###### Abstract

The theory of turbulent Newtonian fluids turns out that the choice of the
boundary condition is a relevant issue, since it can modify the behavior of
the fluid by creating or avoiding a strong boundary layer. In this work we
study stochastic second grade fluids filling a two-dimensional bounded domain,
with the Navier-slip boundary condition (with friction). We prove the well-
posedness of this problem and establish a stability result. Our stochastic
model involves a multiplicative white noise and a convective term with third
order derivatives, which significantly complicate the analysis.

Key words. Stochastic, second grade fluids, solvability, stability.

AMS Subject Classification. 76A05, 76D03, 76F55, 76M35

Well-posedness of stochastic second grade fluids

Nikolai Chemetov111Departamento de Matemática, Faculdade de Ciências da
Universidade de Lisboa. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Fernanda Cipriano222Departamento de Matemática, Faculdade de Ciências e
Tecnologia da Universidade Nova de Lisboa and Centro de Matemática e
Aplicações. E-mail<EMAIL_ADDRESS>
##  1 Introduction

The present work is devoted to the study of the stochastic incompressible
fluids of second grade, which are a special class of non-Newtonian fluids.
Unlike the Newtonian fluids, where only the stretching tensor appears in the
characterization of the stress response to a deformation fluid, here the
Cauchy stress tensor $\mathbb{T}$ of the non-Newtonian fluids is defined by

$\mathbb{T}=-\pi\mathbb{I}+\nu A_{1}+\alpha_{1}A_{2}+\alpha_{2}\,A_{1}^{2},$

where the first term $-\pi\mathbb{I}$ is due to the incompressibility of the
fluid and $\mathbb{A}_{1}$, $\mathbb{A}_{2}$ are the two first Rivlin-Ericksen
tensors (cf. [35])

$\mathbb{A}_{1}(y)=\nabla{y}+\left(\nabla{y}\right)^{\top}\quad\text{and
}\quad\mathbb{A}_{2}(y)=\dot{\mathbb{A}}_{1}(y)+\mathbb{A}_{1}(y)\nabla
y+\left(\nabla y\right)^{\top}\mathbb{A}_{1}(y),$

where $y$ denotes the velocity of the fluid, the superposed dot is the
material time derivative, $\nu$ is the kinematic viscosity of the fluid and
$\alpha_{1}$, $\alpha_{2}$ are constant material moduli. The study developed
in [18] turns out that thermodynamic laws and stability principles impose
$\alpha_{1}\geq 0$ and $\alpha_{1}+\alpha_{2}=0.$ We set $\alpha=\alpha_{1}$
and assume $\alpha_{1}>0.$

It is well known that in turbulent fluids, small random perturbations can
produce relevant macroscopic effects. By this reason, the incorporation of a
stochastic white noise force in the Navier-Stokes equations [6] is widely
recognized as an important step to understand the turbulence phenomena. In
this perspective, we can find in [5] (see Lemma 2.2) a deduction of stochastic
Navier-Stokes equations from fundamental principles, by showing that the
stochastic Navier-Stokes equations are a real physical model. Nowadays, the
stochastic Navier-Stokes equations are quite well understood, see for instance
in [16], [20], [30], [36] and the references therein. In spite of that, there
are few results in the literature about stochastic non-Newtonian fluids [17],
[32], [33], [34]. In this paper we consider the stochastic second grade
equations with multiplicative noise given by

$\left\\{\begin{array}[]{l}\tfrac{\partial}{\partial t}\left(Y-\alpha\Delta
Y\right)=\nu\Delta Y-\mathrm{curl}\left(Y-\alpha\Delta Y\right)\times
Y-\nabla\pi+U+G(t,Y)\,\dot{W}_{t},\vspace{2mm}\\\
\mathrm{div}\,Y=0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{in
}\mathcal{O}\times(0,T),\vspace{2mm}\end{array}\right.$ (1.1)

where $U$ is a body force, $G(t,Y)\,\dot{W}_{t}$ is a multiplicative white
noise and $\mathcal{O}$ is a bounded domain of $\mathbb{R}^{2}$ with a
boundary $\Gamma$.

The study of this system requires suitable boundary conditions on the boundary
$\Gamma$ of the domain. The Dirichlet boundary condition given by

$\begin{array}[]{ll}Y=0&\qquad\mbox{on}\ \Gamma\end{array}$

is accepted as an appropriate boundary condition and is the more usual one.
Another physical relevant boundary condition considered in the literature is
the Navier boundary condition

$Y\cdot\mathrm{n}=0,\qquad\left[2(\mathrm{n}\cdot DY)+\gamma
Y\right]\cdot\mathrm{\tau}=0\qquad\mbox{on}\ \Gamma,$ (1.2)

where $\mathrm{n}=(n_{1},n_{2})$ and $\tau$$=(-n_{2},n_{1})$ are the unit
normal and tangent vectors, respectively, to the boundary $\Gamma$,
$DY=\tfrac{\nabla Y+(\nabla Y)^{\top}}{2}$ is the symmetric part of the
velocity gradient and $\gamma>0$ is a friction coefficient on $\Gamma.$

The stochastic partial differential equations (1.1) with the Dirichlet
boundary condition has been studied in [32] and [34]. In the former paper, the
authors used tightness arguments that conjugated with the Skorohod theorem
provided the existence of a weak stochastic solution, in the sense that the
Brownian motion, being part of the solution, was not given in advance; while
in the second one, the authors proved the existence and uniqueness of a strong
stochastic solution. Let us refer the pioneer papers [31] and [13] (see also
[12]), where the deterministic second grade equations with the Dirichlet
boundary condition were mathematically studied for the first time, and [4]
where the deterministic equations were studied with a particular Navier
boundary condition (without friction, i.e. when $\gamma=0$). The physical
interpretation of these second grade equations can be found in [8], [18],
[19], [21], [23] and [24]. It is relevant to recall that the deterministic
methods are based on the Faedo-Galerkin approximation method and a priori
estimates. Then, compactness arguments can be used to pass to the limit of the
respective approximate equations in the distributional sense. Unfortunately,
for the stochastic partial differential equations a priori estimates are not
enough to pass to the limit of the approximate equations, due to the lack of
regularity on the time and stochastic variables. In order to obtain a strong
stochastic solution we should verify that the sequence of the Galerkin
approximations converges strongly in some adequate topology.

We should mention that even if the Dirichlet boundary condition is widely
accepted as an appropriate boundary condition at the surface of contact
between a fluid and a solid, it is also a source of many problems since it
attaches fluid particles to the boundary, creating a strong boundary layer
(cf. [15], [25], [26], [28]). On the other hand, the Navier boundary condition
allows the slippage of the fluid on the boundary, making it possible to treat
important problems as for instance the boundary layer problem, when the
viscosity $\nu$ and/or the elastic response $\alpha$ tend to zero (cf. [3],
[9]-[11], [14], [27], [29]). However, even if the Navier-slip boundary
condition allows to solve interesting problems, technically, when comparing
with the Dirichlet boundary condition, it requires a more careful mathematical
analysis to show the well-posedness of system (1.1)-(1.2) as well as to
establish stability properties for the solution, since the boundary terms
resulting from integrating by parts of the convective term do not vanish and
should be estimated in an appropriate way.

As far as we know, the stochastic second grade fluid equations with the Navier
boundary condition are studied here for the first time. To show the well-
posedness, as in previous articles, we follow the Faedo-Galerkin approximation
method by taking an appropriate basis. We first deduce uniform estimates for
the approximate solutions that allow to pass to the limit with respect to the
weak topology. In order to show that the limit process is a solution, we adapt
the methods developed in [7] to study the stochastic Navier-Stokes equations.
More precisely, we show that the approximate solutions already converge
strongly up to a certain stopping time, therefore we establish the existence
and uniqueness results for the solution of system (1.1)-(1.2), as a stochastic
process with values in $H^{3}$. We should mention that an analogous reasoning
is considered in [34] to deal with the stochastic second grade fluid equations
with homogeneous Dirichlet boundary condition.

The plan of the present paper is as follows. In Section 2 we state the
functional setting and introduce useful notations. In Section 3 we present
some well known results and relevant lemmas related with the nonlinear term of
(1.1)1, which will be applied in the next sections. The main result concerning
the existence of a strong stochastic solution is established in Section 4.
Finally Section 5 is devoted to the study of the stability property.

##  2 Functional setting and notations

We consider the stochastic second grade fluid model in a bounded and simply
connected domain $\mathcal{O}$ of $\mathbb{R}^{2}$ with a sufficiently regular
boundary $\Gamma$

$\left\\{\begin{array}[]{cc}d(\upsilon(Y))=(\nu\Delta
Y-\mathrm{curl}(\upsilon(Y))\times Y-\nabla\pi+U)\,dt+G(t,Y)\,dW_{t},&\\\
\lx@intercol\mathrm{div}\,Y=0\hfil\lx@intercol&\lx@intercol\mbox{in}\
\mathcal{O}\times(0,T),\hfil\lx@intercol\\\ \lx@intercol
Y\cdot\mathrm{n}=0,\qquad\left[2(\mathrm{n}\cdot DY)+\gamma
Y\right]\cdot\mathrm{\tau}=0\hfil\lx@intercol&\lx@intercol\mbox{on}\
\Gamma\times(0,T),\hfil\lx@intercol\\\ \lx@intercol
Y(0)=Y_{0}\hfil\lx@intercol&\lx@intercol\mbox{in}\
<EMAIL_ADDRESS>(2.1)

where $\nu>0$ is a constant viscosity of the fluid, $\alpha>0$ is a constant
material modulus, the constant $\gamma>0$ is a friction coefficient of
$\Gamma,$ $\ \Delta$ and $\nabla$ respectively denote the Laplacian and the
gradient, $Y=(Y_{1},Y_{2})$ is a 2D velocity field and

$\upsilon(Y)=Y-\alpha\Delta Y.$

The function $\pi$ represents the pressure, $U$ is a distributed mechanical
force and the term

$G(t,Y)\,dW_{t}=\sum_{k=1}^{m}G^{k}(t,Y)\,dW^{k}_{t}$

corresponds to the stochastic perturbation, where
$G(t,Y)=(G^{1}(t,Y),\dots,G^{m}(t,Y))$ has suitable growth assumptions defined
below and $W_{t}=(W^{1}_{t},\dots,W^{m}_{t})$ is a standard
$\mathbb{R}^{m}$-valued Wiener process defined on a complete probability space
$(\Omega,\mathcal{F},P)$ endowed with a filtration
$\left\\{\mathcal{F}_{t}\right\\}_{t\in[0,T]}$. We assume that
$\mathcal{F}_{0}$ contains every $P$-null subset of $\Omega$.

Let $X$ be a real Banach space endowed with the norm
$\left\|\cdot\right\|_{X}.$ We denote by $L^{p}(0,T;X)$ the space of
$X$-valued measurable $p-$integrable functions $y$ defined on $[0,T]$ for
$p\geq 1$.

For $p,r\geq 1$ let $L^{p}(\Omega,L^{r}(0,T;X))$ be the space of processes
$y=y(\omega,t)$ with values in $X$ defined on $\ \Omega\times[0,T],$ adapted
to the filtration $\left\\{\mathcal{F}_{t}\right\\}_{t\in[0,T]}$ , and endowed
with the norms

$\left\|y\right\|_{L^{p}(\Omega,L^{r}(0,T;X))}=\left(\mathbb{E}\left(\int_{0}^{T}\left\|y\right\|_{X}^{r}\,dt\right)^{\frac{p}{r}}\right)^{\frac{1}{p}}\text{
}$

and

$\left\|y\right\|_{L^{p}(\Omega,L^{\infty}(0,T;X))}=\left(\mathbb{E}\sup_{t\in[0,T]}\left\|y\right\|_{X}^{p}\
\right)^{\frac{1}{p}}\quad\text{if }r=\infty,$

where $\mathbb{E}$ is the mathematical expectation with respect to the
probability measure $P.$ As usual in the notation of processes $y=y(\omega,t)$
we normally omit the dependence on $\omega\in\Omega.$

In equation (2.1) the vector product $\times$ for 2D vectors $y=(y_{1},y_{2})$
and $z=(z_{1},z_{2})$ is calculated as $y\times
z=(y_{1},y_{2},0)\times(z_{1},z_{2},0);$ the curl of the vector $y$ is equal
to $\mathrm{curl}\,y=\tfrac{\partial y_{2}}{\partial x_{1}}-\tfrac{\partial
y_{1}}{\partial x_{2}}$ and the vector product of $\mathrm{curl}\,y$ with the
vector $z$ is understood as

$\mathrm{curl}\,y\times z=(0,0,\mathrm{curl}\,y)\times(z_{1},z_{2},0).$

Given two vectors $y,z\in\mathbb{R}^{2}$, $y\cdot z=\sum_{i=1}^{2}y_{i}z_{i}$
stands for the usual scalar product in $\mathbb{R}^{2}$ and given two matrices
$A,B,$ we denote $A\cdot B=\sum_{ij=1}^{2}A_{ij}B_{ij}$.

Let us introduce the following Hilbert spaces

$\begin{array}[]{l}H(\mathrm{curl};\mathcal{O})=\left\\{y\in
L^{2}(\mathcal{O})\mid\mathrm{curl}\,y\in
L^{2}(\mathcal{O}),\quad\mathrm{div}\,y=0\ \text{ in
}\mathcal{O}\right\\},\vspace{2mm}\\\ H=\left\\{y\in
L^{2}(\mathcal{O})\mid\mathrm{div}\,y=0\ \text{ in }\mathcal{O}\ \mbox{ and }\
y\cdot\mathrm{n}=0\ \mbox{ on }\Gamma\right\\},\vspace{2mm}\\\ V=\left\\{y\in
H^{1}(\mathcal{O})\mid\mathrm{div}\,y=0\ \mbox{ in }\ \mathcal{O}\mbox{ and }\
y\cdot\mathrm{n}=0\ \text{ on }\ \Gamma\right\\},\vspace{2mm}\\\
W=\left\\{y\in V\cap H^{2}(\mathcal{O})\mid\left[2(\mathrm{n}\cdot Dy)+\gamma
y\right]\cdot\mathrm{\tau}=0\ \ \mbox{on}\ \Gamma\right\\},\vspace{2mm}\\\
\widetilde{W}=W\cap H^{3}(\mathcal{O}).\end{array}$ (2.2)

We denote by $(\cdot,\cdot)$ the inner product in $L^{2}(\mathcal{O})$ and by
$\|\cdot\|_{2}$ the associated norm. The norm in the space
$H^{p}(\mathcal{O})$ is denoted by $\|\cdot\|_{H^{p}}$. Let us note that
$H(\mathrm{curl};\mathcal{O})$ is a subspace of $H^{1}(\mathcal{O}).$ Let us
denote

$\left(Dy,Dz\right)=\int_{\mathcal{O}}Dy\cdot Dz.$

On the space $V$, we consider the following inner product

$\left(y,z\right)_{V}=\left(\upsilon(y),z\right)=\left(y,z\right)+2\alpha\left(Dy,Dz\right)+\alpha\gamma\int_{\Gamma}y\cdot
z$

and the corresponding norm $\|\cdot\|_{V}.$ We can verify that the norms
$\|\cdot\|_{H^{1}}$ and $\|\cdot\|_{V}$ are equivalent because of the Korn
inequality

$\left\|y\right\|_{H^{1}}\leq
C\left(\left\|Dy\right\|_{2}+\left\|y\right\|_{2}\right),\qquad\forall y\in
H^{1}(\mathcal{O}).$ (2.3)

Here and below, $C$ will denote a generic positive constant that may depend
only on the domain $\mathcal{O},$ the regularity of the boundary $\Gamma$, the
physical constants $\nu$, $\alpha$, $\gamma$ and $K$, defined in (2.5).

Let $B$ be a given Hilbert space with inner product $(\cdot.\cdot)_{B}$. For a
vector

$h=(h^{1},\dots,h^{m})\in B^{m}=\overbrace{B\times...\times B}^{m-times}$

we introduce the norm

$\left\|h\right\|_{B}=\sum_{i=1}^{m}\left\|h_{i}\right\|_{B}$

and the module of the inner product of $h$ and a fixed $v\in B$ as

$|\left(h,v\right)_{B}|=\left(\sum_{k=1}^{m}\left(h^{k},v\right)_{B}^{2}\right)^{1/2}.$
(2.4)

.

Assume that $G(t,y):[0,T]\times V\rightarrow V^{m}$ is Lipschitz on $y$, and
satisfies a linear growth; that is, there exists a positive constant $K$ such
that

$\displaystyle\left\|G(t,y)-G(t,z)\right\|_{V}^{2}$ $\displaystyle\leq
K\left\|y-z\right\|_{V}^{2},$ $\displaystyle\left\|G(t,y)\right\|_{V}$
$\displaystyle\leq K\left(1+\left\|y\right\|_{V}\right),\qquad\forall y,z\in
V,\;t\in[0,T].$ (2.5)

##  3  Preliminary results

Let us introduce the Helmholtz projector
$\mathbb{P}:L^{2}(\mathcal{O})\longrightarrow H$, which is the linear bounded
operator defined by $\mathbb{P}y=\tilde{y}$, where $\tilde{y}\in H$ is
characterized by the Helmholtz decomposition

$y=\tilde{y}+\nabla\phi,\qquad\phi\in H^{1}(\mathcal{O}).$

We recall some useful inequalities, namely, the Poincaré inequality

$\left\|y\right\|_{2}\leq C\left\|\nabla y\right\|_{2}\qquad\mbox{for all}\
y\in V$

and the Sobolev inequality

$\left\|y\right\|_{4}\leq C\left\|\nabla y\right\|_{2}\qquad\mbox{for all}\
y\in V.$

Now, we present the first result of this section. This is a well known and
very important property concerning the Navier boundary conditions (see Lemma
4.1 and Corollary 4.2 in [26]). Let $k$ be the curvature of $\Gamma$.
Parameterizing $\Gamma$ by arc length $s$, the following relation holds

$\tfrac{\partial\mathrm{n}}{\partial\mathrm{\tau}}=\tfrac{d\mathrm{n}}{ds}=k\mathrm{\tau}.$

###### Lemma 3.1

Let $y\in H^{2}(\mathcal{O})\cap V$ be a vector field verifying the Navier
boundary condition. Then

$\mathrm{curl}\,y=g(y)\quad\text{ on}\ \Gamma\quad\text{ with }\
g(y)=\left(2k-\gamma\right)\,y\cdot\mathrm{\tau}.$ (3.1)

Proof. Let us first notice that the anti-symmetric tensor $Ay=\nabla y-(\nabla
y)^{\top}$ can be written in the form

$Ay=\mathrm{curl}\,y{\small\left[\begin{array}[]{lr}0&-1\\\
1&0\end{array}\right].}$

The symmetry of $Dy$ and the anti-symmetry of $Ay$ imply that

$\left(Dy\right)\mathrm{\tau}\cdot\mathrm{n}=\left(Dy\right)\mathrm{n}\cdot\mathrm{\tau}\quad\text{and}\quad\left(Ay\right)\mathrm{\tau}\cdot\mathrm{n}=-\left(Ay\right)\mathrm{n}\cdot\mathrm{\tau}.$

It follows that

$(\nabla
y)\mathrm{\tau}\cdot\mathrm{n}=\left(Dy\right)\mathrm{n}\cdot\mathrm{\tau}-\tfrac{1}{2}\left(Ay\right)\mathrm{n}\cdot\mathrm{\tau}$

which is equivalent to

$\mathrm{curl}\,y=-2(\nabla
y)\mathrm{\tau}\cdot\mathrm{n}+2(Dy)\mathrm{n}\cdot\mathrm{\tau}.$ (3.2)

Taking the derivative of the expression $y\cdot\mathrm{n}=0$ in the direction
of the tangent vector $\mathrm{\tau}$, we deduce

$(\nabla y)\mathrm{\tau}\cdot\mathrm{n}=-k\,y\cdot\mathrm{\tau}.$ (3.3)

The conclusion is then a consequence of (3.2) and (3.3).
$\hfill\hfill\blacksquare\hfill$

Now, we state a formula that can be easily derived by taking integration by
parts

$-\int_{\mathcal{O}}\triangle y\cdot z=-\int_{\Gamma}2(Dy)n\cdot
z+\int_{\mathcal{O}}2\,Dy\cdot Dz,$ (3.4)

that holds for any $y\in H^{2}(\mathcal{O})\cap V$ and $z\in
H^{1}(\mathcal{O})$. Using the boundary conditions, that gives the relation

$-\int_{\mathcal{O}}\triangle y\cdot z=\gamma\int_{\Gamma}y\cdot
z+\int_{\mathcal{O}}2\,Dy\cdot Dz\quad\text{ for any }y\in W\text{ and }z\in
V$ (3.5)

that will be used throughout the article.

Let us consider the following modified Stokes system with Navier boundary
condition

$\left\\{\begin{array}[]{ll}h-\alpha\Delta h+\nabla
p=f,\qquad\mathrm{div}\,h=0&\quad\mbox{in}\ \mathcal{O},\vspace{2mm}\\\
h\cdot\mathrm{n}=0,\qquad\left[2(\mathrm{n}\cdot Dh)+\gamma
h\right]\cdot\mathrm{\tau}=0&\quad\mbox{on}\
\Gamma\vspace{2mm}.\end{array}\right.$ (3.6)

Next, we state a lemma concerning the regularity properties of the solution of
this system.

###### Lemma 3.2

Suppose $f\in H^{m}(\mathcal{O})$, $m=0,1$. Then system (3.6) has a solution
$(h,p)\in H^{m+2}(\mathcal{O})\times H^{m+1}(\mathcal{O})$, moreover the
following estimates hold

$\displaystyle\|h\|_{H^{2}}$ $\displaystyle\leq C\|f\|_{2},$ (3.7)
$\displaystyle\|h\|_{H^{3}}$ $\displaystyle\leq C\|f\|_{H^{1}}.$ (3.8)

Proof. Supposing that $f\in L^{2}(\mathcal{O})$, the existence of the solution
$(h,p)$ with $h$ in $H^{1}(\mathcal{O})$ is given by the Lax-Millgram lemma.
Multiplying (3.6)1 by $h,$ we derive

$\left\|h\right\|_{2}^{2}+\alpha\left(2\left\|Dh\right\|_{2}^{2}+\gamma\|h\|_{L^{2}(\Gamma)}^{2}\right)=(f,h)\leq\left\|f\right\|_{2}\left\|h\right\|_{2},$

which gives

$\left\|h\right\|_{H^{1}}\leq C\left\|f\right\|_{2}.$ (3.9)

On the other hand, applying the operator $\mathrm{curl}$ to system (3.6), we
derive the following system for $u=\mathrm{curl}\,h$

$\left\\{\begin{array}[]{ll}u-\alpha\Delta u=\mathrm{curl}\,f&\quad\mbox{in}\
\mathcal{O},\vspace{2mm}\\\
u=g(h)=\left(2k-\gamma\right)\,h\cdot\mathrm{\tau}&\quad\mbox{on}\
\Gamma.\end{array}\right.$ (3.10)

Let us denote the extension of the unit exterior normal $\mathrm{n}$ (and the
tangent $\mathrm{\tau}=(-n_{2},n_{1})$) on the whole domain
$\overline{\mathcal{O}}$ by the same notation $\mathrm{n}$ (and
$\mathrm{\tau}$). Then the function
$z=u-\left(2k-\gamma\right)\,h\cdot\mathrm{\tau}$ solves the system

$\left\\{\begin{array}[]{ll}z-\alpha\Delta
z=\mathrm{curl}\,f-\left(2k-\gamma\right)\,h\cdot\mathrm{\tau}+\alpha\Delta\left[\left(2k-\gamma\right)\,h\cdot\mathrm{\tau}\right]&\quad\mbox{in}\
\mathcal{O},\vspace{2mm}\\\ z=0&\quad\mbox{on}\ \Gamma.\end{array}\right.$
(3.11)

Multiplying equation (3.11)1 by $z,$ integrating by parts and using (3.9), we
deduce

$\|z\|_{2}+\alpha\|\nabla z\|_{2}\leq
C\left(\|f\|_{2}+\|h\|_{H^{1}}\right)\leq C\|f\|_{2},$

which implies

$\|u\|_{H^{1}}\leq C\left(\|f\|_{2}+\|h\|_{H^{1}}\right)\leq C\|f\|_{2}.$
(3.12)

In addition estimate $(2.3.3.7)$, p. 110 of [22] for system (3.10) gives

$\displaystyle\|u\|_{H^{2}}$ $\displaystyle\leq$ $\displaystyle
C\left(\|\mathrm{curl}\,f\|_{2}+\left\|\left(2k-\gamma\right)\,h\cdot\mathrm{\tau}\right\|_{H^{2-\frac{1}{2}}(\Gamma)}\right)$
(3.13) $\displaystyle\leq$ $\displaystyle
C\left(\|f\|_{H^{1}}+\left\|h\right\|_{H^{2}}\right).$

Since $h$ solves system (3.6), then there exists a stream function $\varphi$
such that $h=\nabla^{\bot}\varphi,$ satisfying the system

$\left\\{\begin{array}[]{ll}\Delta\varphi=u&\quad\mbox{in}\
\mathcal{O},\vspace{2mm}\\\ \varphi=0&\quad\mbox{on}\
\Gamma\end{array}\right.$ (3.14)

and the estimate

$\|\varphi\|_{H^{2+m}}\leq\|u\|_{H^{m}},\qquad m\in\mathbb{N}_{0},$ (3.15)

by Theorem 2.5.1.1 , p. 128 of [22].

Combining (3.12) and (3.15) with $m=1$, we deduce

$\|\varphi\|_{H^{3}}\leq C\|u\|_{H^{1}}\leq C\|f\|_{2},$

hence $h=\nabla^{\bot}\varphi\in H^{2}$ and (3.7) hold. Moreover (3.13) and
(3.15) with $m=2$ imply

$\|\varphi\|_{H^{4}}\leq\|u\|_{H^{2}}\leq
C\left(\|f\|_{H^{1}}+\|h\|_{H^{2}}\right).$

Invoking (3.7), we conclude that $h=\nabla^{\bot}\varphi\in H^{3}$ and (3.8)
hold.

$\hfill\hfill\blacksquare\hfill$

Let us recall that the space $W$ introduced in (2.2) is naturally endowed with
the Sobolev norm $\|\cdot\|_{H^{2}}$. The next result follows directly from
Lemma 5 in [2] and helps to introduce on $W$ an equivalent norm that will be
useful to analyze the stability in Section 5.

###### Lemma 3.3

For each $y\in W$, we have

$\left\|\upsilon(y)-\mathbb{P}\upsilon(y)\right\|_{2}\leq
C\left\|y\right\|_{H^{1}},$ (3.16)
$\left\|\upsilon(y)-\mathbb{P}\upsilon(y)\right\|_{H^{1}}\leq
C\left\|y\right\|_{H^{2}}.$ (3.17)

The next regularity result will be fundamental to establish the well-posedness
of the velocity equation (see Propositions 6 in [4] and Lemma 2.1 in [12] for
similar results).

###### Lemma 3.4

Let $y\in\widetilde{W}$. Then, the following estimates hold

$\displaystyle\left\|y\right\|_{H^{2}}$ $\displaystyle\leq
C\left(\left\|\mathbb{P}\,\upsilon(y)\right\|_{2}+\left\|y\right\|_{H^{1}}\right),$
(3.18) $\displaystyle\left\|y\right\|_{H^{3}}$ $\displaystyle\leq
C\left(\left\|\mathrm{curl}\,\upsilon(y)\right\|_{2}+\left\|y\right\|_{H^{1}}\right).$
(3.19)

Proof. Considering system (3.6) with $f=\upsilon(y),$ then the pair $(y,0)$ is
obviously the solution of such system. Hence estimate (3.7) yields

$\left\|y\right\|_{H^{2}}\leq C\left\|\upsilon(y)\right\|_{2}\leq
C\left(\left\|\upsilon(y)-\mathbb{P}\upsilon(y)\right\|_{2}+\left\|\mathbb{P}\upsilon(y)\right\|_{2}\right).$

Applying (3.16), we deduce (3.18).

Since $~{}\mathrm{curl}\,\upsilon(y)\in L^{2}(\mathcal{O})~{}$ and
$~{}\nabla\cdot\left(\mathrm{curl}\,\upsilon(y)\right)=0$, there exists a
unique vector-potential $\psi\in H^{1}(\mathcal{O})$ such that

$\left\\{\begin{array}[]{ll}\mathrm{{curl}\,}\psi=\mathrm{curl}\,\upsilon(y),\qquad\mathrm{div}\,\psi=0&\quad\mbox{in}\
\mathcal{O},\vspace{2mm}\\\ \psi\cdot\mathrm{n}=0&\quad\mbox{on}\
\Gamma\end{array}\right.$

and

$\left\|\psi\right\|_{H^{1}}\leq
C\left\|\mathrm{curl}\,\upsilon(y)\right\|_{2}.$ (3.20)

It follows that $\mathrm{curl}\left(y-\alpha\Delta y-\psi\right)=0$ and there
exists $~{}\pi\in L^{2}(\mathcal{O}),~{}$ such that

$y-\alpha\Delta y-\psi+\nabla\pi=0.$

Hence $y$ is the solution of the Stokes system (3.6) where $f$ is replaced by
$\psi.$

As a consequence of (3.8), we have

$\left\|y\right\|_{H^{3}}\leq
C\left(\left\|\psi\right\|_{H^{1}}+\left\|y\right\|_{H^{1}}\right).$ (3.21)

Using (3.20) we obtain the claimed result (3.19).

$\hfill\hfill\blacksquare\hfill$

In order to define the solution of equation (2.1)1 in the distributional
sense, we introduce a trilinear functional that is well known in the context
of the Navier-Stokes equations

$b(\phi,z,y)=\left(\phi\cdot\nabla z,y\right),\qquad\forall\phi,z,y\in V.$

In what follows we often will use the following property

$b(\phi,z,y)=-b(\phi,y,z),$ (3.22)

that follows taking integration by parts, knowing that $\phi$ is divergence
free and $(\phi\cdot n)=0$ on $\Gamma$.

Straightforward computations yield the following relation

$\left(\mathrm{curl}\,\upsilon(y)\times
z,\phi\right)=b\left(\phi,z,\upsilon(y)\right)-b\left(z,\phi,\upsilon(y)\right)\quad\quad\forall
y\in\widetilde{W},\;z,\phi\in V.$ (3.23)

In the next lemma, we deduce crucial estimates of major importance to
establish the well-posedness of system (2.1), as well as to prove the
stability property of their solutions. We should mention that some estimates
follow from an adaptation of the method considered in [4] to prove the
uniqueness.

###### Lemma 3.5

Let $y,$ $z,\;\phi\in\widetilde{W}$. Then

$\displaystyle\left|\left(\mathrm{curl}\,\upsilon(y)\times
z,\phi\right)\right|$ $\displaystyle\leq
C\left\|y\right\|_{H^{3}}\left\|z\right\|_{H^{1}}\left\|\phi\right\|_{H^{3}},$
(3.24) $\displaystyle\left|\left(\mathrm{curl}\,\upsilon(y)\times
z,\phi\right)\right|$ $\displaystyle\leq
C\left\|y\right\|_{H^{1}}\left\|z\right\|_{H^{3}}\left\|\phi\right\|_{H^{3}},$
(3.25) $\displaystyle\left|\left(\mathrm{curl}\,\upsilon(y)\times
z,y\right)\right|$ $\displaystyle\leq
C\left\|y\right\|_{H^{1}}^{2}\left\|z\right\|_{H^{3}}.$ (3.26)

Proof. 1st step. The proof of estimate (3.24). We directly can estimate

$\left|\left(\mathrm{curl}\,\upsilon(y)\times
z,\phi\right)\right|\leq\|\phi\|_{\infty}\|z\|_{2}\|\mathrm{curl}\,\upsilon(y)\|_{2}\leq\|\phi\|_{H^{3}}\|z\|_{2}\|y\|_{H^{3}}$

by Sobolev’s embedding $H^{3}(\mathcal{O})\hookrightarrow
L^{\infty}(\mathcal{O}).$ Hence we have (3.24).

2nd step. The proof of estimate (3.25). Equality (3.23) gives

$\left(\mathrm{curl}\,\upsilon(y)\times
z,\phi\right)=b(\phi,z,y)-b(z,\phi,y)-\alpha\left(b(\phi,z,\Delta
y)-b(z,\phi,\Delta y)\right).$ (3.27)

With the help of Sobolev’s embedding $H^{1}(\mathcal{O})\hookrightarrow
L^{4}(\mathcal{O})$, it is easy to see that

$\displaystyle\left|b\left(\phi,z,y\right)-b\left(z,\phi,y\right)\right|$
$\displaystyle=\left|b\left(\phi,z,y\right)+b\left(z,\phi,y\right)\right|$
$\displaystyle\leq\|\phi\|_{4}\|\nabla
z\|_{2}\|y\|_{4}+\|z\|_{4}\|\nabla\phi\|_{2}\|y\|_{4}$ $\displaystyle\leq
C\|\phi\|_{H^{1}}\|z\|_{H^{1}}\|y\|_{H^{1}}.$ (3.28)

Integrating by parts and using the boundary conditions, we derive

$\displaystyle b\left(\phi,z,\Delta y\right)$
$\displaystyle=\sum_{i,j=1}^{2}\int_{\mathcal{O}}\phi_{i}\tfrac{\partial
z_{j}}{\partial x_{i}}\Delta
y_{j}=\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}\phi_{i}\tfrac{\partial
z_{j}}{\partial x_{i}}\tfrac{\partial}{\partial x_{k}}\left(\tfrac{\partial
y_{j}}{\partial x_{k}}-\tfrac{\partial y_{k}}{\partial
x_{j}}\right)=\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}\phi_{i}\tfrac{\partial
z_{j}}{\partial x_{i}}\tfrac{\partial}{\partial x_{k}}(A_{jk}(y))$
$\displaystyle=\sum_{i,j,k=1}^{2}\int_{\Gamma}\phi_{i}\tfrac{\partial
z_{j}}{\partial
x_{i}}A_{jk}(y)n_{k}-\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}\tfrac{\partial}{\partial
x_{k}}\left(\phi_{i}\tfrac{\partial z_{j}}{\partial x_{i}}\right)A_{jk}(y)$
$\displaystyle=\sum_{i,j=1}^{2}\int_{\Gamma}\phi_{i}\tfrac{\partial
z_{j}}{\partial
x_{i}}\,g(y)\tau_{j}-\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}\tfrac{\partial\phi_{i}}{\partial
x_{k}}\tfrac{\partial z_{j}}{\partial
x_{i}}A_{jk}(y)-\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}\phi_{i}\tfrac{\partial^{2}z_{j}}{\partial
x_{k}\partial x_{i}}A_{jk}(y)$ $\displaystyle=I_{1}+I_{2}+I_{3}.$ (3.29)

Again, integrating by parts, it follows that

$\displaystyle I_{1}$
$\displaystyle=\sum_{i,j=1}^{2}\int_{\Gamma}\phi_{i}\tfrac{\partial
z_{j}}{\partial
x_{i}}\,g(y)\tau_{j}=\sum_{i,j=1}^{2}\int_{\mathcal{O}}\phi_{i}\tfrac{\partial
z_{j}}{\partial
x_{i}}\,g(y)\tau_{j}\,\mathrm{div}\,\mathrm{n}+\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}\tfrac{\partial}{\partial
x_{k}}\bigl{(}\phi_{i}\tfrac{\partial z_{j}}{\partial
x_{i}}\,g(y)\tau_{j}\bigr{)}n_{k}$
$\displaystyle=b\left(\phi,z,\mathrm{div}\,\mathrm{n}\,g(y)\mathrm{\tau}\right)+b\left(\left(\mathrm{n}\cdot\nabla\right)\phi,z,\,n_{k}\,g(y)\mathrm{\tau}\right)+\sum_{k=1}^{2}b\left(\phi,\,\tfrac{\partial
z}{\partial x_{k}},n_{k}\,g(y)\mathrm{\tau}\right)$
$\displaystyle+b\left(\phi,z,\,\left(\mathrm{n}\cdot\nabla\right)\bigl{(}g(y)\mathrm{\tau}\bigr{)}\right).$

Then, using Sobolev’s embedding $H^{2}(\mathcal{O})\hookrightarrow
L^{\infty}(\mathcal{O}),$ we easily derive

$\displaystyle\left|b\left(\phi,z,\Delta
y\right)\right|\leq\left|b\left(\phi,z,\mathrm{div}\,\mathrm{n}\,g(y)\mathrm{\tau}\right)\right|+\left|b\left(\left(\mathrm{n}\cdot\nabla\right)\phi,z,\,\,g(y)\mathrm{\tau}\right)\right|+\left|b\left(\phi,z,\,\left(\mathrm{n}\cdot\nabla\right)\bigl{(}g(y)\mathrm{\tau}\bigr{)}\right)\right|$
$\displaystyle+\sum_{k=1}^{2}\left(\left|b\left(\phi,\tfrac{\partial
z}{\partial
x_{k}},\,n_{k}\,g(y)\mathrm{\tau}\right)\right|+\left|b\left(\tfrac{\partial\phi}{\partial
x_{k}},z,A_{\cdot\,k}(y)\right)\right|+\left|b\left(\phi,\tfrac{\partial
z}{\partial x_{k}},A_{\cdot\,k}(y)\right)\right|\right)$ $\displaystyle\leq
C\|z\|_{H^{3}}\|\phi\|_{H^{3}}\|y\|_{H^{1}}.$ (3.30)

By symmetry, it follows that

$\left|b\left(z,\phi,\Delta y\right)\right|\leq
C\|z\|_{H^{3}}\|\phi\|_{H^{3}}\|y\|_{H^{1}}.$ (3.31)

Then (3.25) follows from (3.27)-(3.31).

3d step. The proof of estimate (3.26).  As in the above computations for
(3.29), we obtain

$\displaystyle b\left(z,y,\Delta y\right)$
$\displaystyle=\sum_{i,j=1}^{2}\int_{\Gamma}z_{i}\tfrac{\partial
y_{j}}{\partial
x_{i}}\,g(y)\tau_{j}-\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}\tfrac{\partial
z_{i}}{\partial x_{k}}\tfrac{\partial y_{j}}{\partial
x_{i}}A_{jk}(y)-\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}z_{i}\tfrac{\partial^{2}y_{j}}{\partial
x_{k}\partial x_{i}}A_{jk}(y)$ $\displaystyle=J_{1}+J_{2}+J_{3},$

where

$\displaystyle J_{1}$
$\displaystyle=b\left(z,y,\mathrm{div}\,\mathrm{n}\,g(y)\mathrm{\tau}\right)+b\left(\left(\mathrm{n}\cdot\nabla\right)z,y,\,g(y)\mathrm{\tau}\right)+\sum_{k=1}^{2}b\left(z,\,\tfrac{\partial
y}{\partial x_{k}},n_{k}\,g(y)\mathrm{\tau}\right)$
$\displaystyle+b\left(z,y,\left(\mathrm{n}\cdot\nabla\right)\bigl{(}g(y)\mathrm{\tau}\bigr{)}\right),$
$\displaystyle J_{2}$ $\displaystyle=-\sum_{k=1}^{2}b\left(\tfrac{\partial
z}{\partial x_{k}},y,A_{\cdot\,k}(y)\right)$

and

$\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle-\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}z_{i}\tfrac{\partial}{\partial
x_{i}}\left(\tfrac{\partial y_{j}}{\partial x_{k}}\right)\left(\tfrac{\partial
y_{j}}{\partial x_{k}}-\tfrac{\partial y_{k}}{\partial
x_{j}}\right)=\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}z_{i}\tfrac{\partial}{\partial
x_{i}}\left(\tfrac{\partial y_{j}}{\partial x_{k}}\right)\tfrac{\partial
y_{k}}{\partial x_{j}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}z_{i}\left[\tfrac{\partial}{\partial
x_{i}}\left(\tfrac{\partial y_{j}}{\partial x_{k}}\right)\tfrac{\partial
y_{k}}{\partial x_{j}}+\tfrac{\partial}{\partial x_{i}}\left(\tfrac{\partial
y_{k}}{\partial x_{j}}\right)\tfrac{\partial y_{j}}{\partial
x_{k}}\right]=\sum_{i,j,k=1}^{2}\int_{\mathcal{O}}z_{i}\tfrac{\partial}{\partial
x_{i}}\left(\tfrac{\partial y_{j}}{\partial x_{k}}\tfrac{\partial
y_{k}}{\partial x_{j}}\right)=0.$

Therefore, we derive

$\displaystyle\left|b\left(z,y,\Delta y\right)\right|$
$\displaystyle\leq\left|b\left(z,y,\mathrm{div}\,\mathrm{n}\,g(y)\mathrm{\tau}\right)\right|+\left|b\left(\left(\mathrm{n}\cdot\nabla\right)z,y,\,g(y)\mathrm{\tau}\right)\right|+\sum_{k=1}^{2}\left|b\left(z,\,n_{k}\,g(y)\mathrm{\tau},\tfrac{\partial
y}{\partial x_{k}}\right)\right|$
$\displaystyle+\left|b\left(z,y,\left(\mathrm{n}\cdot\nabla\right)\bigl{(}g(y)\mathrm{\tau}\bigr{)}\right)\right|+\sum_{k=1}^{2}\left|b\left(\tfrac{\partial
z}{\partial x_{k}},y,A_{\cdot\,k}(y)\right)\right|\leq
C\|z\|_{H^{2}}\|y\|_{H^{1}}^{2},$ (3.32)

where we have used that

$\sum_{k=1}^{2}b\left(z,\,\tfrac{\partial y}{\partial
x_{k}},n_{k}\,g(y)\mathrm{\tau}\right)=-\sum_{k=1}^{2}b\left(z,\,n_{k}\,g(y)\mathrm{\tau},\tfrac{\partial
y}{\partial x_{k}}\right)$

by (3.22).

Taking $\ \phi=y$ in (3.29) we have

$b\left(y,z,\Delta y\right)=\sum_{i,j=1}^{2}\int_{\Gamma}y_{i}\tfrac{\partial
z_{j}}{\partial x_{i}}g(y)\tau_{j}-\sum_{k=1}^{2}b\left(\tfrac{\partial
y}{\partial
x_{k}},z,A_{\cdot\,k}(y)\right)-\sum_{k=1}^{2}b\left(y,\tfrac{\partial
z}{\partial x_{k}},A_{\cdot\,k}(y)\right).$

Taking into account the embedding theorems $H^{2}(\mathcal{O})\hookrightarrow
C(\overline{\mathcal{O}}),$ $H^{1}(\mathcal{O})\hookrightarrow L^{2}(\Gamma)$
and $H^{1}(\mathcal{O})\hookrightarrow L^{4}(\mathcal{O}),$ we have

$\displaystyle|b\left(y,z,\Delta y\right)|$
$\displaystyle\leq\sum_{i,j=1}^{2}\int_{\Gamma}\bigl{|}y_{i}\tfrac{\partial
z_{j}}{\partial
x_{i}}g(y)\tau_{j}\bigr{|}+\sum_{k=1}^{2}\left(\bigl{|}b\left(\tfrac{\partial
y}{\partial
x_{k}},z,A_{\cdot\,k}(y)\right)\bigr{|}+\bigl{|}b\left(y,\tfrac{\partial
z}{\partial x_{k}},A_{\cdot\,k}(y)\right)\bigr{|}\right)$ $\displaystyle\leq
C\|y\|_{L^{2}(\Gamma)}^{2}\left\|\nabla
z\right\|_{C(\overline{\mathcal{O}})}+C\left\|\nabla
y\right\|_{2}^{2}\left\|\nabla
z\right\|_{\infty}+\sum_{i,j=1}^{2}\left\|y\right\|_{4}\left\|\tfrac{\partial^{2}z}{\partial
x_{i}\partial x_{k}}\right\|_{4}\left\|\nabla y\right\|_{2}\vspace{2mm}$
$\displaystyle\leq C\left\|y\right\|_{H^{1}}^{2}\left\|z\right\|_{H^{3}}.$
(3.33)

Then (3.26) is a consequence of (3.28) and
(3.32)-(3.33).$\hfill\hfill\blacksquare\hfill$

##  4 Existence of strong solution

The aim of the present section is to establish the existence of a strong
solution for system (2.1) in the probabilistic sense.

###### Definition 4.1

Let

$U\in
L^{2}(\Omega\times(0,T);H(\mathrm{curl};\mathcal{O})),\text{\quad\quad}Y_{0}\in
L^{2}(\Omega,\widetilde{W}).$

A stochastic process $Y\in L^{2}(\Omega,L^{\infty}(0,T;\widetilde{W}))$ is a
strong solution of $(\ref{equation_etat})$, if for a.e.-$P$ and a.e.
$t\in(0,T),$ the following equation holds

$\displaystyle\left(\upsilon(Y(t)),\phi\right)=$
$\displaystyle\int_{0}^{t}\left[-2\nu\left(DY(s),D\phi\right)-\nu\gamma\int_{\Gamma}y\cdot\phi\
dx-\left(\mathrm{curl}\,\upsilon(Y(s))\times Y(s),\phi\right)\right]\,ds$
$\displaystyle+\left(\upsilon(Y(0)),\phi\right)+\int_{0}^{t}\left(U(s),\phi\right)\,ds+\int_{0}^{t}\left(G(s,Y(s)),\phi\right)\,dW_{s}$
(4.1)

for all $\phi\in V$, where the nonlinear term should be understood in the
sense

$\begin{array}[]{ll}\left(\mathrm{curl}\,\upsilon(Y(t))\times
Y(t),\phi\right)&=b\left(\phi,Y(t),\upsilon(Y(s))\right)-b\left(Y(t),\phi,\upsilon(Y(s))\right)\vspace{2mm}\end{array}$

and the stochastic integral is defined by

$\int_{0}^{t}\left(G(s,Y(s)),\phi\right)\,dW_{s}=\sum_{k=1}^{m}\int_{0}^{t}\left(G^{k}(s,Y(s)),\phi\right)\,dW^{k}_{s}.$

Let us formulate our main existence and uniqueness result, which will be shown
in this section.

###### Theorem 4.2

Assume that

$U\in L^{p}(\Omega\times(0,T);H(\mathrm{curl};\mathcal{O})),\qquad Y_{0}\in
L^{p}(\Omega,V)\cap L^{2}(\Omega,\widetilde{W})\qquad\text{for some\ }\ 4\leq
p<\infty.$

Then there exists a unique solution $Y$ to equation $(\ref{var_form_state})$
which belongs to

$L^{2}(\Omega,L^{\infty}(0,T;\widetilde{W}))\cap
L^{p}(\Omega,L^{\infty}(0,T;V)).$

Moreover, the following estimates hold

$\frac{1}{2}\mathbb{E}\sup_{s\in[0,t]}\left\|Y(s)\right\|_{V}^{2}+\mathbb{E}\int_{0}^{t}\left(4\nu\left\|DY\right\|_{2}^{2}\,+2\nu\gamma\left\|Y\right\|_{L^{2}(\Gamma)}^{2}\right)\,ds\leq
C\left(\mathbb{E}\left\|Y_{0}\right\|_{V}^{2}+\mathbb{E}\|U\|_{L^{2}(0,t;L^{2})}^{2}+1\right),$
$\mathbb{E}\sup_{s\in[0,t]}\left\|\mathrm{curl}\,\upsilon\left(Y(s)\right)\right\|_{2}^{2}\leq
C\left(\mathbb{E}\left\|\mathrm{curl}\,\upsilon(Y_{0})\right\|_{2}^{2}+\mathbb{E}\left\|U\right\|_{L^{2}(0,t;H^{1})}^{2}+1\right).$

The proof of the theorem is given by Galerkin’s approximation method. We
consider the inner product of $\widetilde{W}$ defined by

$\left(y,z\right)_{\widetilde{W}}=\left(\mathrm{curl}\upsilon(y),\mathrm{curl}\upsilon(z)\right)+\left(y,z\right)_{V}.$
(4.2)

Taking into account (2.3) and (3.19) the norm $\|\cdot\|_{\widetilde{W}}$
induced by this inner product is equivalent to $\|\cdot\|_{H^{3}}$. The
injection operator $I:\widetilde{W}\rightarrow V$ is a compact operator, then
there exists a basis $\\{e_{i}\\}\subset\widetilde{W}$ of eigenfunctions

$\left(y,e_{i}\right)_{\widetilde{W}}=\lambda_{i}\left(y,e_{i}\right)_{V},\qquad\forall
y\in\widetilde{W},\;i\in\mathbb{N},$ (4.3)

being an orthonormal basis for $V$ and the corresponding sequence
$\\{\lambda_{i}\\}$ of eigenvalues verifies $\lambda_{i}>0$, $\forall
i\in\mathbb{N}$ and $\lambda_{i}\rightarrow\infty$ as $i\rightarrow\infty.$
Let us notice that the ellipticity of equation (4.3) increases the regularity
of their solutions. Hence without loss of generality we can consider
$\\{e_{i}\\}\subset H^{4}$ (see [4]).
In this section, we consider this basis and introduce the Faedo-Galerkin
approximation of system (2.1). Let
$W_{n}=\mathrm{span}\,\\{e_{1},\ldots,e_{n}\\}$ and define

$Y_{n}(t)=\sum_{j=1}^{n}c_{j}^{n}(t)e_{j}$

as the solution of the stochastic differential equation

$\left\\{\begin{array}[]{l}d\left(\upsilon\left(Y_{n}\right),\phi\right)=\left(\left(\nu\Delta
Y_{n}-\mathrm{curl}\left(\upsilon\left(Y_{n}\right)\right)\times
Y_{n}+U\right),\phi\right)\,dt+\left(G(t,Y_{n}),\phi\right)\,dW_{t},\vspace{2mm}\\\
Y_{n}(0)=Y_{n,0},\qquad\qquad\qquad\qquad\qquad\qquad\forall\phi\in
W_{n}.\end{array}\right.$ (4.4)

Here $Y_{n,0}$ denotes the projection of the initial condition $Y_{0}$ onto
the space $W_{n}$.

Let us notice that
$\\{\widetilde{e}_{j}=\frac{1}{\sqrt{\lambda_{j}}}e_{j}\\}_{j=1}^{\infty}$ is
an orthonormal basis for $\widetilde{W}$ and

$Y_{n,0}=\sum_{j=1}^{n}\left(Y_{0},e_{j}\right)_{V}e_{j}=\sum_{j=1}^{n}\left(Y_{0},\widetilde{e}_{j}\right)_{\widetilde{W}}\widetilde{e}_{j},$

then the Parseval’s identity gives

$\left\|Y_{n}(0)\right\|_{V}\leq\left\|Y_{0}\right\|_{V}\quad\text{ and
}\quad\left\|Y_{n}(0)\right\|_{\widetilde{W}}\leq\left\|Y_{0}\right\|_{\widetilde{W}}.$
(4.5)

Equation (4.4) defines a system of stochastic ordinary differential equations
in $\mathbb{R}^{n}$ with locally Lipschitz nonlinearities. Hence there exists
a local-in-time solution $\ Y_{n}$ as an adapted process in the space
$C([0,T_{n}];W_{n})$. The global-in-time existence of $Y_{n}$ follows from
uniform estimates on $n=1,2,...,$ that will be deduced in the next Lemma (a
similar reasoning can be found in [1], [34]).

###### Lemma 4.3

Assume that

$U\in L^{2}(\Omega\times(0,T);H(\mathrm{curl};\mathcal{O})),\qquad Y_{0}\in
L^{2}(\Omega,\widetilde{W}).$

Then problem (4.4) admits a unique solution $Y_{n}\in
L^{2}(\Omega,L^{\infty}(0,T;\widetilde{W}))$. Furthermore, for any
$t\in[0,T]$, the following estimates hold

$\displaystyle\frac{1}{2}\mathbb{E}\sup_{s\in[0,t]}\left\|Y_{n}(s)\right\|_{V}^{2}$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}\left(4\nu\left\|DY_{n}\right\|_{2}^{2}\,+2\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)\,ds$
(4.6) $\displaystyle\leq$ $\displaystyle
C\left(1+\mathbb{E}\left\|Y_{0}\right\|_{V}^{2}+\mathbb{E}\|U\|_{L^{2}(0,t;L^{2})}^{2}\right),$
$\displaystyle\frac{1}{2}\mathbb{E}\sup_{s\in[0,t]}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(s)\right)\right\|_{2}^{2}$
$\displaystyle+\frac{2\nu}{\alpha}\mathbb{E}\int_{0}^{t}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\right\|_{2}^{2}\,ds\leq\mathbb{E}\left\|\mathrm{curl}\,\upsilon\left(Y_{0}\right)\right\|_{2}^{2}$
$\displaystyle+C\mathbb{E}\int_{0}^{t}\left\|\mathrm{curl}\,U\right\|_{2}^{2}\,ds+C\mathbb{E}\int_{0}^{t}\left(1+\left\|Y_{n}\right\|_{V}^{2}\right)\,ds$
(4.7)

and

$\mathbb{E}\sup_{s\in[0,t]}\left\|Y_{n}(s)\right\|_{\widetilde{W}}^{2}\leq
C(\mathbb{E}\left\|Y_{0}\right\|_{\widetilde{W}}^{2}+\mathbb{E}\left\|\,U\right\|_{L^{2}(0,t;H(\mathrm{curl};\mathcal{O}))}^{2}),$
(4.8)

where $C$ are positive constants independent of $n$ (and may depend on the
data of our problem the domain $\mathcal{O},$ the regularity of $\Gamma,$ the
physical constants $\nu$, $\alpha$, $\gamma).$

Proof. For each $n\in\mathbb{N}$, let us consider the sequence
$\\{\tau_{N}^{n}\\}$N∈N of the stopping times

$\tau_{N}^{n}=\inf\\{t\geq 0:\|Y_{n}(t)\|_{H^{3}}\geq N\\}\wedge T_{n}.$

In order to simplify the notation, let us introduce the function

$f(Y_{n})=\left(\nu\Delta
Y_{n}-\mathrm{curl}\left(\upsilon\left(Y_{n}\right)\right)\times
Y_{n}+U\right)\in H^{1}(\mathcal{O}).$

Taking $\phi=e_{i}$ for each $i=1,\dots,n$ in equation (4.4), we obtain

$d\left(Y_{n},e_{i}\right)_{V}=\left(f(Y_{n}),e_{i}\right)\,dt+\left(G(t,Y_{n}),e_{i}\right)\,dW_{t}.$
(4.9)

Step 1. Estimate in the space $V$ for $Y_{n}$, depending on the stopping times
$\tau_{N}^{n}.$

The Itô formula gives

$d\left(Y_{n},e_{i}\right)_{V}^{2}=2\left(Y_{n},e_{i}\right)_{V}\left(f(Y_{n}),e_{i}\right)\,dt+2\left(Y_{n},e_{i}\right)_{V}\left(G(t,Y_{n}),e_{i}\right)\,dW_{t}+|\left(G\left(t,Y_{n}\right),e_{i}\right)|^{2}\,dt,$

where the module in the last term is defined by (2.4). Summing these
equalities over $i=1,\dots,n,$ we obtain

$d\left\|Y_{n}\right\|_{V}^{2}=2\left(f(Y_{n}),Y_{n}\right)\,dt+2\left(G(t,Y_{n}),Y_{n}\right)\,dW_{t}+\sum_{i=1}^{n}|\left(G\left(t,Y_{n}\right),e_{i}\right)|^{2}\,dt.$

We know that

$\displaystyle\left(f(Y_{n}),Y_{n}\right)$
$\displaystyle=-2\nu\left\|DY_{n}\right\|_{2}^{2}-\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2}+\alpha\left(b\left(Y_{n},Y_{n},\Delta
Y_{n}\right)-b\left(Y_{n},Y_{n},\Delta Y_{n}\right)\right)$
$\displaystyle-b\left(Y_{n},Y_{n},Y_{n}\right)+\left(U,Y_{n}\right)$
$\displaystyle=-2\nu\left\|DY_{n}\right\|_{2}^{2}-\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2}+\left(U,Y_{n}\right),$
(4.10)

hence

$\displaystyle d\left\|Y_{n}\right\|_{V}^{2}$ $\displaystyle=$ $\displaystyle
2\left(-2\nu\left\|DY_{n}\right\|_{2}^{2}-\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2}+\left(U,Y_{n}\right)\right)\,dt$
(4.11)
$\displaystyle+2\left(G(t,Y_{n}),Y_{n}\right)\,dW_{t}+\sum_{i=1}^{n}|\left(G\left(t,Y_{n}\right),e_{i}\right)|^{2}\,dt,$

Let $\tilde{G}_{n}$ be the solution of (3.6) for $f=G(t,Y_{n})$. Then

$(\tilde{G}_{n},e_{i})_{V}=(G(t,Y_{n}),e_{i})\quad\text{for }i=1,\dots,n\ $

which implies

$\sum_{i=1}^{n}|\left(G\left(t,Y_{n}\right),e_{i}\right)|^{2}=||\widetilde{G}_{n}||_{V}^{2}\leq
C||G\left(t,Y_{n}\right)||_{2}^{2}\leq C(1+||Y_{n}||_{V}^{2}).$ (4.12)

Here we used the fact that $\widetilde{G}_{n}$ solves the elliptic type
problem (3.6) for $f=G(t,Y_{n})$ and assumption (2.5)${}_{2}.$

Let us take $t\in[0,T]$, the integration over the time interval $(0,s)$,
$0\leq s\leq\tau_{N}^{n}\wedge t$ of equality (4.11) and estimate (4.12) yield

$\displaystyle\left\|Y_{n}(s)\right\|_{V}^{2}+\int_{0}^{s}(4\nu\left\|DY_{n}\right\|_{2}^{2}\,$
$\displaystyle+2\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2})dr\leq\left\|Y_{n}(0)\right\|_{V}^{2}+C(1+\int_{0}^{s}\left\|U\right\|_{2}^{2}\,dr)$
$\displaystyle+\int_{0}^{s}\left\|Y_{n}\right\|_{V}^{2}\,dr+2\int_{0}^{s}\left(G\left(r,Y_{n}\right),Y_{n}\right)\,dW_{r}.$
(4.13)

The Burkholder-Davis-Gundy inequality gives

$\displaystyle\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left|\int_{0}^{s}\left(G\left(r,Y_{n}\right),Y_{n}\right)\,dW_{r}\right|$
$\displaystyle\leq\mathbb{E}\left(\int_{0}^{\tau_{N}^{n}\wedge
t}\left|\left(G\left(s,Y_{n}\right),Y_{n}\right)\right|^{2}\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left\|Y_{n}(s)\right\|_{2}\left(\int_{0}^{\tau_{N}^{n}\wedge
t}\left\|G\left(s,Y_{n}\right)\right\|_{2}^{2}\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq\varepsilon\,\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\|Y_{n}(s)\|_{2}^{2}+C_{\varepsilon}\mathbb{E}\int_{0}^{\tau_{N}^{n}\wedge
t}(1+\left\|Y_{n}\right\|_{V}^{2})\,ds.$

Substituting the last inequality with the chosen $\varepsilon=\frac{1}{2}$ in
(4.13) and considering (4.5), we derive

$\displaystyle\frac{1}{2}\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\|Y_{n}(s)\|_{V}^{2}$ $\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{\tau_{N}^{n}\wedge
t}\left(4\nu\left\|DY_{n}\right\|_{2}^{2}\,+2\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)\,ds\leq\mathbb{E}\left\|Y_{0}\right\|_{V}^{2}$
$\displaystyle+$ $\displaystyle
C\mathbb{E}\int_{0}^{t}(1+\left\|U\right\|_{2}^{2})\,ds+C\mathbb{E}\int_{0}^{\tau_{N}^{n}\wedge
t}\left\|Y_{n}\right\|_{V}^{2}\,ds.$

Hence, if we denote by $1_{[0,\tau_{N}^{n}]}$ the characteristic function of
the interval $[0,\tau_{N}^{n}],$ the function

$f(t)=\mathbb{E}\sup_{s\in[0,t]}1_{[0,\tau_{N}^{n}]}\|Y_{n}(s)\|_{V}^{2}$

fulfills Gronwall´s type inequality

$\frac{1}{2}f(t)\leq
C\int_{0}^{t}f(s)ds+\mathbb{E}\left\|Y_{n}(0)\right\|_{V}^{2}+C\mathbb{E}\int_{0}^{t}(1+\left\|U\right\|_{2}^{2})\,ds,$

which implies

$\displaystyle\frac{1}{2}\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left\|Y_{n}(s)\right\|_{V}^{2}$ $\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{\tau_{N}^{n}\wedge
t}\left(4\nu\left\|DY_{n}\right\|_{2}^{2}\,+2\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)\,ds$
(4.14) $\displaystyle\leq$ $\displaystyle
C\left(1+\mathbb{E}\left\|Y_{0}\right\|_{V}^{2}+\mathbb{E}\|U\|_{L^{2}(0,t;L^{2})}^{2}\right).$

Step 2. $L^{2}$ estimate for $\mathrm{curl}\,$$\upsilon(Y_{n})$, depending on
the stopping times $\tau_{N}^{n}.$

The deduction of this estimate is quite long. Let us first consider the
solutions $\tilde{f}_{n}$ and $\tilde{G}_{n}$ of (3.6) for $f=f(Y_{n})$ and
$f=G(t,Y_{n})$, respectively. Then the following relations hold

$(\tilde{f}_{n},e_{i})_{V}=(f(Y_{n}),e_{i}),\qquad(\tilde{G}_{n},e_{i})_{V}=(G(t,Y_{n}),e_{i}).$
(4.15)

If we use these relations in equality (4.9), we get

$d\left(Y_{n},e_{i}\right)_{V}=\left(\tilde{f}_{n},e_{i}\right)_{V}\,dt+\left(\tilde{G}_{n},e_{i}\right)_{V}\,dW_{t}.$

Multiplying the last identity by $\lambda_{i}$ and using (4.3) in the
resulting equation yields

$d\left(Y_{n},e_{i}\right)_{\widetilde{W}}=(\tilde{f}_{n},e_{i})_{\widetilde{W}}\,dt+(\tilde{G}_{n},e_{i})_{\widetilde{W}}\,dW_{t}.$

On the other hand, the Itô formula gives

$d\left(Y_{n},e_{i}\right)_{\widetilde{W}}^{2}=2\left(Y_{n},e_{i}\right)_{\widetilde{W}}(\tilde{f}_{n},e_{i})_{\widetilde{W}}\,dt+2\left(Y_{n},e_{i}\right)_{\widetilde{W}}(\tilde{G}_{n},e_{i})_{\widetilde{W}}\,dW_{t}+|(\tilde{G}_{n},e_{i})_{\widetilde{W}}|^{2}\,dt.$

Multiplying this equality by $\frac{1}{\lambda_{i}}$ and summing over
$i=1,\dots,n$, we obtain

$d\left\|Y_{n}\right\|_{\widetilde{W}}^{2}=2(\tilde{f}_{n},Y_{n})_{\tilde{W}}\,dt+2(\tilde{G}_{n},Y_{n})_{\widetilde{W}}\,dW_{t}+\sum_{i=1}^{n}\frac{1}{\lambda_{i}}|(\tilde{G}_{n},e_{i})_{\widetilde{W}}|^{2}\,dt,$

that is

$\displaystyle d(\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\|_{2}^{2}$
$\displaystyle+\|Y_{n}\|_{V}^{2})=2\left((\mathrm{curl}\,\upsilon(\tilde{f}_{n}),\mathrm{curl}\,\upsilon(Y_{n}))+(\tilde{f}_{n},Y_{n})_{V}\right)\,dt$
$\displaystyle+2\left((\mathrm{curl}\,\upsilon(\tilde{G}_{n}),\mathrm{curl}\,\upsilon(Y_{n}))+(\tilde{G}_{n},Y_{n})_{V}\right)\,dW_{t}+\sum_{i=1}^{n}\lambda_{i}|(G(t,Y_{n}),e_{i})_{V}|^{2}\,dt$

by the definition of the inner product (4.2). The definition of
$\tilde{f}_{n}$ and $\tilde{G}_{n}$ as solutions of (3.6) implies

$\displaystyle d(\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\|_{2}^{2}$
$\displaystyle+\|Y_{n}\|_{V}^{2})=2\left((\mathrm{curl}\,f(Y_{n}),\mathrm{curl}\,\upsilon(Y_{n}))+(f(Y_{n}),Y_{n})\right)\,dt$
$\displaystyle+2\left((\mathrm{curl}\,G(t,Y_{n}),\mathrm{curl}\,\upsilon(Y_{n}))+(G(t,Y_{n}),Y_{n})\right)\,dW_{t}+\sum_{i=1}^{n}\lambda_{i}|(G(t,Y_{n}),e_{i})_{V}|^{2}\,dt,$

that reduces to

$\displaystyle d\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\|_{2}^{2}$
$\displaystyle=2\left((\mathrm{curl}\,f(Y_{n}),\mathrm{curl}\,\upsilon(Y_{n}))\right)\,dt$
$\displaystyle+2\left((\mathrm{curl}\,G(t,Y_{n}),\mathrm{curl}\,\upsilon(Y_{n}))\right)\,dW_{t}+\sum_{i=1}^{n}(\lambda_{i}-1)|(G(t,Y_{n}),e_{i})|^{2}\,dt,$
(4.16)

taking into account equality (4.11).

Since

$\mathrm{curl}\left(\mathrm{curl}\left(\upsilon(Y_{n})\right)\times
Y_{n}\right)=\left(Y_{n}\cdot\nabla\right)\mathrm{curl}\,\upsilon(Y_{n})\qquad\text{and\quad}\left(\left(Y_{n}\cdot\nabla\right)\mathrm{curl}\,\upsilon(Y_{n}),\mathrm{curl}\,\upsilon(Y_{n})\right)=0,$

we have

$\displaystyle\left(\mathrm{curl}\,f(Y_{n}),\mathrm{curl}\,\upsilon(Y_{n})\right)$
$\displaystyle=\left(\nu\,\mathrm{curl}\,\Delta
Y_{n}+\mathrm{curl}\,U,\mathrm{curl}\,\upsilon(Y_{n})\right)$
$\displaystyle=\left(-\frac{\nu}{\alpha}\,\mathrm{curl}\,\upsilon(Y_{n})+\frac{\nu}{\alpha}\mathrm{curl}\,Y_{n}+\mathrm{curl}\,U,\mathrm{curl}\,\upsilon(Y_{n})\right).$

Substituting this last relation in (4.16), we derive

$\displaystyle
d\left\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\right\|_{2}^{2}$
$\displaystyle+\frac{2\nu}{\alpha}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\right\|_{2}^{2}\,dt=2\left(\frac{\nu}{\alpha}\mathrm{curl}\,Y_{n}+\mathrm{curl}\,U,\mathrm{curl}\,\upsilon(Y_{n})\right)\,dt$
$\displaystyle+2\left((\mathrm{curl}\,G(t,Y_{n}),\mathrm{curl}\,\upsilon(Y_{n}))\right)\,dW_{t}+\sum_{i=1}^{n}(\lambda_{i}-1)|(G(t,Y_{n}),e_{i})|^{2}\,dt.$
(4.17)

Let us take $t\in[0,T]$. By integrating over the time interval $(0,s)$, $0\leq
s\leq\tau^{n}_{N}\wedge t$, taking the supremum and the expectation, we get

$\displaystyle\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(s)\right)\right\|_{2}^{2}$
$\displaystyle+\frac{2\nu}{\alpha}\mathbb{E}\int_{0}^{\tau^{n}_{N}\wedge
t}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\right\|_{2}^{2}\,ds\leq\mathbb{E}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(0)\right)\right\|_{2}^{2}$
$\displaystyle+2\mathbb{E}\,\int_{0}^{\tau^{n}_{N}\wedge
t}\left|\left(\frac{\nu}{\alpha}\mathrm{curl}\,Y_{n}+\mathrm{curl}\,U,\mathrm{curl}\,\upsilon(Y_{n})\right)\right|\,ds$
$\displaystyle+2\mathbb{E}\,\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\left|\int_{0}^{s}\left(\mathrm{curl}\,G(r,Y_{n}),\mathrm{curl}\,\upsilon(Y_{n})\right)\,dW_{r}\right|$
$\displaystyle+\mathbb{E}\,\int_{0}^{\tau^{n}_{N}\wedge
t}\sum_{i=1}^{n}|\lambda_{i}-1|\,|(G(s,Y_{n}),e_{i})|^{2}\,ds.$ (4.18)

Moreover

$\displaystyle 2\mathbb{E}\,\int_{0}^{\tau^{n}_{N}\wedge
t}\left|\left(\frac{\nu}{\alpha}\mathrm{curl}\,Y_{n}+\mathrm{curl}\,U,\mathrm{curl}\,\upsilon(Y_{n})\right)\right|\,ds$
$\displaystyle\leq\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(s)\right)\right\|_{2}\int_{0}^{\tau_{N}\wedge
t}\left(\frac{\nu}{\alpha}\left\|\mathrm{curl}\,Y_{n}\right\|_{2}+\left\|\mathrm{curl}\,U\right\|_{2}\right)\,ds$
$\displaystyle\leq\varepsilon\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(s)\right)\right\|_{2}^{2}+C_{\varepsilon}\mathbb{E}\int_{0}^{\tau^{n}_{N}\wedge
t}(\left\|\mathrm{curl}\,Y_{n}\right\|_{2}^{2}+\left\|\mathrm{curl}\,U\right\|_{2}^{2})\,ds,$

that is

$\displaystyle 2\mathbb{E}\,\int_{0}^{\tau^{n}_{N}\wedge
t}\left|\left(\frac{\nu}{\alpha}\mathrm{curl}\,Y_{n}+\mathrm{curl}\,U,\mathrm{curl}\,\upsilon(Y_{n})\right)\right|\,ds$
$\displaystyle\leq\varepsilon\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(s)\right)\right\|_{2}^{2}+C_{\varepsilon}\mathbb{E}\int_{0}^{\tau^{n}_{N}\wedge
t}(\left\|\mathrm{curl}\,Y_{n}\right\|_{2}^{2}+\left\|\mathrm{curl}\,U\right\|_{2}^{2})\,ds.$
(4.19)

The Burkholder-Davis-Gundy inequality and estimate (2.5)2 imply

$\displaystyle 2\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge t]}$
$\displaystyle\left|\int_{0}^{s}\left(\mathrm{curl}\,G\left(r,Y_{n}\right),\mathrm{curl}\,\upsilon(Y_{n})\right)\,dW_{r}\right|\leq
2\mathbb{E}\left(\int_{0}^{\tau^{n}_{N}\wedge
t}\left|\left(\mathrm{curl}\,G\left(s,Y_{n}\right),\mathrm{curl}\,\upsilon(Y_{n})\right)\right|^{2}\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq 2\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\left\|\mathrm{curl}\,\upsilon(Y_{n}(s))\right\|_{2}\left(\int_{0}^{\tau^{n}_{N}\wedge
t}\left\|\mathrm{curl}\,G\left(s,Y_{n}\right)\right\|_{2}^{2}\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq\varepsilon\,\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\|\mathrm{curl}\,\upsilon(Y_{n})\|_{2}^{2}+C_{\varepsilon}\mathbb{E}\int_{0}^{\tau^{n}_{N}\wedge
t}\left\|G\left(s,Y_{n}\right)\right\|_{V}^{2}\,ds$
$\displaystyle\leq\varepsilon\,\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\|\mathrm{curl}\,\upsilon(Y_{n})\|_{2}^{2}+C_{\varepsilon}\mathbb{E}\int_{0}^{\tau^{n}_{N}\wedge
t}\left(1+\left\|\,Y_{n}(s)\right\|_{V}^{2}\right)\,ds,$

that is

$\displaystyle 2\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge t]}$
$\displaystyle\left|\int_{0}^{s}\left(\mathrm{curl}\,G\left(r,Y_{n}\right),\mathrm{curl}\,\upsilon(Y_{n})\right)\,dW_{r}\right|$
$\displaystyle\leq\varepsilon\,\mathbb{E}\sup_{s\in[0,\tau^{n}_{N}\wedge
t]}\|\mathrm{curl}\,\upsilon(Y_{n})\|_{2}^{2}+C_{\varepsilon}\mathbb{E}\int_{0}^{\tau^{n}_{N}\wedge
t}\left(1+\left\|\,Y_{n}\right\|_{V}^{2}\right)\,ds.$ (4.20)

Substituting (4.19)-(4.20) in (4.18) and choosing $\varepsilon=\frac{1}{4}$,
we obtain

$\displaystyle\frac{1}{2}\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(s)\right)\right\|_{2}^{2}$
$\displaystyle+\frac{2\nu}{\alpha}\mathbb{E}\int_{0}^{\tau_{N}^{n}\wedge
t}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}\right)\right\|_{2}^{2}\,ds\leq\mathbb{E}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(0)\right)\right\|_{2}^{2}$
$\displaystyle+C\mathbb{E}\int_{0}^{t}\left\|\mathrm{curl}\,U\right\|_{2}^{2}\,ds+C\mathbb{E}\int_{0}^{\tau_{N}^{n}\wedge
t}\left(1+\left\|Y_{n}(s)\right\|_{V}^{2}\right)\,ds.$ (4.21)

Step 3. The limit transition, as $N\rightarrow\infty,$ in estimates (4.14),
(4.21).

Since

$\mathbb{E}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(0)\right)\right\|_{2}^{2}\leq
C\mathbb{E}\left\|Y_{n}(0)\right\|_{H^{3}}^{2}\leq
C\mathbb{E}\left\|Y_{0}\right\|_{H^{3}}^{2}\leq C$

and

$\mathbb{E}\int_{0}^{\tau_{N}^{n}\wedge
t}\left(1+\left\|Y_{n}(s)\right\|_{V}^{2}\right)\,ds\leq C$

by (4.5) and (4.14), we obtain

$\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left\|\mathrm{curl}\,\upsilon\left(Y_{n}(s)\right)\right\|_{2}^{2}\leq C.$

Therefore estimates (3.19), (4.14) imply

$\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
T]}\left\|Y_{n}(s)\right\|_{H^{3}}^{2}\leq C,$

where $C$ is a constant independent of $N$ and $n$. Let us fix
$n\in\mathbb{N}$, writing

$\displaystyle\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
T]}\left\|Y_{n}(s)\right\|_{H^{3}}^{2}$
$\displaystyle=\mathbb{E}\left(\sup_{s\in[0,\tau_{N}^{n}\wedge
T]}1_{\\{\tau_{N}^{n}<T\\}}\ \left\|Y_{n}(s)\right\|_{H^{3}}^{2}\right)$
$\displaystyle+\mathbb{E}\left(\sup_{s\in[0,\tau_{N}^{n}\wedge
T]}1_{\\{\tau_{N}^{n}\geq T\\}}\ \left\|Y_{n}(s)\right\|_{H^{3}}^{2}\right)$
$\displaystyle\geq\mathbb{E}\left(\max_{s\in[0,\tau_{N}^{n}]}1_{\\{\tau_{N}^{n}<T\\}}\
\left\|Y_{n}(s)\right\|_{H^{3}}^{2}\right)\geq
N^{2}P\left(\tau_{N}^{n}<T\right),$ (4.22)

we deduce that $P\left(\tau_{N}^{n}<T\right)\rightarrow 0$, as
$N\rightarrow\infty.$ This means that $\tau_{N}^{n}\rightarrow T$ in
probability, as $N\rightarrow\infty$. Then there exists a subsequence
$\\{\tau_{N_{k}}^{n}\\}$ of $\\{\tau_{N}^{n}\\}$ (that may depend on $n$) such
that

$\tau_{N_{k}}^{n}(\omega)\rightarrow T\text{\qquad for a. e. \
}\omega\in\Omega\quad\text{as }k\rightarrow\infty.$

Since $\tau_{N_{k}}^{n}\leq T_{n}\leq T$, we deduce that $T_{n}=T$, so $Y_{n}$
is a global-in-time solution of the stochastic differential equation (4.4). On
the other hand, the sequence $\left\\{\tau_{N}^{n}\right\\}$ of the stopping
times is monotone on $N$ for each fixed $n$, then we can apply the monotone
convergence theorem in order to pass to the limit in inequalities (4.14) and
(4.21) as $N\rightarrow\infty$, deducing estimates (4.6) and (4.7).

Step 4. Estimate in the space $\widetilde{W}$ for $Y_{n}$.  Substituting
estimate (4.6) in (4.7) and using Lemma 3.4, we immediately derive the main
estimate (4.8) of this lemma.

$\hfill\hfill\blacksquare$

In the next lemma, assuming a better integrability for the data $U$, $Y_{0},$
we improve the integrability properties for the solution $Y_{n}$ of problem
(4.4).

###### Lemma 4.4

Assume that

$U\in L^{p}(\Omega\times(0,T);H),\qquad Y_{0}\in
L^{p}(\Omega,V)\qquad\text{for some\ }\ 4\leq p<\infty.$

Then the solution $Y_{n}$ of problem $(\ref{y1})$ belongs to
$L^{p}(\Omega,L^{\infty}(0,T;V))$ and verifies the estimate

$\mathbb{E}\sup_{s\in[0,t]}\left\|Y_{n}(t)\right\|_{V}^{p}\leq
C\mathbb{E}\left\|Y_{0}\right\|_{V}^{p}+C\,(1+\mathbb{E}\int_{0}^{t}\left\|U\right\|_{2}^{p}\,ds),$
(4.23)

where $C$ is a positive constant independent of $n.$

Proof. For each $n\in\mathbb{N},$ let us define the suitable sequence
$\\{\tau_{N}^{n}\\}_{N\in\mathbb{N}}$ of the stopping times

$\tau_{N}^{n}=\inf\\{t\geq 0:\|Y_{n}(t)\|_{V}\geq N\\}\wedge T.$

Applying the Itô formula for the function $\theta(x)=x^{q},\,q\geq 1,$ to
process (4.11), we have

$\displaystyle d\left\|Y_{n}\right\|_{V}^{2q}$
$\displaystyle=q\left\|Y_{n}\right\|_{V}^{2q-2}\left[-\left(4\nu\left\|DY_{n}\right\|_{2}^{2}\,+2\nu\gamma\left\|Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)+2\left(U,Y_{n}\right)+\sum_{i=1}^{n}|\left(G\left(t,Y_{n}\right),e_{i}\right)|^{2}\right]\,dt$
$\displaystyle+2q\left\|Y_{n}\right\|_{V}^{2q-2}\left(G(t,Y_{n}),Y_{n}\right)\,dW_{t}+2q(q-1)\left\|Y_{n}\right\|_{V}^{2q-4}|\left(G(t,Y_{n}),Y_{n}\right)|^{2}\,dt.$

Let us take $t\in[0,T]$. Integrating over the time interval $[0,s]$, $0\leq
s\leq\tau_{N}^{n}\wedge t$, we obtain

$\displaystyle\left\|Y_{n}(s)\right\|_{V}^{2q}$
$\displaystyle\leq\left\|Y_{n}(0)\right\|_{V}^{2q}+q\int_{0}^{s}\left\|Y_{n}\right\|_{V}^{2q-2}\left|2\left(U,Y_{n}\right)+\sum_{i=1}^{n}|\left(G\left(r,Y_{n}\right),e_{i}\right)|^{2}\right|\,dr$
$\displaystyle+2q\left|\int_{0}^{s}\left\|Y_{n}\right\|_{V}^{2q-2}\left(G(r,Y_{n}),Y_{n}\right)\,dW_{r}\right|$
$\displaystyle+2q(q-1)\int_{0}^{s}\left\|Y_{n}\right\|_{V}^{2q-4}|\left(G(r,Y_{n}),Y_{n}\right)|^{2}\,dr.$
(4.24)

From estimate (4.12) we have

$\sum_{i=1}^{n}|\left(G\left(t,Y_{n}\right),e_{i}\right)|^{2}\leq
C(1+||Y_{n}||_{V}^{2}).$

Taking the supremum on $s\in[0,\tau_{N}^{n}\wedge t]$, the expectation in
(4.24), applying Burkholder-Davis-Gundy´s and Young´s inequalities, and
proceeding analogously to (4.13), we obtain

$\displaystyle\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left\|Y_{n}(s)\right\|_{V}^{2q}$
$\displaystyle\leq\mathbb{E}\left\|Y_{n}(0)\right\|_{V}^{2q}+C_{q}\,\mathbb{E}\int_{0}^{t}\left\|U\right\|_{2}^{2q}\,ds$
$\displaystyle+\frac{1}{2}\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left\|Y_{n}(s)\right\|_{V}^{2q}+C\,_{q}(1+\,\mathbb{E}\int_{0}^{\tau_{N}\wedge
t}\left\|Y_{n}\right\|_{V}^{2q}\,ds).$

Using Gronwall´s inequality, we deduce

$\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
t]}\left\|Y_{n}(s)\right\|_{V}^{2q}\leq
C\mathbb{E}\left\|Y_{0}\right\|_{V}^{2q}+C\,(1+\mathbb{E}\int_{0}^{t}\left\|U\right\|_{2}^{2q}\,ds)$
(4.25)

for any $q\geq 1$ and $t\in[0,T]$. Using the fact that

$\mathbb{E}\sup_{s\in[0,\tau_{N}^{n}\wedge
T]}\left\|Y_{n}(s)\right\|_{V}^{2q}\leq C$

with $C$ independent of $n$ and $N$, we may reasoning as in the proof of Lemma
4.3, in order to verify that for each $n$, $\tau_{N}^{n}\rightarrow T$ in
probability, as $N\rightarrow\infty$. Then, there exists a subsequence
$\\{\tau_{N_{k}}^{n}\\}$ of $\\{\tau_{N}^{n}\\}$ (that may depend on $n$) such
that $\tau_{N_{k}}^{n}\rightarrow T$ for a. e. $\omega\in\Omega$, as
$k\rightarrow\infty$. Now, let us consider $q=\frac{p}{2}$. Using the monotone
convergence theorem, we pass to the limit in (4.25) as $k\rightarrow\infty$,
deriving estimate (4.23).$\hfill\hfill\blacksquare$

Proof of Theorem 4.2. Existence. The proof is split into four steps.
Step 1. Estimates and convergences, related with the projection operator.

Let $P_{n}:\widetilde{W}\rightarrow W_{n}$ be the orthogonal projection
defined by

$P_{n}y=\sum_{j=1}^{n}\widetilde{c}_{j}\widetilde{e}_{j}\quad\text{ with
}\quad\text{
}\widetilde{c}_{j}=\left(y,\widetilde{e}_{j}\right)_{\widetilde{W}},\qquad\forall
y\in\widetilde{W},$

where
$\\{\widetilde{e}_{j}=\frac{1}{\sqrt{\lambda_{j}}}e_{j}\\}_{j=1}^{\infty}$ is
the orthonormal basis of $\ \widetilde{W}.$

It is easy to check that

$P_{n}y=\sum_{j=1}^{n}c_{j}e_{j}\quad\text{ with }\quad\text{
}c_{j}=\left(y,e_{j}\right)_{V},\qquad\forall y\in\widetilde{W}.$

By Parseval’s identity we have that

$||P_{n}y||_{V}\leq||y||_{V},\text{ }\quad\forall y\in V,$
$||P_{n}y||_{\widetilde{W}}\leq||y||_{\widetilde{W}}\qquad\mbox{ and }\qquad
P_{n}y\longrightarrow y\qquad\mbox{ strongly in }\ \widetilde{W},\text{
}\quad\forall y\in\widetilde{W}.$

Considering an arbitrary $Z\in L^{2}(\Omega\times(0,T);\widetilde{W})),$ we
have

$||P_{n}Z||_{\widetilde{W}}\leq||Z||_{\widetilde{W}}\qquad\mbox{ and }\qquad
P_{n}Z(\omega,t)\rightarrow Z(\omega,t)\qquad\mbox{ strongly in }\
\widetilde{W},$

which are valid for $P$-a. e. $\omega\in\Omega$ and a.e. $t\in(0,T).$ Hence
Lebesgue’s dominated convergence theorem and the inequality

$||Z||_{V}\leq C||Z||_{\widetilde{W}}\qquad\text{for any }Z\in\widetilde{W}$

imply

$\displaystyle P_{n}Z$ $\displaystyle\longrightarrow$ $\displaystyle
Z\qquad\mbox{ strongly in }\ L^{2}(\Omega\times(0,T);\widetilde{W})),$
$\displaystyle P_{n}Z$ $\displaystyle\longrightarrow$ $\displaystyle
Z\qquad\mbox{ strongly in }\ L^{2}(\Omega\times(0,T);V)).$ (4.26)

Step 2. Passing to the limit in the weak sense.

We have

$\mathbb{E}\sup_{t\in[0,T]}\left\|Y_{n}(t)\right\|_{\widetilde{W}}^{2}\leq
C,\qquad\mathbb{E}\sup_{t\in[0,T]}\left\|Y_{n}(t)\right\|_{V}^{p}\leq C$
(4.27)

for some constants $C$ independent of the index $n$, by estimates (4.8) and
(4.23). Therefore there exists a suitable subsequence $Y_{n}$, which is
indexed by the same index $n$, for simplicity of notations, such that

$\displaystyle Y_{n}$ $\displaystyle\rightharpoonup$ $\displaystyle
Y\qquad\mbox{ *-weakly in }\ L^{2}(\Omega,L^{\infty}(0,T;\widetilde{W})),$
$\displaystyle Y_{n}$ $\displaystyle\rightharpoonup$ $\displaystyle
Y\qquad\mbox{ *-weakly in }\ L^{p}(\Omega,L^{\infty}(0,T;V)).$ (4.28)

Moreover, we have

$\displaystyle P_{n}Y$ $\displaystyle\longrightarrow$ $\displaystyle
Y\qquad\mbox{ strongly in }\ L^{2}(\Omega\times(0,T);\widetilde{W})),$
$\displaystyle P_{n}Y$ $\displaystyle\longrightarrow$ $\displaystyle
Y\qquad\mbox{ strongly in }\ L^{2}(\Omega\times(0,T);V)).$ (4.29)

Let us introduce the operator $B:\widetilde{W}\times
V\rightarrow\widetilde{W}^{\ast},$ defined as

$\left(B\left(y,z\right),\phi\right)=\left(\mathrm{curl}\,\upsilon(y)\times
z,\phi\right)\qquad\mbox{ for any }\ y,\phi\in\widetilde{W}\qquad\mbox{ and }\
z\in V,$

and state some useful properties. Relation (3.23) gives

$\left(B\left(y,z\right),\phi\right)=-\left(B\left(y,\phi\right),z\right),\quad\quad\left(B\left(y,z\right),z\right)=0,$
(4.30)

and (3.24), (3.25) yield

$\|B\left(y,z\right)\|_{\widetilde{W}^{\ast}}\leq
C\left\|z\right\|_{V}\left\|y\right\|_{\widetilde{W}},$ (4.31)
$\|B\left(y,z\right)\|_{\widetilde{W}^{\ast}}\leq
C\left\|y\right\|_{V}\left\|z\right\|_{\widetilde{W}}.$ (4.32)

From (3.26) there exists a fixed constant $C_{1}$ such that

$\|B\left(y,y\right)\|_{\widetilde{W}^{\ast}}\leq
C_{1}\left\|y\right\|_{V}^{2},$ (4.33)

then

$\left\|B(y,y)\right\|_{L^{2}(\Omega\times(0,T);\widetilde{W}^{\ast}))}\leq
C_{1}\left\|y\right\|_{L^{4}(\Omega,L^{4}(0,T;V))}^{2}.$ (4.34)

On the other hand, taking into account (2.5), (4.28), there exist operators
$B^{\ast}(t)$ and $G^{\ast}(t)$, such that

$\displaystyle B(Y_{n},Y_{n})$ $\displaystyle\rightharpoonup$ $\displaystyle
B^{\ast}(t)\qquad\mbox{ weakly in }\
L^{2}(\Omega\times(0,T);\widetilde{W}^{\ast}),$ $\displaystyle G(t,Y_{n})$
$\displaystyle\rightharpoonup$ $\displaystyle G^{\ast}(t)\qquad\mbox{ weakly
in }\ L^{2}(\Omega\times(0,T);V^{m}).$ (4.35)

Passing on the limit $n\rightarrow\infty$ in equation (4.4), we derive that
the limit function $Y$ satisfies the stochastic differential equation

$d\left(\upsilon\left(Y\right),\phi\right)=\left[\left(\nu\Delta
Y+U,\phi\right)-\langle
B^{\ast}(t),\phi\rangle\right]\,dt\vspace{2mm}+\left(G^{\ast}(t),\phi\right)\,dW_{t},\qquad\forall\phi\in\widetilde{W}.$
(4.36)

Step 3. Deduction of strong convergences, as $n\rightarrow\infty$, depending
on the stopping times $\tau_{M}.$

In order to prove that the limit process $Y$ satisfy equation (4.1), we will
adapt the methods in [7] (see also [34]). Let us introduce a sequence
$(\tau_{M})$, $M\in\mathbb{N}$, of stopping times defined by

$\tau_{M}(\omega)=\inf\\{t\geq 0:\|Y(t)\|_{\widetilde{W}}(\omega)\geq
M\\}\wedge T,\qquad\omega\in\Omega.$

Taking the difference of (4.4) and (4.36), we deduce

$\displaystyle d\left(P_{n}Y-Y_{n},e_{i}\right)_{V}$ $\displaystyle=$
$\displaystyle\left[\left(\nu\Delta(Y-Y_{n}),e_{i}\right)+\langle
B(Y_{n},Y_{n})-B^{\ast}(t),e_{i}\rangle\right]\,dt$ (4.37)
$\displaystyle\vspace{2mm}-\left(G(t,Y_{n})-G^{\ast}(t),e_{i}\right)\,dW_{t},$

which is valid for any $e_{i}\in W_{n},$ $i=1,...,n.$

By applying Itô’s formula, equation (4.37) gives

$\displaystyle d(P_{n}Y-Y_{n},e_{i})_{V}^{2}$ $\displaystyle=$ $\displaystyle
2\left(P_{n}Y-Y_{n},e_{i}\right)_{V}\left[\left(\nu\Delta(Y-Y_{n}),e_{i}\right)+\langle
B(Y_{n},Y_{n})-B^{\ast}(t),e_{i}\rangle\right]\,dt\vspace{2mm}$
$\displaystyle-$ $\displaystyle
2\left(P_{n}Y-Y_{n},e_{i}\right)_{V}\left(G(t,Y_{n})-G^{\ast}(t),e_{i}\right)\,dW_{t}+|\left(G(t,Y_{n})-G^{\ast}(t),e_{i}\right)|^{2}\,dt,$

and summing over the index $i$ from $1$ to $n,$ we derive

$\displaystyle
d\left(||P_{n}Y-Y_{n}||_{V}^{2}\right)+\left(4\nu||D(P_{n}Y-Y_{n})||_{2}^{2}+2\nu\gamma\left\|P_{n}Y-Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)dt$
$\displaystyle=2\nu(\Delta(P_{n}Y-Y),P_{n}Y-Y_{n})\,dt$
$\displaystyle+2\langle B(Y_{n},Y_{n})-B^{\ast}(t),P_{n}Y-Y_{n}\rangle\,dt$
$\displaystyle+\sum_{i=1}^{n}|\left(G(t,Y_{n})-G^{\ast}(t),e_{i}\right)|^{2}\,dt-2\left(G(t,Y_{n})-G^{\ast}(t),P_{n}Y-Y_{n}\right)\,dW_{t}.$
(4.38)

Let us notice that

$\displaystyle\langle B(Y_{n},Y_{n})$
$\displaystyle-B^{\ast}(t),P_{n}Y-Y_{n}\rangle=\langle
B(Y_{n},Y_{n})-B(P_{n}Y,P_{n}Y),P_{n}Y-Y_{n}\rangle$ $\displaystyle+\langle
B(P_{n}Y,P_{n}Y)-B(Y,Y),P_{n}Y-Y_{n}\rangle+\langle
B(Y,Y)-B^{\ast}(t),P_{n}Y-Y_{n}\rangle$ $\displaystyle=I_{1}+I_{2}+I_{3}.$
(4.39)

Using (4.30), we derive

$\displaystyle I_{1}$ $\displaystyle=\langle
B(Y_{n},Y_{n})-B(P_{n}Y,P_{n}Y),P_{n}Y-Y_{n}\rangle$ $\displaystyle=\langle
B(Y_{n},Y_{n})-B(Y_{n},P_{n}Y)+B(Y_{n},P_{n}Y)-B(P_{n}Y,P_{n}Y),P_{n}Y-Y_{n}\rangle$
$\displaystyle=-\langle B(Y_{n},P_{n}Y-Y_{n}),P_{n}Y-Y_{n}\rangle-\langle
B(P_{n}Y-Y_{n},P_{n}Y),P_{n}Y-Y_{n}\rangle$ $\displaystyle=\langle
B(P_{n}Y-Y_{n},P_{n}Y-Y_{n}),P_{n}Y\rangle,$

which along with (4.33) implies

$|I_{1}|\leq
C_{1}\left\|Y\right\|_{\widetilde{W}}\left\|P_{n}Y-Y_{n}\right\|_{V}^{2}.$
(4.40)

For the term $I_{2}$, we have

$\displaystyle|I_{2}|$ $\displaystyle=\left|\langle
B(P_{n}Y,P_{n}Y)-B(Y,Y),P_{n}Y-Y_{n}\rangle\right|$
$\displaystyle\leq\left\|B(P_{n}Y,P_{n}Y)-B(Y,Y)\right\|_{\widetilde{W}^{\ast}}\left\|P_{n}Y-Y_{n}\right\|_{\widetilde{W}},$

and for every $\phi\in W$, it follows from (4.31) and (4.32) that

$\displaystyle\|B(P_{n}Y,P_{n}Y)-B(Y,Y)\|_{\widetilde{W}^{\ast}}$
$\displaystyle\leq\|B(P_{n}Y-Y,P_{n}Y)\|_{\widetilde{W}^{\ast}}+\|B(Y,P_{n}Y-Y\|_{\widetilde{W}^{\ast}}$
$\displaystyle\leq C\|Y\|_{\widetilde{W}}\|P_{n}Y-Y\|_{V}.$

and consequently, we obtain

$|I_{2}|\leq
C\left\|Y\right\|_{\widetilde{W}}\left\|P_{n}Y-Y\right\|_{V}\left\|P_{n}Y-Y_{n}\right\|_{\widetilde{W}}.$
(4.41)

On the other hand, denoting by $\widetilde{G}_{n}$, $\widetilde{G}$ and
$\widetilde{G}^{\ast}$ the solutions of the Stokes system (3.6) for
$f=G(t,Y_{n})$, $f=G(t,Y)$ and $f=G^{\ast}(t)$, respectively, we have

$\left(G(t,Y_{n})-G^{\ast}(t),e_{i}\right)=(\widetilde{G}_{n}-\widetilde{G}^{\ast},e_{i})_{V},\qquad
i=1,2,\dots,n.$

Then

$\sum_{i=1}^{n}|\left(G(t,Y_{n})-G^{\ast}(t),e_{i}\right)|^{2}=\|P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}.$

The standard relation $x^{2}=(x-y)^{2}-y^{2}+2xy\;$ allows to write

$\displaystyle\|P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}$
$\displaystyle=\|P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}\|_{V}^{2}-\|P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}$
$\displaystyle+2(P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast},P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast})_{V}.$

From the properties of the solutions of the Stokes system (3.6) and (2.5), we
have

$\|P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}\|_{V}^{2}\leq\|\widetilde{G}_{n}-\widetilde{G}\|_{V}^{2}\leq\|G(t,Y_{n})-G(t,Y)\|_{L^{2}}^{2}\leq
K\left\|Y_{n}-Y\right\|_{V}^{2},$

then, for the fixed constant $C_{2}=2K$, we have

$\displaystyle\|P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}$
$\displaystyle\leq
K\left\|Y_{n}-Y\right\|_{V}^{2}-\|P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}$
$\displaystyle+2(P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast},P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast})_{V}$
$\displaystyle\leq
C_{2}\left\|Y_{n}-P_{n}Y\right\|_{V}^{2}+C\left\|P_{n}Y-Y\right\|_{V}^{2}-\|P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}$
$\displaystyle+2(P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast},P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast})_{V}.$
(4.42)

The positive constants $C_{1}$ and $C_{2}$ in (4.40) and (4.42), are
independent of $n$ and may depend on the data: the domain $\mathcal{O},$ the
regularity of $\Gamma,$ the physical constants $\nu$, $\alpha$, $\gamma$, $K.$

Let us notice that from the convergence results (4.26)-(4.29), (4.35), we can
guess that by passing to the limit in equation (4.38), in a suitable way, as
$n\rightarrow\infty,$ all terms containing $P_{n}Y-Y$ will vanish on the right
hand side of equality (4.38), according to relations (4.26), (4.41) and
(4.42). But the terms with $Y_{n}-P_{n}Y$ will remain. Fortunately, these
terms can be eliminated by the introduction of the auxiliary function

$\xi(t)=e^{-C_{2}t-2C_{1}\int_{0}^{t}\left\|Y\right\|_{\widetilde{W}}ds}.$

Now, applying Itô’s formula in equality (4.38), we get

$\displaystyle
d\left(\xi(t)||P_{n}Y-Y_{n}||_{V}^{2}\right)+\xi(t)\left(4\nu||D(P_{n}Y-Y_{n})||_{2}^{2}+2\nu\gamma\left\|P_{n}Y-Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)dt$
$\displaystyle=2\nu\xi(t)(\Delta(P_{n}Y-Y),P_{n}Y-Y_{n})\,dt$
$\displaystyle+2\xi(t)\langle
B(Y_{n},Y_{n})-B^{\ast}(t),P_{n}Y-Y_{n}\rangle\,dt+\xi(t)\sum_{i=1}^{n}|\left(G(t,Y_{n})-G^{\ast}(t),e_{i}\right)|^{2}\,dt$
$\displaystyle-2\xi(t)\left(G(t,Y_{n})-G^{\ast}(t),P_{n}Y-Y_{n}\right)\,dW_{t}$
$\displaystyle-
C_{2}\xi(t)||P_{n}Y-Y_{n}||_{V}^{2}\,dt-2C_{1}\xi(t)\left\|Y\right\|_{\widetilde{W}}||P_{n}Y-Y_{n}||_{V}^{2}\,dt.$
(4.43)

Integrating it over the time interval $(0,\tau_{M}(\omega)),$ taking the
expectation and applying estimates (4.39), (4.40), (4.42), we deduce

$\displaystyle\mathbb{E}\left(\xi(\tau_{M})||P_{n}Y(\tau_{M})-Y_{n}(\tau_{M})||_{V}^{2}\right)+\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)\|P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}ds$
$\displaystyle+\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)\left(4\nu||D(P_{n}Y-Y_{n})||_{2}^{2}+2\nu\gamma\left\|P_{n}Y-Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)ds$
$\displaystyle\leq
2\nu\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)(\Delta(P_{n}Y-Y),P_{n}Y-Y_{n})\,ds$
$\displaystyle+2\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)I_{2}\,ds+2\mathbb{E}\int_{0}^{\tau_{M}}~{}\xi(s)I_{3}\,ds$
$\displaystyle+\mathbb{E}\int_{0}^{\tau_{M}}~{}\xi(s)\left[C\left\|P_{n}Y-Y\right\|_{V}^{2}+2(P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast},P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast})_{V}\right]\,ds$
$\displaystyle=J_{1}+J_{2}+J_{3}+J_{4}.$

In what follows we show that, for each $M\in\mathbb{N}$, the right hand side
of this inequality goes to zero, as $n\rightarrow\infty$.

Using (4.27)-(4.28) and the properties of the projection $P_{n},$ we have

$\displaystyle|J_{1}|$
$\displaystyle=\bigl{|}2\nu\mathbb{E}\int_{0}^{T}\xi(s)(1_{[0,\tau_{M}]}(s)\Delta(P_{n}Y-Y),P_{n}Y-Y_{n})\,ds\bigr{|}$
$\displaystyle\leq
C\|P_{n}Y-Y\|_{L^{2}(\Omega\times(0,T);H^{2})}\|P_{n}Y-Y_{n}\|_{L^{2}(\Omega\times(0,T);H^{2})}$
$\displaystyle\leq
C\|P_{n}Y-Y\|_{L^{2}(\Omega\times(0,T);H^{2})}\left(\|Y\|_{L^{2}(\Omega\times(0,T);H^{2})}+\|Y_{n}\|_{L^{2}(\Omega\times(0,T);H^{2})}\right)$
$\displaystyle\leq C\|P_{n}Y-Y\|_{L^{2}(\Omega\times(0,T);H^{2})},$

which goes to zero, as $n\rightarrow\infty$, by (4.29). Taking into account
estimates (4.8), (4.41) and knowing that
$1_{[0,\tau_{M}]}(s)\left\|Y(s)\right\|_{\widetilde{W}}\leq M$, P - a. e. in
$\Omega$, we deduce that

$\displaystyle|J_{2}|$ $\displaystyle\leq
2\mathbb{E}\left|\int_{0}^{\tau_{M}}\xi(s)I_{2}\,ds\right|$ $\displaystyle\leq
2\mathbb{E}\int_{0}^{T}\xi(s)1_{[0,\tau_{M}]}(s)\left\|Y\right\|_{\widetilde{W}}\left\|P_{n}Y-Y\right\|_{\widetilde{W}}\left(\left\|Y\right\|_{\widetilde{W}}+\left\|Y_{n}\right\|_{\widetilde{W}}\right)ds|$
$\displaystyle\leq
CM\|P_{n}Y-Y\|_{L^{2}(\Omega\times(0,T);\widetilde{W})}\left(\|Y\|_{L^{2}(\Omega\times(0,T);\widetilde{W})}+\|Y_{n}\|_{L^{2}(\Omega\times(0,T);\widetilde{W})}\right)$
$\displaystyle\leq CM\|P_{n}Y-Y\|_{L^{2}(\Omega\times(0,T);\widetilde{W})},$

which also converges to zero by (4.29).

Convergences (4.28) and (4.29) give that

$P_{n}Y-Y_{n}\rightarrow 0\qquad\text{ weakly in
}L^{2}(\Omega\times(0,T),\widetilde{W}),$

then for any operator $A\in L^{2}(\Omega\times(0,T),\widetilde{W}^{\ast})$ we
have

$\mathbb{E}\int_{0}^{T}\langle A,P_{n}Y-Y_{n}\rangle\,ds\rightarrow
0\qquad\text{as }n\rightarrow\infty.$

Since the function $1_{[0,\tau_{M}]}(s)\xi(s)$ is bounded and independent of
the space variable, we have

$\displaystyle\|1_{[0,\tau_{M}]}(s)\xi(s)\left(B(Y,Y)-B^{\ast}\right)\|_{L^{2}(\Omega\times(0,T),\widetilde{W}^{\ast})}^{2}$
$\displaystyle\leq$ $\displaystyle
C\left(\|B(Y,Y)\|_{L^{2}(\Omega\times(0,T),\widetilde{W}^{\ast})}^{2}+\|B^{\ast}\|_{L^{2}(\Omega\times(0,T),\widetilde{W}^{\ast})}^{2}\right)\leq
C,$

by (4.27), (4.34) and (4.35). Therefore

$\displaystyle J_{3}$
$\displaystyle=2\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)I_{3}\,ds$
$\displaystyle=2\mathbb{E}\int_{0}^{T}\langle
1_{[0,\tau_{M}]}(s)\xi(s)\left(B(Y,Y)-B^{\ast}(s)\right),P_{n}Y-Y_{n}\rangle\,ds\rightarrow
0\qquad\text{as }n\rightarrow\infty.$

We write

$\displaystyle J_{4}$
$\displaystyle=\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)\left[C\left\|P_{n}Y-Y\right\|_{V}^{2}+2(P_{n}\widetilde{G}_{n}-P_{n}\widetilde{G}^{\ast},P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast})_{V}\right]\,ds$
$\displaystyle=C\mathbb{E}\int_{0}^{T}1_{[0,\tau_{M}]}(s)\xi(s)\left\|P_{n}Y-Y\right\|_{V}^{2}ds$
$\displaystyle+C\mathbb{E}\int_{0}^{T}1_{[0,\tau_{M}]}(s)\xi(s)(\widetilde{G}_{n}-\widetilde{G}^{\ast},P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast})_{V}\,ds.$

Due to (4.29), we have

$\left|\mathbb{E}\int_{0}^{T}1_{[0,\tau_{M}]}(s)\xi(s)\left\|P_{n}Y-Y\right\|_{V}^{2}ds\right|\leq\mathbb{E}\int_{0}^{T}\left\|P_{n}Y-Y\right\|_{V}^{2}ds\rightarrow
0,\text{ as }n\rightarrow\infty.$

Now, for each stochastic process $Z\in L^{2}(\Omega\times(0,T),H)$ let us
denote by $\widetilde{Z}$ the solution of the modified Stokes problem (3.6).
We recall that the operator

$A:Z\rightarrow\widetilde{Z}$

is linear and continuous operator from $L^{2}(\Omega\times(0,T),V)$ into
$L^{2}(\Omega\times(0,T),V)$. Applying Proposition A.2 in [7] (see also
references therein), it follows that $A$ is continuous for the weak topology,
namely if $Z_{n}\rightharpoonup Z$ weakly in $L^{2}(\Omega\times(0,T),V)$,
then $\widetilde{Z}_{n}\rightharpoonup\widetilde{Z}$ weakly in
$L^{2}(\Omega\times(0,T),V).$ Due to this property and the convergence result
(4.35), we obtain

$\widetilde{G}_{n}-\widetilde{G}^{\ast}\rightharpoonup\widetilde{G}-\widetilde{G}^{\ast}\text{\qquad\
weakly in }L^{2}(\Omega\times(0,T),V^{m}).$ (4.44)

Moreover, we have $\widetilde{G}-\widetilde{G}^{\ast}\in\widetilde{W}^{m}$ and

$\displaystyle
P_{n}(\widetilde{G}-\widetilde{G}^{\ast})\rightarrow\widetilde{G}-\widetilde{G}^{\ast}\text{\qquad
strongly in }L^{2}(\Omega\times(0,T),\widetilde{W}^{m}),$ $\displaystyle
P_{n}(\widetilde{G}-\widetilde{G}^{\ast})\rightarrow\widetilde{G}-\widetilde{G}^{\ast}\text{\qquad
strongly in }L^{2}(\Omega\times(0,T),V^{m}).$

Then we can verify that

$1_{[0,\tau_{M}]}(s)\xi(s)\,P_{n}(\widetilde{G}-\widetilde{G}^{\ast})\rightarrow
1_{[0,\tau_{M}]}(s)\xi(s)\,(\widetilde{G}-\widetilde{G}^{\ast})\text{\qquad
strongly in }L^{2}(\Omega\times(0,T),V^{m}).$ (4.45)

As a consequence of (4.44) and (4.45), we have

$\displaystyle\mathbb{E}\int_{0}^{T}1_{[0,\tau_{M}]}(s)\xi(s)(\widetilde{G}_{n}-\widetilde{G}^{\ast},P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast})_{V}\,ds$
$\displaystyle=\mathbb{E}\int_{0}^{T}(\widetilde{G}_{n}-\widetilde{G}^{\ast},1_{[0,\tau_{M}]}(s)\xi(s)P_{n}(\widetilde{G}-\widetilde{G}^{\ast}))_{V}\,ds\rightarrow
0,\text{ as }n\rightarrow\infty.$

Collecting all convergence results, we obtain the following strong
convergences, depending on the stopping times $\tau_{M},$

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\left(\xi(\tau_{M})||P_{n}Y(\tau_{M})-Y_{n}(\tau_{M})||_{V}^{2}\right)$
$\displaystyle=0,$
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)\|P_{n}\widetilde{G}-P_{n}\widetilde{G}^{\ast}\|_{V}^{2}ds$
$\displaystyle=0,$
$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\int_{0}^{\tau_{M}}\xi(s)\left(4\nu||D(P_{n}Y-Y_{n})||_{2}^{2}+2\nu\gamma\left\|P_{n}Y-Y_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)ds$
$\displaystyle=0,$

for each $M\in\mathbb{N}$. Since there exists a strictly positive constant
$\mu,$ such that $\mu\leq 1_{[0,\tau_{M}]}(s)\xi(s)\leq 1,$ it follows that

$\lim_{n\rightarrow\infty}\mathbb{E}\int_{0}^{\tau_{M}}||P_{n}Y-Y_{n}||_{V}^{2}ds=0\quad\text{implying}\quad\lim_{n\rightarrow\infty}\mathbb{E}\int_{0}^{\tau_{M}}||Y-Y_{n}||_{V}^{2}ds=0$

by (4.29). In addition, considering (4.26), we have

$\mathbb{E}\int_{0}^{\tau_{M}}\|\widetilde{G}-\widetilde{G}^{\ast}\|_{V}^{2}ds=0.$
(4.46)

Step 4. Identification of  $B^{\ast}(t)$ with $B(Y,Y)$ and  $G^{\ast}(t)$ with
$G(t,Y).$

Now, we are able to show that the limit function $Y$ satisfies equation (4.1).
Integrating equation (4.36) on the time interval $(0,\tau_{M}\wedge t)$, we
derive

$\displaystyle\left(\upsilon\left(Y(\tau_{M}\wedge
t)\right),\phi\right)-\left(\upsilon\left(Y_{0}\right),\phi\right)$
$\displaystyle=$ $\displaystyle\int_{0}^{\tau_{M}\wedge
t}\left[\left(\nu\Delta Y+U,\phi\right)-\langle
B^{\ast}(s),\phi\rangle\right]\,ds$ (4.47)
$\displaystyle+\int_{0}^{\tau_{M}\wedge
t}\left(G^{\ast}(s),\phi\right)\,dW_{s}$

for any $\phi\in\widetilde{W}.$

From (4.46) it follows that

$1_{[0,\tau_{M}]}(t)\widetilde{G}=1_{[0,\tau_{M}]}(t)\widetilde{G}^{\ast}\qquad\text{a.
e. in }\Omega\times(0,T),$

which implies

$1_{[0,\tau_{M}]}(t)G(t,Y)=1_{[0,\tau_{M}]}(t)G^{\ast}(t)\qquad\text{a. e. in
}\Omega\times(0,T)$ (4.48)

by (3.6). Since $B(Y_{n},Y_{n})-B(Y,Y)=B(Y_{n},Y_{n}-Y)-B(Y_{n}-Y,Y),$ using
(4.31)-(4.32), we have

$\|B(Y_{n},Y_{n})-B(Y,Y)\|_{\widetilde{W}^{\ast}}\leq
C\left(\|Y_{n}\|_{\widetilde{W}}+\|Y\|_{\widetilde{W}}\right)\|Y_{n}-Y\|_{V}.$

Then for any $\varphi\in L^{\infty}(\Omega\times(0,T);\widetilde{W}),$ using
(4.27), (4.28)

$\displaystyle\bigl{|}\mathbb{E}\int_{0}^{T}1_{[0,\tau_{M}]}(s)\langle
B(Y_{n},Y_{n})$ $\displaystyle-B(Y,Y),\ \varphi\rangle\,ds\bigr{|}$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T}1_{[0,\tau_{M}]}(s)\left(\|Y_{n}\|_{\widetilde{W}}+\|Y\|_{\widetilde{W}}\right)\|Y_{n}-Y\|_{V}\|\varphi\|_{\widetilde{W}}\,ds$
$\displaystyle\leq
C\|\varphi\|_{L^{\infty}(\Omega\times(0,T),\widetilde{W})}\mathbb{E}\int_{0}^{T}\left(\|Y_{n}\|_{\widetilde{W}}+\|Y\|_{\widetilde{W}}\right)\|Y_{n}-Y\|_{V}\,ds$
$\displaystyle\leq
C\|\varphi\|_{L^{\infty}(\Omega\times(0,T),\widetilde{W})}\left(\mathbb{E}\int_{0}^{\tau_{M}}\|Y_{n}-Y\|_{V}^{2}\,ds\right)^{\frac{1}{2}}\rightarrow
0,\qquad\text{as }n\rightarrow\infty.$

Taking into account (4.35)1 and that the space
$L^{\infty}(\Omega\times(0,T);\widetilde{W})$ is dense in
$L^{2}(\Omega\times(0,T);\widetilde{W}),$ we obtain

$1_{[0,\tau_{M}]}(s)B^{\ast}(s)=1_{[0,\tau_{M}]}(s)B(Y,Y)\qquad\text{a. e. in
}\Omega\times(0,T).$ (4.49)

By introducing identities (4.48), (4.49) in equation (4.47), it follows that

$\displaystyle\left(\upsilon\left(Y(\tau_{M}\wedge
t)\right),\phi\right)-\left(\upsilon\left(Y_{0}\right),\phi\right)$
$\displaystyle=$ $\displaystyle\int_{0}^{\tau_{M}\wedge
t}\left[\left(\nu\Delta Y+U,\phi\right)-\langle B(Y,Y),\phi\rangle\right]\,ds$
(4.50) $\displaystyle+\int_{0}^{\tau_{M}\wedge
t}\left(G(s,Y),\phi\right)\,dW_{s}.$

Now, reasoning as in (4.22) we have $\tau_{M}\rightarrow T$ a. e. in $\Omega$.
We can pass to the limit in each term of equation (4.50) in
$L^{1}(\Omega\times(0,T))$, as $M\rightarrow\infty$ , by applying the Lebesgue
dominated convergence theorem and the Burkholder-Davis-Gundy inequality for
the last (stochastic) term, deriving equation (4.1) a. e. in
$\Omega\times(0,T)$.

Let us notice that the estimates for $Y_{n}$ in Lemmas 4.3 and 4.4 are valid
also for the limit process $Y$, due to convergence (4.28).

The uniqueness of the solution $Y$ follows from the stability result that we
will show in the next section. $\hfill\hfill\blacksquare$

##  5 Stability result for solutions

In this section we will establish a stability property for solutions of the
stochastic second grade fluid model (2.1). In spite of the existence result
with $H^{3}$ space regularity, the difference of two solutions can only be
estimated (with respect to the initial data) in space $H^{2}$. It will be
convenient to introduce the following norm on the space $W$

$\|y\|_{W}=\|y\|_{V}+\|\mathbb{P}\upsilon(y)\|_{2},\qquad y\in W.$

As a consequence of (3.18) and (2.3) this norm $\|\cdot\|_{W}$ is equivalent
to $\|\cdot\|_{H^{2}}$.

###### Theorem 5.1

Assume that for some $\ 4\leq p<\infty$

$U_{1},U_{2}\in L^{p}(\Omega,L^{p}(0,T;H(\mathrm{curl};\mathcal{O}))),\qquad
Y_{1,0},Y_{2,0}\in L^{p}(\Omega,V)\cap L^{2}(\Omega,\widetilde{W})$

and

$Y_{1},Y_{2}\in L^{2}(\Omega,L^{\infty}(0,T;\widetilde{W}))\cap
L^{p}(\Omega,L^{\infty}(0,T;V))$

are corresponding solutions of (2.1) in the sense of the variational equality
$(\ref{var_form_state})$.

Then there exist strictly positive constants $C_{3}$ and $C,$ which depend
only on the data (the domain $\mathcal{O},$ the regularity of $\Gamma,$ the
physical constants $\nu$, $\alpha$, $\gamma$, $K)$ and satisfy the following
estimate

$\mathbb{E}\sup_{s\in[0,t]}\xi(s)\left\|Y_{1}(s)-Y_{2}(s)\right\|_{W}^{2}\leq
C(\mathbb{E}\left\|Y_{1,0}-Y_{2,0}\right\|_{W}^{2}+\mathbb{E}\int_{0}^{t}\xi(s)\left\|U_{1}(s)-U_{2}(s)\right\|_{2}^{2}\,ds)$
(5.1)

with the function $\xi$ defined as

$\xi(t)=e^{-C_{3}\int_{0}^{t}\left(\left\|Y_{1}\right\|_{H^{3}}+\left\|Y_{2}\right\|_{H^{3}}\right)ds}.$

Proof. The process $Y=Y_{1}-Y_{2}$ satisfies the system

$\left\\{\begin{array}[]{ll}d\upsilon(Y)=(\nu\Delta
Y-\mathrm{curl}\,\upsilon(Y)\times Y_{2}-\mathrm{curl}\,\upsilon(Y_{1})\times
Y-\nabla\pi+U)\,dt&\\\
\qquad\qquad+\left(G(t,Y_{1})-G(t,Y_{2})\right)\,dW_{t}&\\\ \nabla\cdot
Y=0&\quad\mbox{in}\ \mathcal{O}\times(0,T),\vspace{2mm}\\\
Y\cdot\mathrm{n}=0,\qquad\left[2(\mathrm{n}\cdot DY)+\gamma
Y\right]\cdot\mathrm{\tau}=0&\quad\mbox{on}\ \Gamma\times(0,T),\vspace{2mm}\\\
Y(0)=Y_{0}=Y_{1,0}-Y_{2,0}&\quad\mbox{in}\ \mathcal{O},\end{array}\right.$
(5.2)

where $\pi=\pi_{1}-\pi_{2}$ and $U=U_{1}-U_{2}$. Applying the operator
$\left(I-\alpha\mathbb{P}\Delta\right)^{-1}$ to equation (5.2)1 we deduce a
stochastic differential equation for $Y$, then with the help of Itô’s formula
we obtain

$\displaystyle d\left\|Y\right\|_{V}^{2}$
$\displaystyle=2\left(\left(\nu\Delta Y-\mathrm{curl}\,\upsilon(Y)\times
Y_{2}-\mathrm{curl}\,\upsilon(Y_{1})\times
Y+U\right),Y\right)\,dt\vspace{2mm}$
$\displaystyle+\|\widetilde{G}_{1}-\widetilde{G}_{2}\|_{V}^{2}\,dt+2\left(G(t,Y_{1})-G(t,Y_{2}),Y\right)\,dW_{t},$
(5.3)

where $\widetilde{G}_{i}$ are the solutions of the modified Stokes problem
(3.6) with $f=G(t,Y_{i}),\,\ i=1,2.$ Hence, using assumption (2.5), we have

$\|\widetilde{G}_{1}-\widetilde{G}_{2}\|_{V}^{2}\leq
C\|G(t,Y_{1})-G(t,Y_{2})\|_{2}^{2}\leq C\|Y\|_{V}^{2}.$

Taking into account property (3.23), estimate (3.26) and the Young inequality,
we derive

$\displaystyle\left\|Y(t)\right\|_{V}^{2}$
$\displaystyle+\int_{0}^{t}\left(4\nu\left\|DY\right\|_{2}^{2}+2\nu\gamma\left\|Y\right\|_{L^{2}(\Gamma)}^{2}\right)\,ds\leq\left\|Y_{0}\right\|_{V}^{2}+C\int_{0}^{t}\left\|Y_{2}\right\|_{H^{3}}\left\|Y\right\|_{Y}^{2}\,ds$
$\displaystyle+\int_{0}^{t}\|U\|_{2}^{2}\,ds+C\int_{0}^{t}\|Y\|_{V}^{2}\,ds+2\int_{0}^{t}\left(G(s,Y_{1})-G(s,Y_{2}),Y\right)\,dW_{s}\vspace{2mm}.$
(5.4)

The Itô formula also gives

$\displaystyle d\left\|\mathbb{P}\upsilon\left(Y\right)\right\|_{2}^{2}$
$\displaystyle=2\left(\nu\Delta Y-\mathrm{curl}\,\upsilon(Y)\times
Y_{2}-\mathrm{curl}\,\upsilon(Y_{1})\times
Y+U,\mathbb{P}\upsilon\left(Y\right)\right)\,dt\vspace{2mm}$
$\displaystyle+\|G(t,Y_{1})-G(t,Y_{2})\|_{2}^{2}\,dt+2\left(G(t,Y_{1})-G(t,Y_{2}),\mathbb{P}\upsilon(Y)\right)\,dW_{t}.$

Estimating the nonlinear term

$\left|\left(\mathrm{curl}\,\upsilon(Y)\times
Y_{2}+\mathrm{curl}\,\upsilon(Y_{1})\times
Y,\mathbb{P}\upsilon(Y)\right)\right|\leq
C_{3}\left(\left\|Y_{2}\right\|_{H^{3}}+\left\|Y_{1}\right\|_{H^{3}}\right)\left(\left\|Y\right\|_{V}^{2}+\left\|\mathbb{P}\upsilon(Y)\right\|_{2}^{2}\right)$

and using (2.5), we deduce

$\displaystyle\left\|\mathbb{P}\upsilon\left(Y(t)\right)\right\|_{2}^{2}+\frac{2\nu}{\alpha}\int_{0}^{t}\left\|\upsilon\left(Y\right)\right\|_{2}^{2}\,ds$
$\displaystyle\leq\left\|\mathbb{P}\upsilon\left(Y_{0}\right)\right\|_{2}^{2}+2\int_{0}^{t}\left(\frac{\nu}{\alpha}Y+U,\mathbb{P}\upsilon\left(Y\right)\right)\,ds$
$\displaystyle+C_{3}\int_{0}^{t}\left(\left\|Y_{2}\right\|_{H^{3}}+\left\|Y_{1}\right\|_{H^{3}}\right)\left(\left\|Y\right\|_{V}^{2}+\left\|\mathbb{P}\upsilon(Y)\right\|_{2}^{2}\right)\,ds$
$\displaystyle+C\int_{0}^{t}\|Y\|_{V}^{2}\,ds+2\int_{0}^{t}\left(G(s,Y_{1})-G(s,Y_{2}),\mathbb{P}\upsilon(Y)\right)\,dW_{s}.$

Summing this inequality with (5.4), we obtain

$\displaystyle\left\|Y(t)\right\|_{V}^{2}+\left\|\mathbb{P}\upsilon\left(Y(t)\right)\right\|_{2}^{2}$
$\displaystyle\leq\left\|Y_{0}\right\|_{V}^{2}+\left\|\mathbb{P}\upsilon\left(Y_{0}\right)\right\|_{2}^{2}$
$\displaystyle+\int_{0}^{t}\|U\|_{2}^{2}\,ds+C\int_{0}^{t}\left(\|Y\|_{V}^{2}+\|\mathbb{P}\upsilon\left(Y\right)\|_{2}^{2}\right)\,ds$
$\displaystyle+C_{3}\int_{0}^{t}\left(\left\|Y_{2}\right\|_{H^{3}}+\left\|Y_{1}\right\|_{H^{3}}\right)\left(\left\|Y\right\|_{V}^{2}+\left\|\mathbb{P}\upsilon(Y)\right\|_{2}^{2}\right)\,ds$
$\displaystyle+2\int_{0}^{t}\left(G(s,Y_{1})-G(s,Y_{2}),Y+\mathbb{P}\upsilon(Y)\right)\,dW_{s}\vspace{2mm}.$

Taking
$\xi(t)=e^{-C_{3}\int_{0}^{t}\left(\left\|Y_{2}\right\|_{H^{3}}+\left\|Y_{2}\right\|_{H^{3}}\right)ds}$
and applying Itô’s formula, then we easily obtain

$\displaystyle\xi(t)\left(\left\|Y(t)\right\|_{V}^{2}+\left\|\mathbb{P}\upsilon\left(Y(t)\right)\right\|_{2}^{2}\right)$
$\displaystyle\leq\left\|Y_{0}\right\|_{V}^{2}+\left\|\mathbb{P}\upsilon\left(Y_{0}\right)\right\|_{2}^{2}$
$\displaystyle+\int_{0}^{t}\xi(s)\|U\|_{2}^{2}\,ds+C\int_{0}^{t}\xi(s)\left(\|Y\|_{V}^{2}+\|\mathbb{P}\upsilon\left(Y\right)\|_{2}^{2}\right)\,ds$
$\displaystyle+2\int_{0}^{t}\xi(s)\left(G(s,Y_{1})-G(s,Y_{2}),Y+\mathbb{P}\upsilon(Y)\right)\,dW_{s}.$
(5.5)

The Burkholder-Davis-Gundy inequality gives

$\displaystyle\mathbb{E}\sup_{s\in[0,t]}\left|\int_{0}^{s}\xi(r)\left(G(r,Y_{1})-G(r,Y_{2}),Y+\mathbb{P}\upsilon(Y)\right)\,dW_{r}\right|$
$\displaystyle\leq\mathbb{E}\left(\int_{0}^{t}\xi^{2}(s)\left\|Y\right\|_{V}^{2}\left\|Y+\mathbb{P}\upsilon(Y)\right\|_{2}^{2}\,ds\right)^{\frac{1}{2}}\vspace{2mm}$
$\displaystyle\leq\varepsilon\mathbb{E}\sup_{s\in[0,t]}\xi(s)\left(\left\|Y\right\|_{V}+\left\|\mathbb{P}\upsilon(Y)\right\|_{2}\right)\vspace{2mm}$
$\displaystyle+C_{\varepsilon}\mathbb{E}\int_{0}^{t}\xi(s)\left(\left\|Y\right\|_{V}+\left\|\mathbb{P}\upsilon(Y)\right\|_{2}\right)\,ds.$

Substituting this inequality with $\varepsilon=\frac{1}{2}$ in (5.5) and
taking the supremum on the time interval $[0,t]$ and the expectation, we
deduce

$\mathbb{E}\sup_{s\in[0,t]}\xi(s)\left\|Y(s)\right\|_{W}^{2}\leq\mathbb{E}\left\|Y_{0}\right\|_{W}^{2}+\mathbb{E}\int_{0}^{t}\xi(s)\|U\|_{2}^{2}\,ds+C\mathbb{E}\int_{0}^{t}\xi(s)\|Y(s)\|_{W}^{2}\,ds.$

Hence Gronwall’s inequality yields (5.1). $\hfill\hfill\blacksquare$

Acknowledgment We would like to thank the anonymous Referees for relevant
suggestions and comments which contributed to improve the article.

The work of F. Cipriano was partially supported by the Fundação para a Ciência
e a Tecnologia (Portuguese Foundation for Science and Technology) through the
project UID/MAT/00297/2013 (Centro de Matemática e Aplicações).

## References

  * [1] S. Albeverio, Z. Brzeźniak, J.-L. Wu, _Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients_. J. Math. Anal. Appl., 371 (2010), 309-322.
  * [2] A. V. Busuioc, D. Iftimie, _A non-Newtonian fluid with Navier boundary conditions_. J. Dynam. Diff. Eq., 18 (2006), 357-379.
  * [3] A. V. Busuioc, D. Iftimie, M. C. Lopes Filho, H. J. Nussenzveig Lopes, _Incompressible Euler as a limit of complex fluid models with Navier boundary conditions_. J. Differential Equations, 252 (2012), 624-640.
  * [4] A. V. Busuioc, T. S. Ratiu, _The second grade fluid and averaged Euler equations with Navier-slip boundary conditions_. Nonlinearity, 16 (2003), 1119-1149.
  * [5] A. Bensoussan,  _Stochastic Navier-Stokes equations._ Acta Appl. Math., 38 (1995), 267-304.
  * [6] A. Bensoussan, R. Temam, _Equations stochastiques du type Navier-Stokes._ J. Funct. Anal., 13 (1973), 195-222.
  * [7] H. I. Breckner, _Approximation and optimal control of the stochastic Navier-Stokes equations._ Ph.D. Thesis, Halle (Saale), 1999.
  * [8] R. Camassa, D. D. Holm, _An integrable shallow water equation with peaked solitons_. Phys. Rev. Lett., 71 (1993), 1661-1664.
  * [9] N.V. Chemetov, F. Cipriano, _Inviscid limit for Navier–Stokes equations indomains with permeable boundaries_. Applied Math. Letters, 33 (2014), 6–11.
  * [10] N.V. Chemetov, F. Cipriano, _Boundary layer problem: Navier-Stokes equationsand Euler equations_. Nonlinear Analysis: Real World Applications, 14 (6) (2013), 2091–2104.
  * [11] N.V. Chemetov, F. Cipriano, _The Inviscid Limit for the Navier–StokesEquations with Slip Condition on Permeable Walls_. J. Nonlinear Science, 23 (5) (2013), 731-750.
  * [12] D. Cioranescu, V. Girault, _Weak and classical solutions of a family of second grade fluids_. Int. J. Nonlinear Mech., 32 (1997), 317-335.
  * [13] D. Cioranescu, E. H. Ouazar, _Existence and uniqueness for fluids of second grade_. Nonlinear Partial Differential Equations and Their Applications (Collège de France Seminar, Paris, 1982/1983). vol. 4, Boston, MAPitman, (1984), 178-197.
  * [14] F. Cipriano, I. Torrecilla, _Inviscid limit for 2D stochastic Navier-Stokes equations_. Journal of Stochastic Processes and their Applications,  125 (6) (2015), 2405-2426.
  * [15] T. Clopeau, A. Mikelic, R. Robert, _On the vanishing viscosity limit for the $2D$ incompressible Navier-Stokes equations with the friction type boundary conditions_. Nonlinearity, 11 (1998), 1625-1636.
  * [16] G. Da Prato, A. Debussche, _2D stochastic Navier–Stokes equations with a time-periodic forcing term._ J. Dynam. Dif. Eq., 20 (2) (2008) 301–335.
  * [17] G. Deugoue, M. Sango, _Weak solutions to stochastic 3D Navier-Stokes-_ $\alpha$_model of turbulence: Asymptotic behavior._ J. Math. Anal. Appl., 384 (2011), 49–62.
  * [18] J. E. Dunn, R. L. Fosdick, _Thermodynamics, stability and boundedness of fluids of complexity $2$ and fluids of second grade_. Arch. Rational Mech. Anal., 56 (1974), 191–252.
  * [19] J. E. Dunn, K. R. Rajagopal, _Fluids of differential type: Critical review and thermodynamical analysis_. Int J. Eng. Sci., 33 (1995), 689–729.
  * [20] F. Flandoli, _An introduction to 3d stochastic fluid dynamics. In SPDE in Hydrodynamic: Recent Progress and Prospects._ Vol. 1942, Lecture Notes in Mathematics. Springer Berlin/Heidelberg, (2008), 51-150.
  * [21] C. Foias, D. D. Holm, E. S. Titi, _The Navier-Stokes- $\alpha$ model of fluid turbulence_. Physica D, 153
  * [22] P. Grisvard, _Elliptic Problems in Nonsmooth Domains._ Pitman Publishing Inc, 1985.
  * [23] D. D. Holm, J. E. Marsden, R. S. Ratiu, _Euler-Poincaré models of ideal fluids with nonlinear dissipation_ , Phys. Rev. Lett. 349 (1998), pp. 4173-4177.
  * [24] D. D. Holm, J. E. Marsden, R. S. Ratiu, _The Euler-Poincaré equations and semi-direct products with applications to continuum theories_ , Adv. Math. 137 (1998), pp. 1-81.
  * [25] D. Iftimie, G. Planas, _Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions_. Nonlinearity, 19 (2006), 899-918.
  * [26] J. P. Kelliher, _Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane_. SIAM J. Math. Anal., 38 (2006), 210-232.
  * [27] J. S. Linshiz, E. S. Titi, _On the convergence rate of the Euler- $\alpha$, an inviscid second-grade complex fluid, model to Euler equations_. J. Stat. Phys., 138 (2010), 305-332.
  * [28] M.C. Lopes Filho, H.J. Nussenzveig Lopes, G. Planas, _On the inviscid limit for $2D$ incompressible flow with Navier friction condition_. SIAM J. Math. Anal., 36 (2005), 1130-1141.
  * [29] M.C. Lopes Filho, H. J. Nussenzveig Lopes, E. S. Titi, A. Zang, _Convergence of the $2D$ Euler-$\alpha$ to Euler equations in the Dirichlet case: Indifference to boundary layers_. Physica D, 292-293 (1) (2015) 51-61.
  * [30] R. Mikulevicius, B. L. Rozovskii,_Global_ $L^{\emph{2}}-$_solutions of stochastic Navier-Stokes equations._ Ann. Probab., 33 (1) (2005), 137-176.
  * [31] E. H. Ouazar, _Sur les Fluides de Second Grade_. Thèse 3ème Cycle, Université Pierre et Marie Curie, 1981.
  * [32] P.A. Razafimandimby, M. Sango, _Weak solutions of a stochastic model for two-dimensional second grade fluids._ Bound. Value Probl. (electr. version) 2010 (2010), 47 pages.
  * [33] P.A. Razafimandimby, M. Sango, _Asymptotic behaviour of solutions of stochastic evolution equations for second grade fluids._ C. R. Acad. Sci. Paris, Ser. I, 348 (13-14) (2010), 787-790.
  * [34] P. A. Razafimandimby, M. Sango, _Strong solution for a stochastic model of two-dimensional second grade fluids: Existence, uniqueness and asymptotic behavior_. Nonlinear Analysis: theory methods & appl., 75 (2012), 4251-4270.
  * [35] R. S. Rivlin, J. L. Ericksen, _Stress-deformation relations for isotropic materials_. Arch. Rational Mech. Anal., 4 (1955), 323-425.
  * [36] M. Sango,_Density dependent stochastic Navier–Stokes equations with non Lipschitz random forcing_. Rev. Math. Phys. 22 (6) (2010), 669–697.