$
$\displaystyle=\|\nabla\Delta\Delta^{-2}_{\mathcal{O}}\operatorname{div}(\Phi(\bm{u}_{1})-\Phi(\bm{u}_{2}))\mathbf{e}_{k}\|_{{2}}$
$\displaystyle\leq
C({\mathcal{O}})\|(\Phi(\bm{u}_{1})-\Phi(\bm{u}_{2}))\mathbf{e}_{k}\|_{{2}}=C({\mathcal{O}})\|\phi_{k}(\bm{u}_{1})-\phi_{k}(\bm{u}_{2})\|_{{2}}$
$\displaystyle\leq C({L,\mathcal{O}})\|\bm{u}_{1}-\bm{u}_{2}\|_{{2}},$

where we have used (2.11). ∎

######  Corollary 3.5.

Let the conditions of Theorem 3.1 be satisfied. Then, for all
$\gamma\in[1,\infty)$

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\pi_{h}(t)\|_{{m}}^{m}\bigg{]}^{\gamma}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\big{\\{}\|\bm{u}(t)\|^{2}_{2}+\kappa{\|\nabla\bm{u}(t)\|^{2}_{2}}\big{\\}}+\sup_{t\in[0,T]}\|\Phi(t)\|^{2}_{\mathcal{L}_{2}}\bigg{]}^{\gamma}$
$\displaystyle\quad+C\mathbb{E}\bigg{[}1+\|\bm{u}_{0}\|^{2}_{2}+\kappa{\|\nabla\bm{u}_{0}\|^{2}_{2}}+\int_{0}^{T}{\|\mathbfcal{H}(t)\|^{r}_{r}}\mathrm{d}t\bigg{]}^{\gamma},$

provided the right hand side of above inequality is finite.

###### Proof.

The proof of this corollary is straightforward from Theorem 3.1-(1). ∎

######  Corollary 3.6.

Let the conditions of Theorem 3.1 be satisfied, and assume that the following
decomposition holds:

$\displaystyle{\mathbfcal{H}}={\mathbfcal{H}}_{1}+{\mathbfcal{H}}_{2},$

where
$\mathbfcal{H}_{1}\in\mathrm{L}^{r_{1}}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{r_{1}}(0,T;{\mathrm{L}^{r_{1}}(\mathcal{O})^{d\times
d}}))$,
${\mathbfcal{H}}_{2}\in\mathrm{L}^{r_{2}}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{r_{2}}(0,T;{\mathrm{L}^{r_{2}}(\mathcal{O})^{d\times
d}}))$ and
$\operatorname{div}{\mathbfcal{H}}_{2}\in\mathrm{L}^{r_{2}}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{r_{2}}(0,T;{\mathrm{L}^{r_{2}}(\mathcal{O})^{d}}))$.
Then, we have

$\displaystyle\pi_{\mathbfcal{H}}=\pi_{1}+\pi_{2},$

and there holds for all $\gamma\in[1,\infty)$,

$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\|\pi_{1}(t)\|^{r_{1}}_{r_{1}}\mathrm{d}t\bigg{]}^{\gamma}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\int_{0}^{T}{\|\mathbfcal{H}_{1}(t)\|^{r_{1}}_{r_{1}}}\mathrm{d}t\bigg{]}^{\gamma},$
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\|\pi_{2}(t)\|^{r_{2}}_{r_{2}}\mathrm{d}t\bigg{]}^{\gamma}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\int_{0}^{T}{\|\mathbfcal{H}_{2}(t)\|^{r_{2}}_{r_{2}}}\mathrm{d}t\bigg{]}^{\gamma},$
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\|\nabla\pi_{2}(t)\|_{r_{2}}^{r_{2}}\mathrm{d}t\bigg{]}^{\gamma}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\int_{0}^{T}\big{\\{}\|{\mathbfcal{H}}_{2}(t)\|^{r_{2}}_{r_{2}}+\|\operatorname{div}{\mathbfcal{H}}_{2}(t)\|^{r_{2}}_{r_{2}}\big{\\}}\mathrm{d}t\bigg{]}^{\gamma}.$

###### Proof.

Here $\pi_{1}$ and $\pi_{2}$ are the unique solutions of the following
equations defined on $\mathbb{P}\otimes\lambda$, a.e.:

$\displaystyle\int_{\mathcal{O}}\pi_{1}(t)\Delta\psi\,\mathrm{d}\bm{x}$
$\displaystyle=-\int_{\mathcal{O}}{\mathbfcal{H}}_{1}(t):\nabla^{2}\psi\,\mathrm{d}\bm{x},\qquad{\pi_{1}\in\Delta\mathrm{W}_{0}^{2,r_{1}}(\mathcal{O})},$
$\displaystyle\int_{\mathcal{O}}\pi_{2}(t)\Delta\psi\,\mathrm{d}\bm{x}$
$\displaystyle=-\int_{\mathcal{O}}{\mathbfcal{H}}_{2}(t):\nabla^{2}\psi\,\mathrm{d}\bm{x},\qquad{\pi_{2}\in\Delta\mathrm{W}_{0}^{2,r_{2}}(\mathcal{O})}.$

By [63, Lemma 2.3], and once that
$\mathbfcal{H}_{1}\in\mathrm{L}^{r_{1}}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{r_{1}}(0,T;{\mathrm{L}^{r_{1}}(\mathcal{O})^{d\times
d}}))$,
${\mathbfcal{H}}_{2}\in\mathrm{L}^{r_{2}}(\Omega,\mathscr{F},\break\mathbb{P};\mathrm{L}^{r_{2}}(0,T;{\mathrm{L}^{r_{2}}(\mathcal{O})^{d\times
d}}))$ and
$\operatorname{div}{\mathbfcal{H}}_{2}\in\mathrm{L}^{r_{2}}(\Omega,\mathscr{F},\mathbb{P};\mathrm{L}^{r_{2}}(0,T;{\mathrm{L}^{r_{2}}(\mathcal{O})^{d}}))$,
the estimates are straightforward. ∎

##  4\. Solvability of The Approximate System

In this section, we establish the solvability of an auxiliary problem that
regularizes the problem (1.1)-(1.4) with the following stabilization term in
the momentum equation:

$\alpha\mathbf{a}(\bm{u}),\qquad\mathbf{a}(\bm{u}):=|\bm{u}|^{q-2}\bm{u},\qquad\alpha>0,\qquad
1<q<\infty.$ (4.1)

Given $\alpha>0$, we consider the problem

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
&\mathrm{d}(\mathrm{I}-\kappa\Delta)\bm{u}=\big{\\{}\operatorname{div}\mathbf{A}(\bm{u})-\operatorname{div}(\bm{u}\otimes\bm{u})+\nabla\pi-\alpha\mathbf{a}(\bm{u})+\bm{f}\big{\\}}\mathrm{d}t+\Phi(\bm{u})\mathrm{d}\mathrm{W}(t),\\\
&\bm{u}(0)=\bm{u}_{0},\end{aligned}}\right.$ (4.2)

depending on the initial $\Lambda_{0}$ and forcing $\Lambda_{\bm{f}},$ laws in
the conditions of (2.9) and (2.10), respectively, and for the operator
$\mathbf{A}(\bm{u})$ defined in (2.5). The exponent $q$ in (4.1) is chosen in
such a way that the convective term becomes a compact perturbation (see (4.23)
below). For that purpose, we choose

$q\geq\max\\{2p^{\prime},3\\},$ (4.3)

and thus a solution $\bm{u}$ is expected in the following space:

$\displaystyle\bm{\mathcal{V}}_{p,q}:=\mathrm{L}^{2}({\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{\infty}(0,T;\mathbf{V}))\cap\mathrm{L}^{p}({\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{p}(0,T;{\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}})\cap\mathrm{L}^{q}({\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{q}(0,T;{\mathrm{L}^{q}(\mathcal{O})^{d}})).$

###  4.1. Approximate solutions

We construct a solution to the problem (4.2) as a limit of suitable Galerkin
approximations. Let $s$ be the smallest positive integer such that
${\mathrm{W}^{s,2}_{0}(\mathcal{O})^{d}}\hookrightarrow{\mathrm{W}^{1,\infty}_{0}(\mathcal{O})}^{d}$,
for $s>1+\frac{d}{2}$, and let us consider the space $\mathbf{V}_{s}$
associated to ${\mathrm{W}^{s,2}_{0}(\mathcal{O})^{d}}$ defined in Section 2.
By means of separability, there exists a basis
$\big{\\{}\mathds{\bm{\psi}}_{k}\big{\\}}_{k\in\mathbb{N}}$ of
$\mathbf{V}_{s}$, formed by the eigenfunctions of a suitable spectral problem,
that is, orthogonal in ${\mathrm{L}^{2}(\mathcal{O})^{d}}$ and that can be
made orthonormal in ${\mathrm{W}^{s,2}_{0}(\mathcal{O})^{d}}$ (see [48,
Theorem A.4.11]). Given $n\in\mathbb{N}$, let us consider the $n-$dimensional
space $\mathbf{X}^{n}=\mathrm{span}\\{\bm{\psi}_{1},\dots,\bm{\psi}_{n}\\}$.
For each $n\in\mathbb{N}$, we search for approximate solutions of the form

$\displaystyle\bm{u}_{n}(x,t)=\sum_{k=1}^{n}c_{k}^{n}(t)\bm{\psi}_{k}(x),\quad\bm{\psi}_{k}\in\mathbf{X}^{n},$
(4.4)

where the coefficients $c_{1}^{n}(t),\dots,c_{n}^{n}(t)$ are solutions of the
following $n$ stochastic ordinary differential equations

$\displaystyle\mathrm{d}\Big{[}\big{(}\bm{u}_{n}(t),\bm{\psi}_{k}\big{)}+\kappa\big{(}\nabla\bm{u}_{n}(t),\nabla\bm{\psi}_{k}\big{)}\Big{]}$
$\displaystyle=\Big{[}\big{(}\bm{u}_{n}(t)\otimes\bm{u}_{n}(t):\nabla\bm{\psi}_{k}\big{)}-\nu\big{<}|\mathbf{D}(\bm{u}_{n}(t))|^{p-2}\mathbf{D}(\bm{u}_{n}(t)):\mathbf{D}(\bm{\psi}_{k})\big{>}+\big{(}\bm{f}(t),\bm{\psi}_{k}\big{)}\Big{]}\mathrm{d}t$
$\displaystyle\qquad-\alpha\big{(}\mathbf{a}(\bm{u}_{n}(t)),\bm{\psi}_{l}\big{)}+{\Phi(\bm{u}_{n}(t))}\mathrm{d}\mathrm{W}_{n}(t),\bm{\psi}_{k}\big{)},\qquad
k=1,\dots,n,$ (4.5)

supplemented with the initial conditions

$\displaystyle\bm{u}_{n}(0)=\bm{u}_{0}^{n},\quad\mbox{in}\ \mathcal{O},$ (4.6)

where $\bm{u}_{0}^{n}=P^{n}(\bm{u}_{0})$, with $P^{n}$ denoting the orthogonal
projection $P^{n}:\mathbf{V}\longrightarrow\mathbf{X}^{n}$ so that

$\displaystyle\bm{u}_{n}(0,x)=\sum_{k=1}^{n}c_{k}^{n}(0)\bm{\psi}_{k}(x),\quad
c_{k}^{n}(0)=c_{k,0}^{n}:=(\bm{u}_{0},\bm{\psi}_{k}),\quad k=1,\dots,n.$

In (4.1), we assume that the approximate cylindrical Wiener process
$\big{\\{}\mathrm{W}_{n}(\cdot)\big{\\}}$ has the form

$\mathrm{W}_{n}(t)=\sum_{k=1}^{n}\mathbf{e}_{k}\beta_{k}(t),$

and note that (4.1) is to be understood $\mathbb{P}-$a.s., and for all
$t\in(0,T]$. Observe that the stochastic system (4.1)-(4.6) can be written in
the matrix form as follows,

$\displaystyle\mathbf{C}\mathrm{d}\mathbf{c}(t)=\mathbf{b}(t)\mathrm{d}t+\mathbf{G}(t)\mathrm{d}\bm{\beta}(t),\quad\mathbf{c}(0)=\mathbf{c}_{n,0},$
(4.7)

where
$\mathbf{C}=\mathopen{}\mathclose{{}\left\\{a^{n}_{lm}}\right\\}_{l,m=1}^{n}$,
$\mathbf{b}(t)=\\{b_{l}^{n}(t)\\}_{l=1}^{n}$,
$\mathbf{c}(t)=\\{c_{m}^{n}(t)\\}_{m=1}^{n}$,
$\mathbf{G}(t)=\mathopen{}\mathclose{{}\left\\{g^{n}_{lm}(t)}\right\\}_{l,m=1}^{n}$
and $\bm{\beta}(t)=\\{\beta_{m}^{n}(t)\\}_{m=1}^{n}$, with

$\displaystyle a_{lm}^{n}$
$\displaystyle:=\mathopen{}\mathclose{{}\left(\bm{\psi}_{l},\bm{\psi}_{m}}\right)+\kappa\mathopen{}\mathclose{{}\left(\nabla\bm{\psi}_{l},\nabla\bm{\psi}_{m}}\right),$
$\displaystyle b_{l}^{n}(t)$
$\displaystyle:=\big{(}\bm{f}(t),\bm{\psi}_{l}\big{)}+\big{(}\bm{u}^{n}_{n}(t)\otimes\bm{u}^{n}_{n}(t):\nabla\bm{\psi}_{l}\big{)}-\nu\big{<}|\mathbf{D}(\bm{u}_{n}(t))|^{p-2}\mathbf{D}(\bm{u}_{n}(t)):\mathbf{D}(\bm{\psi}_{l})\big{>}$
$\displaystyle\quad-\alpha\big{(}\mathbf{a}(\bm{u}_{n}(t)),\bm{\psi}_{l}\big{)},$
$\displaystyle\mathbf{c}(t)$
$\displaystyle:=(c_{1}^{n}(t),\dots,c_{n}^{n}(t))\qquad\mbox{and}\qquad\mathbf{c}_{0}={(c_{1,0}^{n},\dots,c_{n,0}^{n})},$
$\displaystyle g^{n}_{lm}(t)$
$\displaystyle:=\big{(}\Phi(\bm{u}_{n}(t))\mathbf{e}_{l},\bm{\psi}_{m}\big{)},$
$\displaystyle\bm{\beta}(t)$
$\displaystyle:=(\beta_{1}^{n}(t),\dots,\beta_{n}^{n}(t)).$

Taking into account that the family
$\big{\\{}\bm{\psi}_{k}\big{\\}}_{k\in\mathbb{N}}$ is linearly independent in
$\mathbf{V}$, $\big{(}\mathbf{C}\bm{\xi},\bm{\xi}\big{)}>0$ for all
$\bm{\xi}\in\mathbb{R}^{n}\setminus\\{0\\}$, we can write (4.7) in the form

$\displaystyle\mathrm{d}\mathbf{c}(t)=\overline{\mathbf{b}(t)}\mathrm{d}t+\overline{\mathbf{G}(t)}\mathrm{d}\bm{\beta}(t),\quad\mathbf{c}(0)=\mathbf{c}_{n,0},$
(4.8)

where $\overline{\mathbf{b}(t)}=\mathbf{C}^{-1}\mathbf{b}(t)$ and
$\overline{\mathbf{G}(t)}=\mathbf{C}^{-1}\mathbf{G}(t)$.

If both the coefficients $\mathbf{b}(\cdot)$ and $\mathbf{G}(\cdot)$ are
globally Lipschitz-continuous, one can use the classical existence results for
stochastic differential equations. Our case does not fall in that category, so
we have to use the monotonicity method and verify the conditions of the
vectorial version of Lemma 2.1 as follows:

$\displaystyle\langle\mathbf{b}(\cdot,\bm{u}_{n})-\mathbf{b}(\cdot,\bm{v}_{n}),\bm{u}_{n}-\bm{v}_{n}\rangle$
$\displaystyle=\big{(}\bm{u}_{n}\otimes\bm{u}_{n}-\bm{v}_{n}\otimes\bm{v}_{n}:\nabla(\bm{u}_{n}-\bm{v}_{n})\big{)}$
$\displaystyle\quad-\nu\langle|\mathbf{D}(\bm{u}_{n})|^{p-2}\mathbf{D}(\bm{u}_{n})-|\mathbf{D}(\bm{v}_{n})|^{p-2}\mathbf{D}(\bm{v}_{n}):\mathbf{D}(\bm{u}_{n}-\bm{v}_{n})\rangle$
$\displaystyle\quad-\alpha\big{(}|\bm{u}_{n}|^{q-2}\bm{u}_{n}-|\bm{v}_{n}|^{q-2}\bm{v}_{n},\bm{u}_{n}-\bm{v}_{n}\big{)}$
$\displaystyle\leq\big{(}\bm{u}_{n}\otimes\bm{u}_{n}-\bm{v}_{n}\otimes\bm{v}_{n}:\nabla(\bm{u}_{n}-\bm{v}_{n})\big{)},$

where we have used (2.4). If $\|\bm{u}_{n}\|_{2}\leq R$ and
$\|\bm{v}_{n}\|_{2}\leq R$, there exists a constant $C(R,n)>0$ such that

$\displaystyle\langle\mathbf{b}(t,\bm{u}_{n})-\mathbf{b}(t,\bm{v}_{n}),\bm{u}_{n}-\bm{v}_{n}\rangle\leq
C(R,n)\|\bm{u}_{n}-\bm{v}_{n}\|^{2}_{2}.$

This, together with the Lipschitz-continuity of $\mathbf{G}$ (cf. (2.11)),
gives us the weak monotonicity property in the sense of Lemma 2.1-(2). Using
the fact $\langle\bm{u}_{n}\otimes\bm{u}_{n}:\nabla\bm{u}_{n}\rangle=0$, we
find

$\displaystyle\langle\mathbf{b}(\cdot,\bm{u}_{n}),\bm{u}_{n}\rangle$
$\displaystyle=-\nu\langle|\mathbf{D}(\bm{u}_{n})|^{p-2}\mathbf{D}(\bm{u}_{n}):\mathbf{D}(\bm{u}_{n})\rangle-\alpha\big{(}|\bm{u}_{n}|^{q-2}\bm{u}_{n},\bm{u}_{n}\big{)}+\big{(}\bm{f},\bm{u}_{n}\big{)}$
$\displaystyle\leq C(1+\|\bm{f}\|_{2}\|\bm{u}_{n}\|_{2})\leq
C(1+\|\bm{f}\|_{2})(1+\|\bm{u}_{n}\|^{2}_{2}).$

Using the linear growth of $\mathbf{G}$ cf. (2.11), we find

$\displaystyle\langle\mathbf{b}(t,\bm{u}_{n}),\bm{u}_{n}\rangle+\|\mathbf{G}\|_{\mathcal{L}_{2}}^{2}\leq
C(1+\|\bm{f}\|_{2})(1+\|\bm{u}_{n}\|^{2}_{2}).$

Since the term $\int_{0}^{T}(1+\|\bm{f}\|_{2})\mathrm{d}t$ is finite,
$\mathbb{P}-$a.s., this yields the weak coercivity property in the sense of
Lemma 2.1-(3). By Lemma 2.1, we obtain the existence of a unique strong
solution
$\mathbf{C}\mathbf{c}\in\mathrm{L}^{2}(\Omega,\mathscr{F},\mathbb{P};\mathrm{C}([0,T^{*}];\mathbb{R}^{n}))$
to the stochastic system (4.7), which implies

$\displaystyle\sup_{t\in[0,T^{*}]}\|\mathbf{C}\mathbf{c}(t)\|_{\mathbb{R}^{n}}<+\infty.$

Our aim is to show that
$\sup\limits_{t\in[0,T^{*}]}\|\mathbf{c}(t)\|_{\mathbb{R}^{n}}<+\infty$, which
we obtain as follows:

$\displaystyle\sup_{t\in[0,T^{*}]}\|\mathbf{c}(t)\|_{\mathbb{R}^{n}}$
$\displaystyle=\sup_{t\in[0,T^{*}]}\|\mathbf{C}^{-1}\mathbf{C}\mathbf{c}(t)\|_{\mathbb{R}^{n}}\leq\|\mathbf{C}^{-1}\|_{\mathcal{L}(\mathbb{R}^{n};\mathbb{R}^{n})}\sup_{t\in[0,T^{*}]}\|\mathbf{C}\mathbf{c}(t)\|_{\mathbb{R}^{n}}<+\infty,$

since $\mathbf{C}^{-1}$ is a bounded operator. Thus, we obtain the existence
of a unique strong solution to the stochastic system (4.6).

###  4.2. Uniform estimates

In this subsection, we establish the uniform energy estimate for the solution
of the finite-dimensional approximate system corresponding to the system
(4.2).

######  Theorem 4.1.

Let $p\in(1,\infty)$ and assume that (2.11) and (4.3) are verified. Assume, in
addition, that (2.9) and (2.10) hold with $\gamma=2$. Then there exists a
positive constant $C,$ neither depending on $n$ nor on $\alpha$ such that

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\sup_{t\in(0,T)}\mathopen{}\mathclose{{}\left\\{\|\bm{u}_{n}(t)\|^{2}_{2}+2\kappa{\|\nabla\bm{u}_{n}(t)\|_{2}^{2}}}\right\\}+4C({p,\mathcal{O}})\nu\int_{0}^{T}{\|\nabla\bm{u}_{n}(t)\|_{p}^{p}}\mathrm{d}t+2\alpha\int_{0}^{T}\|\bm{u}_{n}(t)\|^{q}_{q}\mathrm{d}t}\right]$
$\displaystyle\leq
C(\kappa)\mathopen{}\mathclose{{}\left\\{\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\int_{\mathbf{V}}\|\bm{z}\|_{\mathbf{V}}^{2}\mathrm{d}\Lambda_{0}(\bm{z})+\int_{\mathbf{L}^{2}(\mathcal{O}_{T})}\|\mathbf{g}\|_{\mathbf{L}^{2}(\mathcal{O}_{T})}^{2}\mathrm{d}\Lambda_{\bm{f}}(\mathbf{g})+C(K)T}\right\\}e^{\frac{C(K)T}{\eta_{1}}},$
(4.9)

where $C$ is independent of $\alpha$ and $\eta_{1}$ is the first eigenvalue of
the Dirichlet Laplacian.

###### Proof.

Define a sequence of stopping times $\\{\tau_{N}^{n}\\}_{N\in\mathbb{N}}$ as
follows:

$\displaystyle\tau_{N}^{n}:=\inf_{t\in[0,T]}\\{t:\|\nabla\bm{u}_{n}(t)\|_{2}\geq
N\\}.$ (4.10)

Applying finite-dimensional Itô’s formula to the process
$\|(\mathrm{I}-\kappa\Delta)^{\frac{1}{2}}\bm{u}_{n}(\cdot)\|^{2}_{2}$, we
obtain, $\mathbb{P}-$a.s.,

$\displaystyle\|\bm{u}_{n}(t\wedge\tau_{N}^{n})\|^{2}_{2}+\kappa\|\nabla\bm{u}_{n}(t\wedge\tau_{N}^{n})\|_{2}^{2}$
$\displaystyle=\|\bm{u}_{n}(0)\|^{2}_{2}+\kappa\|\nabla\bm{u}_{n}(0)\|_{2}^{2}-2\nu\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\mathbf{A}(\bm{u}_{n}):\mathbf{D}(\bm{u}_{n})\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad-2\alpha\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\mathbf{a}(\bm{u}_{n})\cdot\bm{u}_{n}\,\mathrm{d}\bm{x}\mathrm{d}s+2\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}(s)\cdot\bm{u}_{n}\,\mathrm{d}\bm{x}$
$\displaystyle\quad\quad+\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\mathrm{d}\bigg{\langle}\int_{0}^{\cdot}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}\bigg{\rangle}_{s}\,\mathrm{d}\bm{x}$
$\displaystyle\leq\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\|\nabla\bm{u}_{n}(0)\|_{2}^{2}-2\nu\int_{0}^{t}\|\mathbf{D}(\bm{u}_{n}(s))\|_{p}^{p}\mathrm{d}s-2\alpha\int_{0}^{t\wedge\tau_{N}^{n}}\|\bm{u}_{n}(s)\|^{q}_{q}\mathrm{d}s$
$\displaystyle\quad+2\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\bm{f}\cdot\bm{u}_{n}\,\mathrm{d}\bm{x}\mathrm{d}s+2\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}(s)\cdot\bm{u}_{n}\mathrm{d}\bm{x}$
$\displaystyle\quad+\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\mathrm{d}\bigg{\langle}\int_{0}^{\cdot}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}\bigg{\rangle}_{s}\,\mathrm{d}\bm{x},$
(4.11)

where we have used Poincaré’s inequality. Taking expectation on both sides and
then using Korn’s inequality (cf. [38] or [3, Lemma 2.1]), we find

$\displaystyle\mathbb{E}\bigg{[}\|\bm{u}_{n}(t\wedge\tau_{N}^{n})\|^{2}_{2}+\kappa\|\nabla\bm{u}_{n}(t\wedge\tau_{N}^{n})\|_{2}^{2}+2C({p,\mathcal{O}})\nu\int_{0}^{t\wedge\tau_{N}^{n}}\|\nabla\bm{u}_{n}(s)\|_{p}^{p}\mathrm{d}s\bigg{]}$
$\displaystyle\qquad+2\alpha\mathbb{E}\bigg{[}\int_{0}^{t\wedge\tau_{N}^{n}}\|\bm{u}_{n}(s)\|^{q}_{q}\mathrm{d}s\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\|\nabla\bm{u}_{n}(0)\|_{2}^{2}+2\underbrace{\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\bm{f}\cdot\bm{u}_{n}\,\mathrm{d}\bm{x}\mathrm{d}s}_{:=I_{1}(t\wedge\tau_{N}^{n})}+\underbrace{\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\mathrm{d}\bigg{\langle}\int_{0}^{\cdot}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}\bigg{\rangle}_{s}\,\mathrm{d}\bm{x}}_{:=I_{2}(t\wedge\tau_{N}^{n})}$
$\displaystyle\quad+2\underbrace{\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}(s)\cdot\bm{u}_{n}\mathrm{d}\bm{x}}_{:=I_{3}(t\wedge\tau_{N}^{n})}\bigg{]}$
$\displaystyle\leq\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\mathbb{E}\big{[}\|\nabla\bm{u}_{n}(0)\|_{2}^{2}\big{]}+2\mathbb{E}\big{[}I_{1}(t\wedge\tau_{N}^{n})\big{]}+\mathbb{E}\big{[}I_{2}(t\wedge\tau_{N}^{n})\big{]}+2\mathbb{E}\big{[}I_{3}(t\wedge\tau_{N}^{n})\big{]}.$

We estimate $\mathbb{E}\big{[}|I_{1}(t\wedge\tau_{N}^{n})|\big{]}$ using
Young’s and Poincaré’s inequalities as follows:

$\displaystyle\mathbb{E}\big{[}|I_{1}(t\wedge\tau_{N}^{n})|\big{]}$
$\displaystyle\leq\varepsilon\mathbb{E}\bigg{[}\int_{0}^{t\wedge\tau_{N}^{n}}\|\bm{u}_{n}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}+C(\varepsilon)\mathbb{E}\bigg{[}\int_{0}^{t\wedge\tau_{N}^{n}}\|\bm{f}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}$
$\displaystyle\leq\frac{\varepsilon}{\eta_{1}}\mathbb{E}\bigg{[}\int_{0}^{t\wedge\tau_{N}^{n}}{\|\nabla\bm{u}_{n}(s)\|_{2}^{2}}\mathrm{d}s\bigg{]}+C(\varepsilon)\mathbb{E}\bigg{[}\int_{0}^{t\wedge\tau_{N}^{n}}\|\bm{f}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}.$

We know that $\mathbb{E}\big{[}I_{3}(t\wedge\tau_{N}^{n})\big{]}=0$, since the
term $I_{3}(t\wedge\tau_{N}^{n})$ is a martingale with zero expectation. Now,
we consider the term $\mathbb{E}\big{[}I_{2}(t\wedge\tau_{N}^{n})\big{]}$ and
estimate it with the help of Itô’s isometry, (2.11) and Poincaré’s inequality
as,

$\displaystyle\mathbb{E}\big{[}I_{2}(t\wedge\tau_{N}^{n})\big{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sum_{k=1}^{n}\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}\big{|}\Phi(\bm{u}_{n})\mathbf{e}_{k}\big{|}^{2}\mathrm{d}\bm{x}\mathrm{d}s\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\sum_{k\in\mathbb{N}}\int_{0}^{t\wedge\tau_{N}^{n}}\int_{\mathcal{O}}|\phi_{k}(\bm{u}_{n})|^{2}\,\mathrm{d}\bm{x}\mathrm{d}s\bigg{]}\leq
C(K)\mathbb{E}\bigg{[}T+{\frac{1}{\eta_{1}}\int_{0}^{t\wedge\tau_{N}^{n}}\|\nabla\bm{u}_{n}(s)\|^{2}_{2}}\mathrm{d}s\bigg{]}.$

Combining the above estimates and using it in (4.2), we find

$\displaystyle\mathbb{E}\bigg{[}\|\bm{u}_{n}(t\wedge\tau_{N}^{n})\|^{2}_{2}+\kappa{\|\nabla\bm{u}_{n}(t\wedge\tau_{N}^{n})\|_{2}^{2}}+2C({p,\mathcal{O}})\nu\int_{0}^{t\wedge\tau_{N}^{n}}{\|\nabla\bm{u}_{n}(s)\|_{p}^{p}}\mathrm{d}s+2\alpha\int_{0}^{t\wedge\tau_{N}^{n}}\|\bm{u}_{n}(s)\|^{q}_{q}\mathrm{d}s\bigg{]}$
$\displaystyle\leq\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\mathbb{E}\big{[}{\|\nabla\bm{u}_{n}(0)\|_{2}^{2}}\big{]}+C(K,T)+\frac{1}{\eta_{1}}(C(K)+\varepsilon)\mathbb{E}\bigg{[}\int_{0}^{t\wedge\tau_{N}^{n}}{\|\nabla\bm{u}_{n}(s)\|_{2}^{2}}\mathrm{d}s\bigg{]}$
$\displaystyle\quad+C(\varepsilon)\mathbb{E}\bigg{[}\int_{0}^{t\wedge\tau_{N}^{n}}\|\bm{f}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}.$

An application of Gronwall’s inequality with a proper choice of $\varepsilon$
in the above inequality yields

$\displaystyle\mathbb{E}\bigg{[}{\|\nabla\bm{u}_{n}(t\wedge\tau_{N}^{n})\|_{2}^{2}}\bigg{]}$
$\displaystyle\leq
C(\kappa)\bigg{\\{}\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\mathbb{E}\big{[}{\|\nabla\bm{u}_{n}(0)\|_{2}^{2}}\big{]}+C\mathbb{E}\bigg{[}\int_{0}^{T}\|\bm{f}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}+C(K,T)\bigg{\\}}e^{\frac{C(K)T}{\eta_{1}}}.$
(4.12)

Taking limit $N\to\infty$ in (4.2) and using the monotone convergence theorem,
we get

$\displaystyle\mathbb{E}\bigg{[}{\|\nabla\bm{u}_{n}(t)\|_{2}^{2}}\bigg{]}$
$\displaystyle\leq
C(\kappa)\bigg{\\{}\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\mathbb{E}\big{[}{\|\nabla\bm{u}_{0}\|_{2}^{2}}\big{]}+C\mathbb{E}\bigg{[}\int_{0}^{T}\|\bm{f}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}+C(K,T)\bigg{\\}}e^{\frac{C(K)T}{\eta_{1}}},$
(4.13)

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

Taking supremum from 0 to $T\wedge\tau_{N}^{n}$ and then expectation in (4.2),
we find

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}\big{\\{}\|\bm{u}_{n}(t)\|^{2}_{2}+\kappa{\|\nabla\bm{u}_{n}(t)\|_{2}^{2}}\big{\\}}+2C({p,\mathcal{O}})\nu\int_{0}^{T\wedge\tau_{N}^{n}}{\|\nabla\bm{u}_{n}(s)\|_{p}^{p}}\mathrm{d}s\bigg{]}$
$\displaystyle\qquad+2\alpha\mathbb{E}\bigg{[}\int_{0}^{T\wedge\tau_{N}^{n}}\|\bm{u}_{n}(s)\|^{q}_{q}\mathrm{d}s\bigg{]}$
$\displaystyle\leq\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\mathbb{E}\big{[}{\|\nabla\bm{u}_{0}\|_{2}^{2}}\big{]}+C(\varepsilon)\mathbb{E}\bigg{[}\int_{0}^{T}\|\bm{f}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}+C(K)T$
$\displaystyle\quad+\frac{1}{\eta_{1}}(C(K)+\varepsilon)\mathbb{E}\bigg{[}\int_{0}^{T\wedge\tau_{N}^{n}}{\|\nabla\bm{u}_{n}(s)\|_{2}^{2}}\mathrm{d}s\bigg{]}+2\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}I_{3}(t)\bigg{]}.$
(4.14)

We estimate the final term from the right hand side of (4.2) using the BDG
(see [19, Theorem 1.1]), Hölder’s and Young’s inequalities as follows:

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}|I_{3}(t)|\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}\bigg{|}\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}(s)\cdot\bm{u}_{n}(s)\,\mathrm{d}\bm{x}\bigg{|}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}\bigg{|}\int_{0}^{t}\sum_{k=1}^{n}\int_{\mathcal{O}}\Phi(\bm{u}_{n})\mathbf{e}_{k}\mathrm{d}\beta_{k}(s)\cdot\bm{u}_{n}\,\mathrm{d}\bm{x}\bigg{|}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}\bigg{|}\int_{0}^{t}\sum_{k=1}^{n}\int_{\mathcal{O}}\phi_{k}(\bm{u}_{n})\mathrm{d}\beta_{k}(s)\cdot\bm{u}_{n}\,\mathrm{d}\bm{x}\bigg{|}\bigg{]}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\int_{0}^{T\wedge\tau_{N}^{n}}\sum_{k=1}^{n}\bigg{(}\int_{\mathcal{O}}\phi_{k}(\bm{u}_{n})\cdot\bm{u}_{n}\,\mathrm{d}\bm{x}\bigg{)}^{2}\mathrm{d}s\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\int_{0}^{T\wedge\tau_{N}^{n}}\sum_{k=1}^{n}\bigg{(}\int_{\mathcal{O}}|\bm{u}_{n}|^{2}\mathrm{d}\bm{x}\int_{\mathcal{O}}|\phi_{k}(\bm{u}_{n})|^{2}\mathrm{d}\bm{x}\bigg{)}\mathrm{d}s\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq
C(K)\mathbb{E}\bigg{[}1+\int_{0}^{T\wedge\tau_{N}^{n}}\bigg{(}\int_{\mathcal{O}}|\bm{u}_{n}|^{2}\mathrm{d}\bm{x}\bigg{)}^{2}\mathrm{d}s\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq\varepsilon\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}\|\bm{u}_{n}(s)\|^{2}_{2}\bigg{]}+C(\varepsilon,K)\mathbb{E}\bigg{[}T+\int_{0}^{T\wedge\tau_{N}^{n}}\|\bm{u}_{n}(s)\|^{2}_{2}\mathrm{d}s\bigg{]}.$
(4.15)

Substituting the above estimate in (4.2) with a proper choice of
$\varepsilon$, and an application of Gronwall’s inequality yields

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T\wedge\tau_{N}^{n}]}\big{\\{}\|\bm{u}_{n}(t)\|^{2}_{2}+2\kappa{\|\nabla\bm{u}_{n}(t)\|_{2}^{2}}\big{\\}}+4C({p,\mathcal{O}})\nu\int_{0}^{T\wedge\tau_{N}^{n}}{\|\nabla\bm{u}_{n}(t)\|_{p}^{p}}\mathrm{d}t\bigg{]}$
$\displaystyle\qquad+2\alpha\mathbb{E}\bigg{[}\int_{0}^{T\wedge\tau_{N}^{n}}\|\bm{u}_{n}(t)\|^{q}_{q}\mathrm{d}t\bigg{]}$
$\displaystyle\leq
C(\kappa)\bigg{\\{}\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\int_{\mathbf{V}}\|\bm{z}\|_{\mathbf{V}}^{2}\mathrm{d}\Lambda_{0}(\bm{z})+2C\int_{\mathbf{L}^{2}(\mathcal{O}_{T})}\|\mathbf{g}\|_{\mathbf{L}^{2}(\mathcal{O}_{T})}^{2}\mathrm{d}\Lambda_{\bm{f}}(\mathbf{g})+C(K,T)\bigg{\\}}e^{\frac{C(K)T}{\eta_{1}}}.$

Finally, passing $N\to\infty$ in the above inequality, we arrive at

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\mathopen{}\mathclose{{}\left\\{\|\bm{u}_{n}(t)\|^{2}_{2}+2\kappa{\|\nabla\bm{u}_{n}(t)\|_{2}^{2}}}\right\\}+4C({p,\mathcal{O}})\nu\int_{0}^{T}{\|\nabla\bm{u}_{n}(t)\|_{p}^{p}}\mathrm{d}t+2\alpha\int_{0}^{T}\|\bm{u}_{n}(t)\|^{q}_{q}\mathrm{d}t\bigg{]}$
$\displaystyle\leq
C(\kappa)\mathopen{}\mathclose{{}\left\\{\bigg{(}\frac{1}{\eta_{1}}+\kappa\bigg{)}\int_{\mathbf{V}}\|\bm{z}\|_{\mathbf{V}}^{2}\mathrm{d}\Lambda_{0}(\bm{z})+\int_{\mathbf{L}^{2}(\mathcal{O}_{T})}\|\mathbf{g}\|_{\mathbf{L}^{2}(\mathcal{O}_{T})}^{2}\mathrm{d}\Lambda_{\bm{f}}(\mathbf{g})+C(K,T)}\right\\}e^{\frac{C(K)T}{\eta_{1}}},$

by assuming the right-hand side is finite, which is true due to (2.9) and
(2.10).∎

######  Theorem 4.2.

Let $p\in(1,\infty)$ and assume the conditions (2.11) and (4.3) are satisfied.
Assume, in addition, that (2.9) and (2.10) hold with $\gamma=2$. Then there
exists a _martingale solution_ to the approximate system (4.2),

$\displaystyle((\overline{\Omega},\overline{\mathscr{F}},\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]},\overline{\mathbb{P}}),\overline{\bm{u}},\overline{\bm{u}}_{0},\overline{\bm{f}},\overline{\mathrm{W}}),$

in the following sense:

  1. (1)

$(\overline{\Omega},\overline{\mathscr{F}},\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]},\overline{\mathbb{P}})$
is a stochastic basis, with a complete right-continuous filtration
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}$;

  2. (2)

$\overline{\mathrm{W}}$ is a cylindrical
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}-$adapted Wiener process;

  3. (3)

$\overline{\bm{u}}$ is a progressively
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}-$measurable stochastic process
with $\overline{\mathbb{P}}-$a.s. paths
$t\mapsto\overline{\bm{u}}(t,\omega)\in\mathrm{L}^{\infty}(0,T;\mathbf{V})\cap\mathrm{L}^{p}(0,T;{\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}})\cap\mathrm{L}^{q}(0,T;{\mathrm{L}^{q}(\mathcal{O})^{d}})$,
with a continuous modification having paths in $\mathrm{C}([0,T];\mathbf{V})$;

  4. (4)

$\overline{\bm{u}}_{0}$ is progressively
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}-$measurable on the probability
space $(\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}})$, with
$\overline{\mathbb{P}}-$a.s. paths
$\overline{\bm{u}}_{0}(\omega)\in\mathbf{V}$ and
$\Lambda_{0}=\overline{\mathbb{P}}\circ\overline{\bm{u}}_{0}^{-1}$ in the
sense of (2.6);

  5. (5)

$\overline{\bm{f}}$ is a
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}-$adapted stochastic process,
with $\overline{\mathbb{P}}-$a.s. paths
$\overline{\bm{f}}(t,\omega)\in\mathbf{L}^{2}(\mathcal{O}_{T})$ and
$\Lambda_{\overline{\bm{f}}}=\overline{\mathbb{P}}\circ\overline{\bm{f}}^{-1}$
in the sense of (2.7);

  6. (6)

for every $\bm{\varphi}\in{\mathrm{C}_{0}^{\infty}(\mathcal{O})^{d}}$, with
$\operatorname{div}\bm{\varphi}=0$ in $\mathcal{O}_{T}$, and for all
$t\in[0,T],$ the following identity holds $\overline{\mathbb{P}}-$a.s.:

$\displaystyle\int_{\mathcal{O}}\overline{\bm{u}}(t)\cdot\bm{\varphi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\overline{\bm{u}}(t):\nabla\bm{\varphi}\,\mathrm{d}\bm{x}-\int_{0}^{t}\int_{\mathcal{O}}\overline{\bm{u}}\otimes\overline{\bm{u}}:\mathds{\nabla}\bm{\varphi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad+\alpha\int_{0}^{t}\int_{\mathcal{O}}|\overline{\bm{u}}|^{q-2}\overline{\bm{u}}\cdot{\bm{\varphi}}\,\mathrm{d}\bm{x}\mathrm{d}s+\nu\int_{0}^{t}\int_{\mathcal{O}}|\mathbf{D}(\overline{\bm{u}})|^{p-2}\mathbf{D}(\overline{\bm{u}}):\mathbf{D}\bm{\varphi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\int_{\mathcal{O}}\overline{\bm{u}}_{0}\cdot\bm{\varphi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\overline{\bm{u}}_{0}:\nabla\bm{\varphi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}\overline{\bm{f}}\cdot\bm{\varphi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}\Phi(\overline{\bm{u}})\mathrm{d}\overline{\mathrm{W}}(s)\cdot\bm{\varphi}\,\mathrm{d}\bm{x}.$
(4.16)

In addition, there exists a positive constant $C$, not depending on $\alpha$,
such that

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\sup_{t\in[0,T]}\mathopen{}\mathclose{{}\left\\{\|\overline{\bm{u}}(t)\|^{2}_{2}+2\kappa{\|\nabla\overline{\bm{u}}(t)\|_{2}^{2}}}\right\\}+4\nu\int_{0}^{T}{\|\nabla\overline{\bm{u}}(t)\|_{p}^{p}}\mathrm{d}t+2\alpha\int_{0}^{T}\|\overline{\bm{u}}(t)\|^{q}_{q}\mathrm{d}t}\right]$
$\displaystyle\leq
C\mathopen{}\mathclose{{}\left\\{\bigg{(}\frac{1}{\lambda_{1}}+\kappa\bigg{)}\int_{\mathbf{V}}\|\overline{\bm{z}}\|_{\mathbf{V}}^{2}\mathrm{d}\Lambda_{0}(\overline{\bm{z}})+\int_{\mathbf{L}^{2}(\mathcal{O}_{T})}\|\overline{\mathbf{g}}\|_{\mathbf{L}^{2}(\mathcal{O}_{T})}^{2}\mathrm{d}\Lambda_{\overline{\bm{f}}}(\overline{\mathbf{g}})+C(K,T)}\right\\}.$
(4.17)

###### Proof.

For the sake of better comprehension, the proof of Theorem 4.2 shall be split
through the following sections:

###  4.3. Weak convergence

In view of (4.1), the Banach-Alaoglu theorem gives the existence of functions
$\bm{u}$, $\bm{w}$, $\mathbf{S}$ and $\Psi$ such that, for some subsequences
still labeled by the same subscript,

$\displaystyle\bm{u}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\bm{u},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{\infty}(0,T;\mathbf{V})\big{)},$
(4.18) $\displaystyle\bm{u}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\bm{u},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{p}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{p}(0,T;{\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}}\big{)},$
(4.19) $\displaystyle\bm{u}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\bm{u},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{q}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{q}(0,T;{\mathrm{L}^{q}(\mathcal{O})^{d}})\big{)},$
(4.20) $\displaystyle\mathbf{a}(\bm{u}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\mathbf{a},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{q^{\prime}}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{q^{\prime}}(0,T;{\mathrm{L}^{q^{\prime}}(\mathcal{O})^{d}})\big{)},$
(4.21) $\displaystyle\bm{u}_{n}\otimes\bm{u}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\bm{w},\quad$
$\displaystyle\text{in}\quad{\mathrm{L}^{\frac{q}{2}}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{\frac{q}{2}}(0,T;\mathrm{L}^{\frac{q}{2}}(\mathcal{O})^{d\times
d}))},$ (4.22) $\displaystyle\mathbf{A}(\bm{u}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\mathbf{S},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{p^{\prime}}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{p^{\prime}}(0,T;{\mathrm{L}^{p^{\prime}}(\mathcal{O})^{d\times
d}})\big{)},$ (4.23) $\displaystyle\mathbf{A}(\bm{u}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\mathbf{S},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{p^{\prime}}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{p^{\prime}}(0,T;{\mathrm{W}^{-1,p^{\prime}}(\mathcal{O})^{d\times
d}})\big{)},$ (4.24) $\displaystyle\Phi(\bm{u}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\Psi,\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}\big{(}{\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{2}(0,T;{\mathcal{L}_{2}(\mathbf{U},\mathrm{L}^{2}(\mathcal{O})^{d})})\big{)}.$
(4.25)

Our aim is to establish that

$\displaystyle\bm{w}=\bm{u}\otimes\bm{u},\ \ \mathbf{S}=\mathrm{A}(\bm{u}),\
\text{ and }\ \Psi=\Phi(\bm{u}).$

###  4.4. Compactness

From the Subsection 4.3, we have the weak convergence results (4.18)-(4.25),
which are not enough to pass to the limit in the nonlinear terms and in the
noise coefficient appearing in our model. In order to pass these terms to the
limit, we need some compactness arguments.

We test (4.1) with $\bm{\phi}\in\bm{\mathcal{V}}$, so that $\mathbb{P}-$a.s.,

$\displaystyle\int_{\mathcal{O}}\bm{u}_{n}(t)\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}_{n}(t):\nabla\bm{\phi}\,\mathrm{d}\bm{x}\equiv\int_{\mathcal{O}}\bm{u}_{n}(t)\cdot{P^{n}_{s}}(\bm{\phi})\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}_{n}(t):\nabla
P^{n}_{s}(\bm{\phi})\,\mathrm{d}\bm{x}$
$\displaystyle=\int_{\mathcal{O}}\bm{u}_{0}\cdot
P^{n}_{s}(\bm{\phi})\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}_{0}(t):\nabla
P^{n}_{s}(\bm{\phi})\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}\mathbf{G}_{n}:\nabla
P^{n}_{s}(\bm{\phi})\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}(s)\cdot
P^{n}_{s}(\bm{\phi})\,\mathrm{d}\bm{x},$ (4.26)

where $P^{n}_{s}$ denotes the projection into the $n-$dimensional space
$\mathbf{X}^{n}$ with respect to the $\mathbf{V}^{s}$ inner product, and

$\displaystyle\mathbf{G}_{n}:=\bm{u}_{n}\otimes\bm{u}_{n}+\nabla\Delta^{-1}\mathbf{a}(\bm{u}_{n})-\nu\mathbf{A}(\bm{u}_{n})+\mathbf{F},$
(4.27)

with $\mathbf{F}$ chosen in
$\mathrm{L}^{2}(0,T;\mathrm{W}^{1,2}(\mathcal{O})^{d\times d})$ in such a way
that $\operatorname{div}\mathbf{F}=-\bm{f}$ in the weak sense.

######  Claim 1.

$\bm{w}=\bm{u}\otimes\bm{u}$ and $\Psi=\Phi(\bm{u})$.

###### Proof of Claim 1.

Using the energy estimate obtained in (4.1) and the definitions of
$\mathbf{A}$ and $\mathbf{a}$ (see (2.5) and (4.1)), and observing (4.3), we
find

$\displaystyle\mathbf{G}_{n}\in\mathrm{L}^{q_{0}}({\Omega,\mathscr{F},\mathbb{P}};\mathrm{L}^{q_{0}}(0,T;{\mathrm{L}^{q_{0}}(\mathcal{O})^{d\times
d}})),\qquad{q_{0}:=\min\mathopen{}\mathclose{{}\left\\{p^{\prime},q^{\prime}}\right\\}>1},$
(4.28)

uniformly in $n$. Note that in fact it should be
$q_{0}:=\min\mathopen{}\mathclose{{}\left\\{p^{\prime},q^{\prime},\frac{q}{2}}\right\\}>1$,
but once that $q\geq 3$ (see assumption (4.3)), we have
$q_{0}=\min\mathopen{}\mathclose{{}\left\\{p^{\prime},q^{\prime}}\right\\}$.
Let us define the functional

$\displaystyle\mathcal{H}(t,\bm{\phi}):=\int_{0}^{t}\int_{\mathcal{O}}\mathbf{G}_{n}:\nabla
P^{n}_{s}(\bm{\phi})\,\mathrm{d}\bm{x}\mathrm{d}s,\qquad{\bm{\phi}\in\bm{\mathcal{V}}}.$

As $1+\frac{2}{q_{0}}>\frac{1}{q_{0}}-\frac{1}{2}$ implies the embedding
$\mathrm{W}^{\tilde{s},q_{0}}(\mathcal{O})\hookrightarrow\mathrm{W}^{s,2}(\mathcal{O})$
for $\tilde{s}\geq s+d\big{(}1+\frac{2}{q_{0}}\big{)}$, we can use (4.28) to
show that

$\displaystyle\mathbb{E}\bigg{[}\|\mathcal{H}\|_{\mathrm{W}^{1,q_{0}}(0,T;\mathrm{W}_{\sigma}^{-\tilde{s},q_{0}}(\mathcal{O}))}\bigg{]}\leq
C.$ (4.29)

The above estimate can be justified as follows (cf. [16, Section 4]):

$\displaystyle\bigg{\|}\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{H}(t,\cdot)\bigg{\|}_{\mathrm{L}^{q_{0}}(0,T;\mathrm{W}_{\sigma}^{-\tilde{s},q_{0}}(\mathcal{O}))}$
$\displaystyle=\bigg{\|}\sup_{\|\bm{\phi}\|_{\tilde{s},q_{0}^{\prime}}\leq
1}\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{H}(t,\bm{\phi})\bigg{\|}_{\mathrm{L}^{q_{0}}(0,T)}=\bigg{\|}\sup_{\|\bm{\phi}\|_{\tilde{s},q_{0}^{\prime}}\leq
1}\int_{\mathcal{O}}\mathbf{G}_{n}(t):\nabla
P_{\tilde{s}}^{n}(\bm{\phi})\,\mathrm{d}\bm{x}\bigg{\|}_{\mathrm{L}^{q_{0}}(0,T)}$
$\displaystyle\leq\bigg{\|}\sup_{\|\bm{\phi}\|_{\tilde{s},q_{0}^{\prime}}\leq
1}\|\mathbf{G}_{n}(t)\|_{q_{0}}\|\nabla
P_{\tilde{s}}^{n}(\bm{\phi})\|_{q_{0}^{\prime}}\bigg{\|}_{\mathrm{L}^{q_{0}}(0,T)}\leq
C\bigg{(}\int_{0}^{T}\|\mathbf{G}_{n}(t)\|_{q_{0}}^{q_{0}}\mathrm{d}t\bigg{)}^{\frac{1}{q_{0}}}.$

For all $0\leq s<t\leq T$, we have

$\displaystyle\mathbb{E}\bigg{[}\bigg{\|}\int_{0}^{t}\Phi(\bm{u}_{n}(\ell))\mathrm{d}\mathrm{W}_{n}(\ell)-\int_{0}^{s}\Phi(\bm{u}_{n}(\ell))\mathrm{d}\mathrm{W}_{n}(\ell)\bigg{\|}_{2}^{q}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\bigg{\|}\int_{s}^{t}\Phi(\bm{u}_{n}(\ell))\mathrm{d}\mathrm{W}_{n}(\ell)\bigg{\|}_{2}^{q}\bigg{]}=\mathbb{E}\bigg{[}\bigg{\|}\int_{s}^{t}\sum_{j=1}^{\infty}\Phi(\bm{u}_{n}(\ell))\mathbf{e}_{i}\mathrm{d}\beta_{j}^{n}\bigg{\|}_{2}^{q}\bigg{]}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\bigg{(}\int_{s}^{t}\sum_{j=1}^{\infty}\|\phi_{k}(\bm{u}_{n}(\ell))\|^{2}_{2}\mathrm{d}\ell\bigg{)}^{\frac{q}{2}}\bigg{]}\leq
C|t-s|^{\frac{q}{2}}\bigg{\\{}\mathbb{E}\bigg{[}\sup_{\ell\in[0,T]}\big{\\{}1+\|\bm{u}_{n}(\ell)\|^{2}_{2}\big{\\}}\bigg{]}\bigg{\\}}^{\frac{q}{2}}\leq
C|t-s|^{\frac{q}{2}},$

where we have used the BDG and Young’s inequalities and the energy estimate
(4.1). By the Kolmogorov continuity criterion, there exists a modification
which has $\mathbb{P}-$a.s. Hölder continuous paths such that

$\displaystyle\mathbb{E}\bigg{[}\bigg{\|}\int_{0}^{t}\Phi(\bm{u}_{n}(s))\mathrm{d}\mathrm{W}_{n}(s)\bigg{\|}_{\mathrm{C}^{\mu}([0,T];\mathrm{L}^{2}(\mathcal{O})^{d})}\bigg{]}\leq
C,\quad\mu:=\theta-\frac{1}{q},$ (4.30)

for $\frac{1}{q}<\theta<\frac{1}{2}$ if $q>2$, which is the case due to
assumption (4.3). Observing that
$\mathrm{W}_{0}^{\tilde{s},q_{0}^{\prime}}(\mathcal{O})^{d}\hookrightarrow\mathrm{W}_{0}^{1,2}(\mathcal{O})^{d}\hookrightarrow\mathrm{L}^{2}(\mathcal{O})^{d}$,
$1<q_{0}<\infty$, implies
$\mathrm{L}^{2}(\mathcal{O})^{d}\hookrightarrow\mathrm{W}^{-\tilde{s},q_{0}}(\mathcal{O})^{d}$,
by the above inequality, we have

$\displaystyle\mathbb{E}\bigg{[}\bigg{\|}\int_{0}^{t}\Phi(\bm{u}_{n}(s))\mathrm{d}\mathrm{W}_{n}(s)\bigg{\|}_{\mathrm{C}^{\mu}([0,T];\mathrm{W}_{0}^{-\tilde{s},q_{0}}(\mathcal{O})^{d})}\bigg{]}\leq
C.$ (4.31)

Collecting the information from (4.29) and (4.31) in (4.4), and still using
(2.9), we arrive at

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\|(\mathrm{I}-\kappa\Delta)\bm{u}_{n}\|_{\mathrm{C}^{\mu}([0,T];\mathrm{W}_{\sigma}^{-\tilde{s},q_{0}}(\mathcal{O})^{d})}}\right]\leq
C,$

for some positive constant $C$ that does not depend on $n$. This in turn
implies for some $\eta>0$ that

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\|(\mathrm{I}-\kappa\Delta)\bm{u}_{n}\|_{\mathrm{W}^{\eta,q_{0}}(0,T;\mathrm{W}_{\sigma}^{-\tilde{s},q_{0}}(\mathcal{O})^{d})}}\right]\leq
C.$ (4.32)

From (4.32), we conclude that

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\|\bm{u}_{n}\|_{\mathrm{W}^{\eta,q_{0}}(0,T;\mathrm{W}_{\sigma}^{2-\tilde{s},q_{0}}(\mathcal{O})^{d})}}\right]\leq
C.$ (4.33)

Observing that $\tilde{s}\geq s+d\big{(}1+\frac{2}{q_{0}}\big{)}$ implies
$2-\tilde{s}<0$.

Let

$\mathbf{Z}_{p,q}:=\mathrm{L}^{\infty}(0,T;\mathbf{V})\cap\mathrm{L}^{p}(0,T;\mathrm{W}^{1,p}_{0}(\mathcal{O})^{d})\cap\mathrm{L}^{q}(0,T;\mathrm{L}^{q}(\mathcal{O})^{d}).$

By the Sobolev embedding theorem, we infer

$\mathbf{Z}_{p,q}\hookrightarrow\mathrm{L}^{\infty}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d})\cap\mathrm{L}^{p}(0,T;\mathrm{W}^{1,p}_{0}(\mathcal{O})^{d}),\qquad\rho\leq\min\mathopen{}\mathclose{{}\left\\{2^{\ast},q}\right\\}.$

By a version of the Aubin-Lions compactness lemma (see [29, Theorem 2.1]), we
further have

$\mathrm{W}^{\eta,q_{0}}(0,T;\mathrm{W}_{\sigma}^{2-\tilde{s},q_{0}}(\mathcal{O})^{d})\cap\mathrm{L}^{\infty}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d})\cap\mathrm{L}^{p}(0,T;\mathrm{W}^{1,p}_{0}(\mathcal{O})^{d})\hookrightarrow\hookrightarrow\mathrm{L}^{\rho}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d}),$

for $q_{0}\leq\rho<p^{\ast}$. In view of this, and attending to the definition
of $q_{0}$ in (4.28), one has

$\displaystyle\mathrm{W}^{\eta,q_{0}}(0,T;\mathrm{W}_{\sigma}^{2-\tilde{s},q_{0}}(\mathcal{O})^{d})\cap\mathbf{Z}_{p,q}\hookrightarrow\hookrightarrow\mathrm{L}^{\rho}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d}),$
(4.34)

for

$\min\mathopen{}\mathclose{{}\left\\{p^{\prime},q^{\prime}}\right\\}<\rho<\min\mathopen{}\mathclose{{}\left\\{2^{\ast},p^{\ast},q}\right\\},\qquad
p>\frac{2d}{d+2}.$ (4.35)

Now, we define the space

$\displaystyle\mathfrak{V}:=\mathrm{L}^{\rho}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d})\otimes\mathrm{C}([0,T];\mathbf{U}_{0})\otimes\mathbf{V}\otimes\mathbf{L}^{2}(\mathcal{O}_{T}).$

In the sequel, we use the following notations:

  1. (1)

$\varrho_{\bm{u}_{n}}$ denotes the law of $\bm{u}_{n}$ on
$\mathrm{L}^{\rho}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d})$;

  2. (2)

$\varrho_{\mathrm{W}}$ denotes the law of $\mathrm{W}$ on
$\mathrm{C}([0,T];\mathbf{U}_{0})$;

  3. (3)

$\varrho_{n}$ denotes the joint law of $\bm{u}_{n},\mathrm{W},\bm{u}_{0}$ and
$\bm{f}$ on the space $\mathfrak{V}$.

Let us consider a ball $B_{N}$ in the space
$\mathrm{W}^{\eta,q_{0}}([0,T];\mathrm{W}_{\sigma}^{2-\tilde{s},q_{0}}(\mathcal{O})^{d})\cap\mathbf{Z}_{p,q}$
and denote its complement by $B_{N}^{c}$. Using the uniform estimates (4.1)
and (4.33), we find

$\displaystyle\varrho_{\bm{u}_{n}}(B_{N}^{c})$
$\displaystyle=\mathbb{P}\mathopen{}\mathclose{{}\left(\|\bm{u}_{n}\|_{\mathrm{W}^{\eta,q_{0}}(0,T;\mathrm{W}_{\sigma}^{2-\tilde{s},q_{0}}(\mathcal{O})^{d})}+\|\bm{u}_{n}\|_{\mathbf{Z}_{p,q}}\geq
N}\right)$
$\displaystyle\leq\frac{1}{N}\mathbb{E}\mathopen{}\mathclose{{}\left[\|\bm{u}_{n}\|_{\mathrm{W}^{\eta,q_{0}}(0,T;\mathrm{W}_{\sigma}^{2-\tilde{s},q_{0}}(\mathcal{O})^{d})}+\|\bm{u}_{n}\|_{\mathbf{Z}_{p,q}}}\right]\leq\frac{C}{N}.$

For any fixed $\xi>0$, we can find $N(\xi)$ such that

$\displaystyle\varrho_{\bm{u}_{n}}(B_{N(\xi)})\geq 1-\frac{\xi}{4}.$

Since, the law $\varrho_{\mathrm{W}}$ is tight as being a Radon measure on the
Polish space $\mathrm{C}([0,T];\mathbf{U}_{0})$, then there exists a compact
subset $K_{\xi}\subset\mathrm{C}([0,T];\mathbf{U}_{0})$ such that
$\varrho_{\bm{u}_{n}}(K_{\xi})\geq 1-\frac{\xi}{4}$. By the same reasoning, we
are able to find compact subsets of $\mathbf{V}$ and
$\mathbf{L}^{2}(\mathcal{O}_{T})$, such that their measures $\Lambda_{0}$ and
$\Lambda_{\bm{f}}$ are greater than $1-\frac{\xi}{4}$. Therefore, we can find
a compact subset $\mathfrak{V}_{\xi}\subset\mathfrak{V}$ such that
$\varrho_{n}(\mathfrak{V}_{\xi})\geq 1-\xi$. Hence,
$\\{\varrho_{n}\\}_{n\in\mathbb{N}}$ is tight in the same space. An
application of Prokhorov’s theorem (see [34, Theorem 2.6]) yields
$\varrho_{n}$ is relatively weakly compact, which implies that $\varrho_{n}$
has a weakly convergent subsequence with weak limit $\varrho$.

###  4.5. Existence of a probability space
$(\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}})$

Applying Skorohod’s representation theorem (see [34, Theorem 2.7]) to ensure
the existence of another probability space
$(\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}})$, a random
sequence
$(\overline{\bm{u}}_{n},\overline{\mathrm{W}}_{n},\overline{\bm{u}}_{0}^{n},\overline{\bm{f}}_{n})$
and a random variable
$(\overline{\bm{u}},\overline{\mathrm{W}},\overline{\bm{u}}_{0},\overline{\bm{f}})$
on the probability space
$(\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}})$, taking
values in $\mathfrak{V}$ such that the following holds:

  1. (1)

The laws of sequence of random variables
$(\overline{\bm{u}}_{n},\overline{\mathrm{W}}_{n},\overline{\bm{u}}_{0}^{n},\overline{\bm{f}}_{n})$
and the random variable
$(\overline{\bm{u}},\overline{\mathrm{W}},\overline{\bm{u}}_{0},\overline{\bm{f}})$
under the new probability measure $\overline{\mathbb{P}}$ coincide with
$\varrho_{n}$ and $\varrho:=\lim\limits_{n\to\infty}\varrho_{n}$,
respectively;

  2. (2)

The following convergence results hold true

$\displaystyle\overline{\bm{u}}_{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\bm{u}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{\rho}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d}),$
$\displaystyle\overline{\mathrm{W}}_{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\mathrm{W}},\quad$
$\displaystyle\text{in}\quad\mathrm{C}([0,T];\mathbf{U}_{0}),$
$\displaystyle\overline{\bm{u}}_{0}^{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\bm{u}}_{0},\quad$
$\displaystyle\text{in}\quad\mathbf{V},$ $\displaystyle\overline{\bm{f}}_{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\bm{f}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}(0,T;\mathrm{L}^{2}(\mathcal{O})^{d}),$

$\overline{\mathbb{P}}-$a.s.;

  3. (3)

The convergence results (4.19) and (4.22) still hold for the functions defined
on the newly constructed probability space
$(\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}})$. Moreover,
for any finite $\beta,$ we have

$\displaystyle\overline{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\|\overline{\mathrm{W}}_{n}(t)\|_{\mathbf{U}_{0}}^{\beta}\bigg{]}=\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\overline{\mathrm{W}}(t)\|_{\mathbf{U}_{0}}^{\beta}\bigg{]}.$

Applying Vitali’s convergence theorem, we have for some subsequences still
labeled by the same subscript

$\displaystyle\overline{\mathrm{W}}_{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\mathrm{W}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}({\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{C}([0,T];\mathbf{U}_{0})),$
(4.36) $\displaystyle\overline{\bm{u}}_{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\bm{u}},\quad$
$\displaystyle\text{in}\quad{\mathrm{L}^{\rho}({\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{\rho}(0,T;\mathrm{L}_{\sigma}^{\rho}(\mathcal{O})^{d}))},$
(4.37) $\displaystyle\overline{\bm{u}}_{0}^{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\bm{u}}_{0},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}({\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathbf{V}),$
(4.38) $\displaystyle\overline{\bm{f}}_{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\bm{f}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}({\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{2}(0,T;{\mathrm{L}^{2}(\mathcal{O})^{d}})),$
(4.39)

for $\rho$ and $p$ in the conditions of (4.35). Now, we need to define the
filtration on the newly constructed probability space. First, we define an
operator of restriction to the interval $[0,T]$ denoted by $\mathbf{h}_{t}$
acting on various path spaces. More precisely, let $\mathbf{X}$ denote any of
the spaces $\mathrm{L}^{\rho}(0,T;\mathrm{L}^{\rho}(\mathcal{O})^{d})$,
$\mathrm{L}^{2}(0,T;\mathrm{L}^{2}(\mathcal{O})^{d})$, or
$\mathrm{C}([0,T];\mathbf{U}_{0})$, and for $t\in[0,T]$, we define

$\displaystyle\mathbf{h}_{t}:\mathbf{X}\to\mathbf{X}\big{|}_{[0,t]},\quad
g\mapsto g\big{|}_{[0,t]}.$ (4.40)

The mapping ${\bf h}_{t}$ is continuous. Let us define the
$\overline{\mathbb{P}}-$augmented canonical filtration by
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}$ of the process
$(\overline{\bm{u}},\overline{\bm{f}},\overline{\mathrm{W}})$, that is,

$\displaystyle\overline{\mathscr{F}}_{t}=\sigma\big{(}\sigma(\mathbf{h}_{t}\overline{\bm{u}},\mathbf{h}_{t}\overline{\bm{f}},\mathbf{h}_{t}\overline{\mathrm{W}})\big{)}\cup\\{M\in\overline{\mathscr{F}};\overline{\mathbb{P}}(M)=0\\}),\quad
t\in[0,T].$

Next, we are going to show that our approximate equations also hold in the new
probability space.

###  4.6. Validity of the approximate equations in
$(\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}})$

The method we employ in this subsection has been already discussed in several
works (see for e.g. [14, 33, 53]). The main aim of this method is to identify
the quadratic variation of the martingale as well as cross variation with the
limit Wiener process obtained through compactness. Note that the laws of
$\overline{\mathrm{W}}_{n}$ and $\overline{\mathrm{W}}$ are the same. As a
consequence of the new probability space, we can find a collection of mutually
independent real-valued $\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}-$Wiener
processes $\\{\overline{\beta}_{j}^{n}\\}_{j\in\mathbb{N}}$ such that
$\overline{\mathrm{W}}_{n}=\sum\limits_{j\in\mathbb{N}}\mathbf{e}_{j}\overline{\beta}_{j}^{n}$,
that is, there exists a collection of mutually independent real-valued
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}-$Wiener processes
$\\{\overline{\beta}_{j}\\}_{j\in\mathbb{N}}$ such that
$\overline{\mathrm{W}}=\sum\limits_{j\in\mathbb{N}}\mathbf{e}_{j}\overline{\beta}_{j}$.
We used the abbreviation $\overline{\mathrm{W}}_{n,n}$ for
$\sum\limits_{j=1}^{n}\mathbf{e}_{j}\overline{\beta}_{j}^{n}$. Let us consider
$\bm{\phi}\in{\bm{\mathcal{V}}}$ and define the functionals for $t\in[0,T]$

$\displaystyle N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{t}$
$\displaystyle=\int_{\mathcal{O}}\bm{u}_{n}(t)\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\bm{u}_{n}(t):\nabla\bm{\phi}\,\mathrm{d}\bm{x}-\int_{\mathcal{O}}\bm{u}_{0}^{n}\cdot\bm{\phi}\,\mathrm{d}\bm{x}-\kappa\int_{\mathcal{O}}\nabla\bm{u}_{0}^{n}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}\bm{f}\cdot
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x}\mathrm{d}s+\int_{0}^{t}\int_{\mathcal{O}}\bm{u}_{n}\otimes\bm{u}_{n}:\nabla
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad+\nu\int_{0}^{t}\int_{\mathcal{O}}\mathbf{A}(\bm{u}_{n}):\mathbf{D}(P^{n}(\bm{\phi}))\,\mathrm{d}\bm{x}\mathrm{d}s,$
$\displaystyle M(\bm{u}_{n})_{t}$
$\displaystyle=\sum_{j=1}^{n}\int_{0}^{t}\bigg{(}\int_{\mathcal{O}}\phi_{j}(\bm{u}_{n})\cdot
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x}\bigg{)}^{2}\mathrm{d}s,$ $\displaystyle
M_{j}(\bm{u}_{n})_{t}$
$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\phi_{j}(\bm{u}_{n})\cdot
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x}\mathrm{d}s.$

Let us denote the increment
$N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{t}-N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{s}$ by
$N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{s,t}$ and similarly for
$M(\bm{u}_{n})_{s,t}$ and $M_{j}(\bm{u}_{n})_{s,t}$. Observe that our proof
will be over once we prove that the process $N(\overline{\bm{u}}_{n})$ is an
$\\{\overline{\mathscr{F}}_{t}\\}_{t\in[0,T]}-$martingale and its quadratic
and cross variations satisfy

$\displaystyle\langle
N(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}})\rangle=M(\overline{\bm{u}}_{n}),\quad\text{
and }\quad\langle
N(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}}),\overline{\beta}_{j}\rangle=M_{j}(\overline{\bm{u}}_{n}),$
(4.41)

respectively. In that situation, we have

$\displaystyle\bigg{\langle}N(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}})-\int_{0}^{\cdot}\int_{\mathcal{O}}\Phi(\overline{\bm{u}}_{n})\mathrm{d}\overline{\mathrm{W}}_{n,n}\cdot
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x}\bigg{\rangle}=0,$ (4.42)

which implies the required result in the new probability space. Let us
establish (4.41). To finish this proof, we use the uniform estimate and we
claim that the mappings

$\displaystyle(\bm{u}_{n},\bm{u}_{0},\bm{f})\mapsto
N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{t},\quad\bm{u}_{n}\mapsto
M(\bm{u}_{n})_{t},\quad\text{and}\quad\bm{u}_{n}\mapsto
M_{j}(\bm{u}_{n})_{t},$

are well defined and measurable on a subspace of the path space, where the
joint law of $(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}})$
is supported, that is, the uniform estimate (4.1) of Theorem 4.1 holds. In the
case of $M(\bm{u}_{n})_{t},$ we have by (2.11) and the continuity of $P^{n}$
in the space $\mathrm{L}^{2}(\mathcal{O})^{d}$

$\displaystyle\sum_{j=1}^{n}\int_{0}^{t}\bigg{(}\int_{\mathcal{O}}\phi_{j}(\bm{u}_{n})\cdot
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x}\bigg{)}^{2}\mathrm{d}s$ $\displaystyle\leq
C(\bm{\phi})\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathcal{O}}|\phi_{j}(\bm{u}_{n})|^{2}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\leq
C(\bm{\phi},K)\bigg{(}T+\int_{0}^{T}\|\bm{u}_{n}(t)\|^{2}_{2}\mathrm{d}t\bigg{)},$

which is finite in view of the uniform estimate (4.1). The rest two mappings
$N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{t}$ and $M_{j}(\bm{u}_{n})_{t}$ can be
tackled in the same manner. Therefore the following random variables have the
same laws:

$\displaystyle N(\bm{u}_{n},\bm{u}_{0},\bm{f})$ $\displaystyle\sim
N(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}}),$
$\displaystyle M(\bm{u}_{n})$ $\displaystyle\sim M(\overline{\bm{u}}_{n}),$
$\displaystyle M_{j}(\bm{u}_{n})$ $\displaystyle\sim
M_{j}(\overline{\bm{u}}_{n}).$

For any fixed times $s,t\in[0,T]$ with $s<t$, let us define a continuous
function by

$\displaystyle h:\mathfrak{V}\big{|}_{[0,s]}\to[0,1].$

Since

$\displaystyle
N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{t}=\int_{0}^{t}\int_{\mathcal{O}}\Phi(\bm{u}_{n})\mathrm{d}\mathrm{W}_{n}(s)\cdot
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x}=\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathcal{O}}\phi_{j}(\bm{u}_{n})\mathrm{d}\beta_{j}\cdot
P^{n}(\bm{\phi})\,\mathrm{d}\bm{x},$

is a square integrable $\\{\mathscr{F}_{t}\\}-$martingale, we conclude that

$\displaystyle\big{[}N(\bm{u}_{n},\bm{u}_{0},\bm{f})\big{]}^{2}-M(\bm{u}_{n}),\quad\text{and}\quad
N(\bm{u}_{n})\beta_{j}-M_{j}(\bm{u}_{n}),$

are $\\{\mathscr{F}_{t}\\}-$martingales. Again, consider the restriction of a
function to the interval $[0,s]$ which is denoted by $\mathbf{h}_{s}$. Due to
the equality of laws, we find

$\displaystyle\overline{\mathbb{E}}$
$\displaystyle\bigg{[}h\big{(}\mathbf{h}_{s}\overline{\bm{u}}_{n},\mathbf{h}_{s}\overline{\mathrm{W}}_{n},\mathbf{h}_{s}\overline{\bm{f}},\overline{\bm{u}}_{0}\big{)}N(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}})_{s,t}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}h\big{(}\mathbf{h}_{s}\bm{u}_{n},\mathbf{h}_{s}\mathrm{W},\mathbf{h}_{s}\bm{f},\bm{u}_{0}\big{)}N(\bm{u}_{n},\bm{u}_{0},\bm{f})_{s,t}\bigg{]}=\text{0},$
$\displaystyle\overline{\mathbb{E}}$
$\displaystyle\bigg{[}h\big{(}\mathbf{h}_{s}\overline{\bm{u}}_{n},\mathbf{h}_{s}\overline{\mathrm{W}}_{n},\mathbf{h}_{s}\overline{\bm{f}},\overline{\bm{u}}_{0}\big{)}\big{(}\big{[}N(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}})\big{]}_{s,t}^{2}-M(\overline{\bm{u}}_{n})_{s,t}\big{)}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}h\big{(}\mathbf{h}_{s}\bm{u}_{n},\mathbf{h}_{s}\mathrm{W},\mathbf{h}_{s}\bm{f},\bm{u}_{0}\big{)}\big{(}\big{[}N(\bm{u}_{n},\bm{u}_{0},\bm{f})\big{]}_{s,t}^{2}-M(\bm{u}_{n})_{s,t}\big{)}\bigg{]}=\text{0},$
$\displaystyle\overline{\mathbb{E}}$
$\displaystyle\bigg{[}h\big{(}\mathbf{h}_{s}\overline{\bm{u}}_{n},\mathbf{h}_{s}\overline{\mathrm{W}}_{n},\mathbf{h}_{s}\overline{\bm{f}},\overline{\bm{u}}_{0}\big{)}\big{(}\big{[}N(\overline{\bm{u}}_{n},\overline{\bm{u}}_{0},\overline{\bm{f}})\overline{\beta}_{j}^{n}\big{]}_{s,t}-M_{j}(\overline{\bm{u}}_{n})_{s,t}\big{)}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}h\big{(}\mathbf{h}_{s}\bm{u}_{n},\mathbf{h}_{s}\mathrm{W},\mathbf{h}_{s}\bm{f},\bm{u}_{0}\big{)}\big{(}\big{[}N(\bm{u}_{n},\bm{u}_{0},\bm{f})\beta_{j}\big{]}_{s,t}-M_{j}(\bm{u}_{n})_{s,t}\big{)}\bigg{]}=\text{0}.$

Thus, we proved (4.41) and hence (4.42), which implies that on the new
probability space
$(\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}})$, we have
the equations for $j=1,\ldots,n,$

$\displaystyle\int_{\mathcal{O}}\mathrm{d}$
$\displaystyle\overline{\bm{u}}_{n}\cdot\bm{\psi}_{j}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\mathrm{d}\nabla\overline{\bm{u}}_{n}:\nabla\bm{\psi}_{j}\,\mathrm{d}\bm{x}+\nu\int_{\mathcal{O}}\mathbf{A}(\overline{\bm{u}}_{n}):\mathbf{D}(\bm{\psi}_{j})\,\mathrm{d}\bm{x}\mathrm{d}t+\alpha\int_{\mathcal{O}}\mathbf{a}(\overline{\bm{u}}_{n})\cdot\bm{\psi}_{j}\,\mathrm{d}\bm{x}\mathrm{d}t$
$\displaystyle=\int_{\mathcal{O}}\overline{\bm{u}}_{n}\otimes\overline{\bm{u}}_{n}:\nabla\bm{\psi}_{j}\,\mathrm{d}\bm{x}\mathrm{d}t+\int_{\mathcal{O}}\overline{\bm{f}}\cdot\bm{\psi}_{j}\,\mathrm{d}\bm{x}\mathrm{d}t+\int_{\mathcal{O}}\Phi(\overline{\bm{u}}_{n})\mathrm{d}\overline{\mathrm{W}}_{n,n}\cdot\bm{\psi}_{j}\,\mathrm{d}\bm{x},$
(4.43) $\displaystyle\overline{\bm{u}}_{n}(0)$
$\displaystyle=P^{n}\overline{\bm{u}}_{0},$ (4.44)

and the following convergence results:

$\displaystyle\overline{\bm{u}}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\overline{\bm{u}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{\infty}(0,T;\mathbf{V})\big{)},$
(4.45) $\displaystyle\overline{\bm{u}}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\overline{\bm{u}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{p}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{p}(0,T;{\mathrm{W}_{0}^{1,p}(\mathcal{O})^{d}}\big{)},$
(4.46) $\displaystyle\overline{\bm{u}}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\overline{\bm{u}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{q}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{q}(0,T;{\mathrm{L}^{q}(\mathcal{O})^{d}})\big{)},$
(4.47) $\displaystyle\mathbf{a}(\overline{\bm{u}}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\mathbf{a}(\overline{\bm{u}}),\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{q^{\prime}}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{q^{\prime}}(0,T;{\mathrm{L}^{q^{\prime}}(\mathcal{O})^{d}})\big{)}$
(4.48) $\displaystyle\overline{\bm{u}}_{n}\otimes\overline{\bm{u}}_{n}$
$\displaystyle\xrightharpoonup[n\to\infty]{}\overline{\bm{u}}\otimes\overline{\bm{u}},\quad$
$\displaystyle\text{in}\quad{\mathrm{L}^{\frac{q}{2}}}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{\frac{q}{2}}(0,T;\mathrm{L}^{\frac{q}{2}}(\mathcal{O})^{d\times
d})\big{)},$ (4.49) $\displaystyle\mathbf{A}(\overline{\bm{u}}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\overline{\mathbf{S}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{p^{\prime}}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{p^{\prime}}(0,T;{\mathrm{L}^{p^{\prime}}(\mathcal{O})^{d\times
d}})\big{)},$ (4.50) $\displaystyle\mathbf{A}(\overline{\bm{u}}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\overline{\mathbf{S}},\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{p^{\prime}}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{p^{\prime}}(0,T;{\mathrm{W}^{-1,p^{\prime}}(\mathcal{O})^{d\times
d}})\big{)},$ (4.51) $\displaystyle\Phi(\overline{\bm{u}}_{n})$
$\displaystyle\xrightharpoonup[n\to\infty]{}\Phi(\overline{\bm{u}}),\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}\big{(}{\overline{\Omega},\overline{\mathscr{F}},\overline{\mathbb{P}}};\mathrm{L}^{2}(0,T;\mathcal{L}_{2}(\mathbf{U};{\mathrm{L}^{2}(\mathcal{O})^{d}}))\big{)}.$
(4.52)

Using (4.36)-(4.39) and (4.45)-(4.52), we obtain the following limit equation,
for all $t\in[0,T]$, $\overline{\mathbb{P}}-$a.s.:

$\displaystyle\int_{\mathcal{O}}$
$\displaystyle\overline{\bm{u}}(t)\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\overline{\bm{u}}{(t)}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}+\nu\int_{0}^{t}\int_{\mathcal{O}}\overline{\mathbf{S}}:\mathbf{D}(\bm{\phi})\,\mathrm{d}\bm{x}\mathrm{d}s+\alpha\int_{0}^{t}\int_{\mathcal{O}}\mathbf{a}(\overline{\bm{u}})\cdot\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\int_{\mathcal{O}}\overline{\bm{u}}_{0}\cdot\bm{\phi}\,\mathrm{d}\bm{x}+\kappa\int_{\mathcal{O}}\nabla\overline{\bm{u}}_{0}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}+\int_{0}^{t}\int_{\mathcal{O}}\overline{\bm{u}}\otimes\overline{\bm{u}}:\nabla\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s+\int_{0}^{t}\int_{\mathcal{O}}\overline{\bm{f}}\cdot\bm{\phi}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle\quad+\int_{\mathcal{O}}\int_{0}^{t}\Phi(\overline{\bm{u}})\mathrm{d}\overline{\mathrm{W}}(s)\cdot\bm{\phi}\,\mathrm{d}\bm{x},$
(4.53)

for all $\bm{\phi}\in\mathrm{C}_{0,\sigma}^{\infty}(\mathcal{O})^{d}$. The
stochastic terms need some justification while passing to the limit. From
(4.36), from one hand, and (2.11) and (4.37), on the other, we have

$\displaystyle\overline{\mathrm{W}}_{n}$
$\displaystyle\xrightarrow[n\to\infty]{}\overline{\mathrm{W}},\quad$
$\displaystyle\text{in}\quad\mathrm{C}([0,T];\mathbf{U}_{0}),$
$\displaystyle\Phi(\overline{\bm{u}}_{n})$
$\displaystyle\xrightarrow[n\to\infty]{}\Phi(\overline{\bm{u}}),\quad$
$\displaystyle\text{in}\quad\mathrm{L}^{2}(0,T;\mathcal{L}_{2}(\mathbf{U};{\mathrm{L}^{2}(\mathcal{O})^{d}})),$

in probability. By [26, Lemma 2.1], these convergence results imply

$\displaystyle\int_{0}^{t}\Phi(\overline{\bm{u}}_{n})\mathrm{d}\overline{\mathrm{W}}_{n}(s)\xrightarrow[n\to\infty]{}\int_{0}^{t}\Phi(\overline{\bm{u}})\mathrm{d}\overline{\mathrm{W}}(s),\quad\text{in}\quad\mathrm{L}^{2}(0,T;\mathrm{L}^{2}(\mathcal{O})^{d}),$

in probability. Thus, we are able to pass to the limit in the stochastic term.
This concludes the proof of Claim 1. ∎

The proof of the next claim shall be done by using the theory of monotone
operators.

######  Claim 2.

$\displaystyle\overline{\mathbf{S}}=\mathbf{A}(\overline{\bm{u}}).$ (4.54)

###### Proof Claim 2.

Applying infinite-dimensional Itô’s formula (see [32, Theorem 2.1] (see
Subsection 4.8 below for a proper justification) to the process
$\|(\mathrm{I}-\kappa\Delta)^{\frac{1}{2}}\overline{\bm{u}}(\cdot)\|^{2}_{2}$
and using the fact
$\int_{\mathcal{O}}\overline{\bm{u}}\otimes\overline{\bm{u}}:\nabla\overline{\bm{u}}\mathrm{d}x=0$
in (4.6), we get $\mathbb{P}-$a.s.,

$\displaystyle\|\overline{\bm{u}}(t)\|^{2}_{2}+\kappa{\|\nabla\overline{\bm{u}}(t)\|_{2}^{2}}+2\nu\int_{0}^{t}\int_{\mathcal{O}}\overline{\mathbf{S}}:\mathbf{D}(\overline{\bm{u}})\,\mathrm{d}\bm{x}\mathrm{d}s+\alpha\int_{0}^{t}\int_{\mathcal{O}}\mathbf{a}(\overline{\bm{u}})\cdot\overline{\bm{u}}\,\mathrm{d}\bm{x}\mathrm{d}s$
$\displaystyle=\|\overline{\bm{u}}_{0}\|^{2}_{2}+\kappa{\|\nabla\overline{\bm{u}}_{0}\|_{2}^{2}}+2\int_{0}^{t}\int_{\mathcal{O}}\overline{\bm{f}}\cdot\overline{\bm{u}}\,\mathrm{d}\bm{x}\mathrm{d}s+2\int_{0}^{t}\int_{\mathcal{O}}\Phi(\overline{\bm{u}})\mathrm{d}\overline{\mathrm{W}}(s)\cdot\overline{\bm{u}}\,\mathrm{d}\bm{x}$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}\mathrm{d}\bigg{\langle}\int_{0}^{\cdot}\Phi(\overline{\bm{u}})\mathrm{d}\overline{\mathrm{W}}\bigg{\rangle}_{s}\,\mathrm{d}\bm{x},$

for all $t\in[0,T]$. Applying finite-dimensional Itô’s formula now to the
process
$\|(\mathrm{I}-\kappa\Delta)^{\frac{1}{2}}\overline{\bm{u}}_{n}(\cdot)\|^{2}_{2}$,
subtracting the former from the later, and then taking expectation to the
obtained equation, we get

$\displaystyle
2\nu\overline{\mathbb{E}}\bigg{[}\int_{0}^{T}\int_{\mathcal{O}}(\mathbf{A}(\overline{\bm{u}}_{n})-\mathbf{A}(\overline{\bm{u}})):\mathbf{D}\big{(}\overline{\bm{u}}_{n}-\overline{\bm{u}}\big{)}\,\mathrm{d}\bm{x}\mathrm{d}s\bigg{]}$
$\displaystyle\quad+2\alpha\overline{\mathbb{E}}\bigg{[}\int_{0}^{T}\int_{\mathcal{O}}(\mathbf{a}(\overline{\bm{u}}_{n})-\mathbf{a}(\overline{\bm{u}}))\cdot(\overline{\bm{u}}_{n}-\overline{\bm{u}})\,\mathrm{d}\bm{x}\mathrm{d}s\bigg{]}$