# Theory of weak asymptotic autonomy of pullback stochastic weak attractors
and its applications to 2D stochastic Euler equations driven by multiplicative
noise

Kush Kinra111Funding: CSIR, India, File No. 09/143(0938)/2019-EMR-I
<EMAIL_ADDRESS>Manil T. Mohan222Funding: DST-INSPIRE: IFA17-MA110
maniltmohan<EMAIL_ADDRESS>1Department of Mathematics,
Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee,
Uttarakhand 247667, India

###### Abstract

The two dimensional stochastic Euler equations (EE) perturbed by a linear
multiplicative noise of Itô type on the bounded domain $\mathcal{O}$ have been
considered in this work. Our first aim is to prove the existence of global
weak (analytic) solutions for stochastic EE when the divergence free initial
data $\boldsymbol{u}^{*}\in\mathbb{H}^{1}(\mathcal{O})$, and the external
forcing
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{H}^{1}(\mathcal{O}))$.
In order to prove the existence of weak solutions, a vanishing viscosity
technique has been adopted. In addition, if $\mathrm{curl}\
\boldsymbol{u}^{*}\in\mathrm{L}^{\infty}(\mathcal{O})$ and $\mathrm{curl}\
\boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$,
we establish that the global weak (analytic) solution is unique. This work
appears to be the first one to discuss the existence and uniqueness of global
weak (analytic) solutions for stochastic EE driven by linear multiplicative
noise. Secondly, we prove the existence of a pullback stochastic weak
attractor for stochastic non-autonomous EE using the abstract theory available
in the literature. Finally, we propose an abstract theory for weak asymptotic
autonomy of pullback stochastic weak attractors. Then we consider the 2D
stochastic EE perturbed by a linear multiplicative noise as an example to
discuss that how to prove the weak asymptotic autonomy for concrete stochastic
partial differential equations. As EE do not contain any dissipative term, the
results on attractors (deterministic and stochastic) are available in the
literature for dissipative (or damped) EE only. Since we are considering
stochastic EE without dissipation, all the results of this work for 2D
stochastic EE perturbed by a linear multiplicative noise are totally new.

###### keywords:

2D stochastic Euler equations, pullback stochastic weak attractors, weak
asymptotic autonomy of pullback stochastic weak attractors, bounded domains.
MSC2020: Primary 35B41, 35Q35; Secondary 37L55, 37N10, 35R60

##  1 Introduction

The motion of gases and liquids are described by fluid dynamics. If the
material density of the fluid is a constant (that is, the divergence free
condition is satisfied by the fluid velocity), then we call it as an
incompressible fluid. The Motion of a viscous incompressible fluid flow is
described by the well-known Navier-Stokes equations. In the case of inviscid
flow (zero viscosity), the motion of the ideal incompressible fluid in a
bounded domain $\mathcal{O}\subset\mathbb{R}^{2}$ is described by the Euler
equations (EE)

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\partial\boldsymbol{u}}{\partial
t}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\nabla p&=\boldsymbol{f},\ \ \
\text{ in }\ \mathcal{O}\times(\tau,\infty),\\\
\nabla\cdot\boldsymbol{u}&=0,\hskip 14.22636pt\text{ in }\
\mathcal{O}\times[\tau,\infty),\\\ \end{aligned}}\right.$ (1.1)

where $\boldsymbol{u}(x,t)\in\mathbb{R}^{2}$ stands for the velocity field,
$p(x,t)\in\mathbb{R}$ denotes the pressure field and
$\boldsymbol{f}(x,t)\in\mathbb{R}^{2}$ is an external forcing, for all
$(x,t)\in\mathcal{O}\times(\tau,\infty)$.

###  1.1 The stochastic model

This work aims to study the dynamics of the stochastic counterpart of the
system (1.1) on a bounded domain. Let $\mathcal{O}\subset\mathbb{R}^{2}$ be a
bounded domain with $\mathrm{C}^{2}$-boundary $\partial\mathcal{O}$. Given
$\tau\in\mathbb{R}$, we consider the following non-autonomous 2D stochastic EE
on $\mathcal{O}$:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\/\mathrm{d}\/\boldsymbol{u}+\mathopen{}\mathclose{{}\left[(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\nabla
p}\right]\/\mathrm{d}\/t&=\boldsymbol{f}\/\mathrm{d}\/t+\sigma\boldsymbol{u}\/\mathrm{d}\/\mathrm{W},\
\text{ in }\ \mathcal{O}\times(\tau,\infty),\\\
\nabla\cdot\boldsymbol{u}&=0,\hskip 65.44133pt\text{ in }\
\mathcal{O}\times[\tau,\infty),\\\ \boldsymbol{u}\cdot\boldsymbol{n}&=0,\hskip
65.44133pt\text{ on }\ \partial\mathcal{O}\times[\tau,\infty),\\\
\boldsymbol{u}|_{t=\tau}&=\boldsymbol{u}_{\tau},\hskip 59.75078pt\text{ in }\
\mathcal{O},\\\ \end{aligned}}\right.$ (1.2)

where the coefficient $\sigma>0$ is known as the _noise intensity_ and
$\boldsymbol{n}$ is the outward normal to $\partial\mathcal{O}$. Here, the
stochastic integral is understood in the sense of Itô and
$\mathrm{W}=\mathrm{W}(t,\omega)$ is a one-dimensional two-sided Wiener
process defined on a filtered probability space
$(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\in\mathbb{R}},\mathbb{P})$ (see
Subsection 3.2 below).

###  1.2 Literature survey

The derivation of the Euler equations can be found in the several books and
articles, for example, see [21, 50, 52], etc. The author in [38] proved that
there exists a unique classical solution of the non-stationary EE on a bounded
domain of the plane. The author in [45] proved the existence of a unique
global classical solution of 2D EE in a time-dependent domain (see [35] for
existence and uniqueness of weak solutions). The global well-posedness
(existence, uniqueness and continuous dependence on the data) of the system
(1.1) in $\mathbb{L}^{2}(\mathcal{O})$ and $\mathbb{H}^{1}(\mathcal{O})$ is
not known till date. Several mathematicians have taken Euler equations into
consideration and established that there exists a weak solution to the system
(1.1) with divergence free initial data
$\boldsymbol{u}_{\tau}\in\mathbb{H}^{1}(\mathcal{O})$ (cf. [2, 8, 37] etc.).
In addition, if the divergence free initial data
$\boldsymbol{u}_{\tau}\in\mathbb{H}^{1}(\mathcal{O})$ and $\mathrm{curl\
}\boldsymbol{u}_{\tau}\in\mathrm{L}^{\infty}(\mathcal{O})$, then there exists
a unique weak solution to the system (1.1) (cf. [2, 69, 70] etc.). For 3D EE,
we refer the interested readers to the works [28, 40, 57], etc., and
references therein. L. Onsager in 1949 conjectured that the energy of an ideal
incompressible flow will conserve if it is Hölder continuous with exponent
greater than 1/3, and is known as _Onsager’s conjecture_. If the domain is
either torus or the whole space (that is, in the absence of physical
boundary), the Onsager’s conjecture is proved in [22]. Also, the works [3, 4]
deal with the Onsager’s conjecture on bounded domains.

Let us now discuss the solvability results of 2D stochastic Euler equations.
The existence of solutions for 2D stochastic EE on torus is studied in the
work [13] using nonstandard analysis. The authors in the works [5, 7, 12, 39]
proved the existence of weak/martingale/strong solutions (in probabilistic
sense) for 2D stochastic EE on bounded domains. In addition, the uniqueness of
solutions of 2D stochastic EE driven by additive noise is also shown in [7,
39]. The local and global existence of smooth solutions for stochastic EE
driven by multiplicative noise have been demonstrated in [33]. Moreover, in
[33], the authors considered the stochastic EE with a linear multiplicative
noise of Itô type separately and proved the existence of a global smooth
solution. Some authors (for instance, see [26, 44, 51], etc.) have considered
2D as well as 3D stochastic EE driven by Lévy noise and proved the existence
of local smooth solutions. For 3D stochastic EE, we refer the interested
readers to the works [10, 36], etc., also. Recently, the existence of
invariant measures is established for 2D stochastic dissipative EE driven by
additive noise on a torus in [9]. For 2D stochastic EE driven by a linear
multiplicative noise, the existence of invariant measures will be addressed in
a future work.

The concepts of global (pullback) attractors of deterministic autonomous (non-
autonomous) dynamical systems (DS) can be found in the books [15, 16, 54, 59],
etc. The notion of attractors of deterministic DS was generalized to
global/pullback random attractors for autonomous/non-autonomous compact/non-
compact stochastic/random DS in the articles [11, 23, 24, 61], etc. The
dynamics of deterministic dissipative EE can be found in the works [17, 18,
19, 20, 72], etc. Later, these abstract theories have been extensively applied
in the articles [41, 62, 64, 67], etc. to several physically relevant
stochastic models.

###  1.3 Motivations

In order to apply the classical theory of global/pullback attractors
(deterministic or random), one needs to define a semigroup (or evolution
process) on a metric space $(\mathrm{X},\/\mathrm{d}\/_{\mathrm{X}})$ which is
continuous on $\mathrm{X}$. In this work, we are considering a stochastic DS
associated with stochastic EE perturbed by a white noise which is neither
dissipative nor has regularizing effects. We show the existence of a unique
global weak (analytic) solution (see Definition 3.2 below) with initial data
from space
$\mathcal{W}=\\{\boldsymbol{v}\in\mathbb{V}:\nabla\land\boldsymbol{v}\in\mathrm{L}^{\infty}(\mathcal{O})\\}$
(see Subsection 3.1 for functional spaces). Furthermore, the stochastic flow
is neither continuous nor compact with respect to the topology
$\/\mathrm{d}\/_{\mathcal{W}}(\boldsymbol{u},\boldsymbol{v})=\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb{V}}+\|\nabla\land\boldsymbol{u}-\nabla\land\boldsymbol{v}\|_{\mathrm{L}^{\infty}(\mathcal{O})}$.
But the stochastic flow is compact with respect to an another topology
$\delta(\boldsymbol{u},\boldsymbol{v})=\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb{H}}$
on $\mathcal{W}$ which is weaker than $\/\mathrm{d}\/_{\mathcal{W}}$ in some
sense. For such kind of stochastic flows, the author in [6] came up with a
notion of attractor with respect to a pair of topologies
$(\mathrm{d}_{\mathcal{W}},\delta)$ known as stochastic weak attractor (SWA)
where the attractor is $\mathrm{d}_{\mathcal{W}}$-bounded and $\delta$-compact
(see Theorem 2.5 below). As the uniqueness holds for the stochastic EE when
the initial data is in $\mathcal{W}$, the stochastic flow generated by
stochastic EE satisfies the evolution property. But it does not enjoys the
continuity in phase space $\mathcal{W}$ which is used to prove the invariance
property of attractors in the classical theory of global/pullback random
attractors. Despite of continuity in phase space $\mathcal{W}$, the author in
[6] considered an another assumption which assures the invariance and the
existence of a $\mathrm{d}_{\mathcal{W}}$-bounded and $\delta$-compact SWA
(see Hypothesis 2.1 below). Moreover, this abstract theory was applied to
prove the existence of SWA for 2D stochastic dissipative EE driven by additive
noise in a bounded domain ([6, Theorem 4.6]). We use the abstract theory
developed in [6] to prove the existence of a minimal SWA for 2D stochastic
Euler equations perturbed by a linear multiplicative noise (see Theorem 4.6
below).

The effect of time-dependent forcing creates the non-autonomous feature of
dynamical systems which could be the most important feature distinguishing
from autonomous. Fundamentally, if the forcing $\boldsymbol{f}(x,t)$ (time-
dependent) in (1.2) asymptotically approaches to a time-independent forcing
$\boldsymbol{f}_{\infty}$ in some sense, then the random dynamics of non-
autonomous system converges to the random dynamics of autonomous system. This
phenomena is called the asymptotic autonomy or asymptotically autonomous
robustness of random dynamics. From now onward, we use the terminology
pullback deterministic (stochastic) weak attractors for non-autonomous
deterministic (stochastic) dynamical systems and deterministic (stochastic)
weak attractors for autonomous deterministic (stochastic) dynamical systems.
Our ambition is to establish the asymptotically autonomous robustness of
pullback stochastic weak attractors of (1.2) when the non-autonomous forcing
$\boldsymbol{f}_{\infty}$ satisfies the following hypotheses:

######  Hypothesis 1.1.

$\boldsymbol{f}(\cdot)$ converges to $\boldsymbol{f}_{\infty}$ in the
following sense:

$\displaystyle\lim_{\tau\to+\infty}\int_{0}^{\infty}\|\boldsymbol{f}(t+\tau)-\boldsymbol{f}_{\infty}\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}\/\mathrm{d}\/t=0.$

######  Hypothesis 1.2.

For the time-dependent external forcing
$\boldsymbol{f}\in\mathrm{L}^{2}_{\emph{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$
and
$\nabla\land\boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$,
there exist two numbers $\delta_{1},\delta_{2}\in[0,\frac{\sigma^{2}}{2})$
such that for all $t\in\mathbb{R}$

$\displaystyle\sup_{s\geq\tau}\int_{-\infty}^{0}\bigg{[}e^{\delta_{1}t}\|\boldsymbol{f}(t+s)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+e^{\delta_{1}t}\|\nabla\land\boldsymbol{f}(t+s)\|^{2}_{\mathrm{L}^{2}(\mathcal{O})}+e^{\delta_{2}t}\|\nabla\land\boldsymbol{f}(t+s)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\bigg{]}\/\mathrm{d}\/t<+\infty,$
(1.3)

where $\nabla\land\boldsymbol{f}=\mathrm{curl}\boldsymbol{f}$.

######  Remark 1.3.

1\. An example of Hypotheses 1.1 and 1.2 is
$\boldsymbol{f}(x,t)=\boldsymbol{f}_{\infty}(x)e^{t}+\boldsymbol{f}_{\infty}(x)$
with $\boldsymbol{f}_{\infty}\in\mathbb{L}^{2}(\mathcal{O})$ and
$\nabla\land\boldsymbol{f}_{\infty}\in\mathrm{L}^{\infty}(\mathcal{O})$.

2\. Hypothesis 1.1 implies that
$\sup\limits_{s\geq\tau}\int_{-\infty}^{0}e^{kt}\|\boldsymbol{f}(t+s)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}\/\mathrm{d}\/t<+\infty$
for all $k>0$ and $\tau\in\mathbb{R}$ (cf. [65, Lemma 4.1]). Therefore the
first term in the integral is finite for
$\delta_{1}\in(0,\frac{\sigma^{2}}{2})$ under Hypothesis 1.1.

The concept of asymptotically autonomous robustness of pullback random
attractors has been considered in several articles, see [42, 49, 63, 66, 68,
71], etc. for 2D Navier-Stokes equations, 2D MHD equations, $g$-Navier-Stokes
equations and Brinkman-Forchheimer equations etc. As we have already
discussed, the classical theory for pullback random attractors depends on the
continuity of the flow map in the phase space $\mathcal{W}$ (cf. [61]). But
the flow map of the system (1.2) is not continuous in $\mathcal{W}$. Motivated
from the work [6], we introduce a notion of asymptotic autonomy of pullback
stochastic weak attractors relative to a pair of
$(\mathrm{d}_{\mathcal{W}},\delta)$ topologies or weak asymptotic autonomy of
pullback stochastic weak attractors. By the weak asymptotic autonomy of
pullback stochastic weak attractors, we means that the minimal pullback
stochastic weak attractors
$\\{\mathscr{A}(t,\omega)\\}_{t\in\mathbb{R},\omega\in\Omega}$
($\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact) converges
towards the minimal stochastic weak attractor
$\\{\mathscr{A}_{\infty}(\omega)\\}_{\omega\in\Omega}$
($\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact) in the following
sense:

$\displaystyle\lim_{t\to+\infty}\mathrm{dist}_{\mathcal{W}}^{\delta}(\mathscr{A}(t,\omega),\mathscr{A}_{\infty}(\omega))=0,\
\ \mathbb{P}\text{-a.e. }\omega\in\Omega,$ (1.4)

where, $\mathrm{dist}_{\mathcal{W}}^{\delta}(\cdot,\cdot)$ denotes the
Hausdorff semi-distance between two non-empty subsets of the Banach space
$\mathcal{W}$ with respect to the metric $\delta$, that is, for non-empty sets
$A,B\subset\mathcal{W}$

$\mathrm{dist}_{\mathcal{W}}^{\delta}(A,B)=\sup_{a\in A}\inf_{b\in
B}\delta(a,b).$

We prove that three sufficient conditions guarantee the weak asymptotic
autonomy of pullback stochastic weak attractors, see Theorem 2.5 below. Then
we apply this abstract theory to 2D stochastic EE driven by a linear
multiplicative noise to show how to check the sufficient conditions for
concrete stochastic partial differential equations.

###  1.4 Novelties

To the best of our knowledge, there are no works available in the literature
for the random dynamics of 2D stochastic EE. Most of the works available in
the literature related to the random dynamics and asymptotic analysis are for
2D deterministic/stochastic dissipative EE, cf. [6, 8, 9, 19, 20, 72] etc.
Since we are not considering any kind of damping term in stochastic EE, the
linear multiplicative structure of the Itô type noise coefficient helps us to
prove the results of this work. This work appears to be the first one
concerning the random dynamics of 2D stochastic EE perturbed by a linear
multiplicative noise (both Itô and Stratonovich noise). The major aims and
novelties of this work are:

  * (i)

Existence and uniqueness of the global weak (analytic) solution of stochastic
Euler system (1.2) by using a vanishing viscosity method and compactness
arguments.

  * (ii)

Existence of a minimal pullback stochastic weak attractor for the non-
autonomous and a minimal stochastic weak attractor for the autonomous
stochastic Euler system (1.2). We achieve this goal by using the abstract
theory developed in [6].

  * (iii)

Weak asymptotic autonomy of pullback stochastic weak attractors of the
stochastic Euler system (1.2). In order to obtain this, we first establish an
abstract theory of weak asymptotic autonomy of pullback stochastic weak
attractors in Section 2 and then apply this to the system (1.2).

###  1.5 Difficulties and approaches

Since Euler equations do not have any regularizing term, the existence of
global weak (analytic) solutions has been proved by using a vanishing
viscosity method. Firstly, we convert the stochastic Euler system (1.2) into
an equivalent pathwise deterministic system (see (3.5) below) with the help of
an stationary change of variable (see (3.4) below). Next, we consider a
Navier-Stokes type system (see system (3.11) below) and obtain the existence
of global weak (analytic) solutions to the Euler system (1.2) as viscosity
coefficient approaches to zero. We prove the existence of global weak
(analytic) solutions of the stochastic Euler system (1.2) when the divergence
free initial data $\boldsymbol{u}_{\tau}\in\mathbb{H}^{1}(\mathcal{O})$, and
external forcing
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$
and
$\nabla\land\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{2}(\mathcal{O}))$.
In order to obtain the uniqueness of weak solutions, we need more regular
divergence free initial data and external forcing. We establish the existence
of a unique global weak (analytic) solution when divergence free initial data
$\boldsymbol{u}_{\tau}\in\mathbb{H}^{1}(\mathcal{O})$,
$\nabla\land\boldsymbol{u}_{\tau}\in\mathrm{L}^{\infty}(\mathcal{O})$,
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$
and
$\nabla\land\boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$.

For the existence of pullback stochastic weak attractors, we borrowed the
abstract theory introduced in [6]. The linear structure of Itô type
multiplicative noise plays a crucial role to show the existence of
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact absorbing set. On
contrast to the works available in the literature [8, 9], the presence of
linear multiplicative noise helped us to prove the results for the system
(1.2) without a linear damping term. Since the weak asymptotic autonomy
concept is not available in the literature, we develop an abstract theory in
Section 2 and then apply it to show the weak asymptotic autonomy of pullback
stochastic weak attractors for the system (1.2).

###  1.6 Outline

The rest of the sections are organized as follows: In the next section, we
propose an abstract theory for weak asymptotic autonomy of pullback stochastic
weak attractors. In Section 3, we prove the existence of a unique global weak
(analytic) solution of the stochastic Euler system (1.2). The existence of
minimal pullback stochastic weak attractors for the stochastic Euler system
(1.2) is established in Section 4. In light of the abstract theory developed
in Section 2, we demonstrate the weak asymptotic autonomy of pullback
stochastic weak attractors in the final section.

##  2 Abstract theory

The abstract theory of weak global attractors for autonomous dynamical systems
was developed in [8]. Later, the author in [6] extended this theory to non-
autonomous deterministic (stochastic) dynamical systems named as deterministic
(stochastic) weak attractors. Let us first recall the abstract theory
available in the work [6] for pullback stochastic weak attractors. Then we
develop an abstract theory for weak asymptotic autonomy of pullback stochastic
weak attractors.

###  2.1 Pullback stochastic weak attractors and stochastic weak attractors

Let $(\mathcal{W},\/\mathrm{d}\/_{\mathcal{W}})$ be a complete metric space,
$\delta$ be an another metric on $\mathcal{W}$ and
$(\Omega,\mathscr{F},\mathbb{P})$ be a probability space. Let
$\\{S(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}:\mathcal{W}\to\mathcal{W}$
be a family of mappings which satisfies the following hypothesis:

######  Hypothesis 2.1.

Let $\\{S(t,\tau,\omega)\\}_{t\geq\tau,\
\omega\in\Omega}:\mathcal{W}\to\mathcal{W}$ be a family of mappings,
satisfying for $\mathbb{P}$-a.e. $\omega\in\Omega,$ the following two
properties:

  * (i)

If a sequence $\\{\boldsymbol{w}_{n}\\}_{n\in\mathbb{N}}$ is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-convergent to
$\boldsymbol{w}$ in $\mathcal{W},$ then $S(t,\tau,\omega)\boldsymbol{w}_{n}$
is $\delta$-convergent to $S(t,\tau,\omega)\boldsymbol{w}$, for all
$t\geq\tau$.

  * (ii)

_Evolution property_ :

$\displaystyle
S(t,r,\omega)S(r,\tau,\omega)\boldsymbol{w}=S(t,\tau,\omega)\boldsymbol{w},\text{
for all }\tau\leq r\leq t\text{ and for all }\boldsymbol{w}\in\mathcal{W}.$
(2.1)

Let $\mathscr{B}(\mathcal{W})$ be the $\sigma$-algebra of the metric space
$\mathcal{W}$ and $\mathcal{H}$ be a separable metric space such that
$\mathcal{W}\subset\mathcal{H}$ with continuous embedding. The metric $\delta$
is the metric endowed on $\mathcal{H}$ and $\mathcal{W}$ is closed in
$\mathcal{H}$. Let us assume that the following hypothesis holds:

######  Hypothesis 2.2.

For all $t\in\mathbb{R}$ and $\boldsymbol{w}\in\mathcal{W}$, the mapping
$(\tau,\omega)\mapsto S(t,\tau,\omega)\boldsymbol{w}$ is measurable from
$((-\infty,t]\times\Omega,\mathscr{B}((-\infty,t])\times\mathscr{F})$ to
$(\mathcal{W},\mathscr{B}(\mathcal{H})\cap\mathcal{W})$.

######  Definition 2.3.

For given $t\in\mathbb{R}$ and $\omega\in\Omega$, a bi-parametric set
$\mathcal{B}(t,\omega)$ is called _$\/\mathrm{d}\/_{\mathcal{W}}$ -bounded
absorbing set_ at time $t$ if for all $\/\mathrm{d}\/_{\mathcal{W}}$-bounded
set $B\subset\mathcal{W}$, there exists a time $\tau_{0}(B)$ depending on $B$
such that

$\displaystyle S(t,\tau,\omega)B\subset\mathcal{B}(t,\omega),\ \ \text{ for
all }\ \tau\leq\tau_{0}.$

######  Definition 2.4.

A family of mappings $\\{S(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$ is
called _asymptotically $\/\mathrm{d}\/_{\mathcal{W}}/\delta$ -compact_ if we
can find a measurable set $\widetilde{\Omega}\subset\Omega$ with
$\mathbb{P}(\widetilde{\Omega})=1$ such that for all $t\in\mathbb{R}$ and
$\omega\in\widetilde{\Omega}$, there exists a
$\/\mathrm{d}\/_{\mathcal{W}}$-absorbing set $\mathcal{B}(t,\omega)$ with
$\delta$-compactness.

The following theorem provides the existence of a measurable pullback
stochastic weak attractor proved in [6].

######  Theorem 2.5 ([6, Theorem 3.2]).

Let Hypotheses 2.1 and 2.2 hold. Assume that for each $t\in\mathbb{R}$, there
exists a measurable set $\Omega_{t}\subset\Omega$ with
$\mathbb{P}(\Omega_{t})=1$, such that for all $\omega\in\Omega_{t}$, there
exists a $\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact absorbing
set. Then for $\mathbb{P}$-a.e. $\omega\in\Omega_{t}$, the set

$\mathscr{A}(t,\omega)=\overline{\bigcup_{B\subset\mathcal{W}}\Omega^{\delta}(B,t,\omega)}^{\delta},\
\ \text{ where }\ \
\Omega^{\delta}(B,t,\omega)=\bigcap_{r<t}\overline{\bigcup_{\tau<r}S(t,\tau,\omega)B}^{\delta},$

is a measurable pullback stochastic weak attractor, that is,

  * (i)

$\mathscr{A}(t,\omega)$ is non-empty, it is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact,

  * (ii)

$\mathscr{A}(t,\omega)$ is invariant under the mapping $S(\cdot,\cdot,\cdot)$,
that is,

$S(t,\tau,\omega)\mathscr{A}(\tau,\omega)=\mathscr{A}(t,\omega),\ \text{ for
all }\ t\geq\tau,$

  * (iii)

For every $\/\mathrm{d}\/_{\mathcal{W}}$-bounded set $B\subset\mathcal{W}$,

$\lim_{\tau\to-\infty}\delta(S(t,\tau,\omega)B,\mathscr{A}(t,\omega))=0,\
\text{ for all }\ t\geq\tau,$

  * (iv)

$\mathscr{A}(t,\omega)$ is measurable with respect to the
$\mathbb{P}$-completion of $\mathscr{F}$.

  * (v)

$\mathscr{A}(t,\omega)$ is the minimal $\/\mathrm{d}\/_{\mathcal{W}}$-bounded
and $\delta$-closed set with property $\mathrm{(iii)}$, that is, if
$\widehat{\mathscr{A}}(t,\omega)$ is an another
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-closed set with the
property $\mathrm{(iii)}$, then
$\mathscr{A}(t,\omega)\subset\widehat{\mathscr{A}}(t,\omega)$.

###### Proof.

Properties $\mathrm{(i)}$-$\mathrm{(iv)}$ have been proved in [6, Theorem
3.2]. Let us prove the property $\mathrm{(v)}$. Note that if
$\widehat{\mathscr{A}}(t,\omega)$ satisfies property $\mathrm{(iii)}$, then we
have $\Omega^{\delta}(B,t,\omega)\subset\widehat{\mathscr{A}}(t,\omega)$ for
all $\/\mathrm{d}\/_{\mathcal{W}}$-bounded sets $B\subset\mathcal{W}$. Since
$\widehat{\mathscr{A}}(t,\omega)$ is $\/\mathrm{d}\/_{\mathcal{W}}$-bounded
and $\delta$-closed,
$\mathscr{A}(t,\omega)\subset\widehat{\mathscr{A}}(t,\omega)$. ∎

Let us now discuss the abstract theory for stochastic weak attractors.

######  Hypothesis 2.6.

There exists a group $\\{\theta_{t}\\}_{t\in\mathbb{R}}$ of measure preserving
transformations of $(\Omega,\mathscr{F},\mathbb{P})$ such that

$\displaystyle
T(t,\tau,\omega)\boldsymbol{w}=T(t-\tau,0,\theta_{\tau}\omega)\boldsymbol{w},\
\ \ \mathbb{P}\text{-a.s.},$

for all $t\geq\tau$ and $\boldsymbol{w}\in\mathcal{W}$.

If a family of mappings $\\{T(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$
satisfies Hypothesis 2.6, then the attractor does not depend on the time
parameter. The following theorem assures the existence of a measurable
stochastic weak attractor when Hypothesis 2.6 is satisfied (see [23, Theorem
2.2] also).

######  Theorem 2.7.

Let Hypotheses 2.1, 2.2 and 2.6 hold. Assume that for $\mathbb{P}$-a.e.
$\omega\in\Omega$, there exists a $\/\mathrm{d}\/_{\mathcal{W}}$-bounded and
$\delta$-compact absorbing set at time $0$. Then for $\mathbb{P}$-a.e.
$\omega\in\Omega$, the set

$\mathscr{A}_{\infty}(\omega)=\overline{\bigcup_{B\subset\mathcal{W}}\Omega^{\delta}(B,\omega)}^{\delta},\
\ \text{ where }\ \
\Omega^{\delta}(B,\omega)=\bigcap_{r<0}\overline{\bigcup_{\tau<r}T(0,\tau,\omega)B}^{\delta},$

is a measurable stochastic weak attractor, that is,

  * (i)

$\mathscr{A}_{\infty}(\omega)$ is non-empty, it is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact,

  * (ii)

$\mathscr{A}_{\infty}(\omega)$ is invariant under the mapping
$T(\cdot,\cdot,\cdot)$, that is,

$T(t,\tau,\omega)\mathscr{A}_{\infty}(\theta_{\tau}\omega)=\mathscr{A}_{\infty}(\theta_{t}\omega),\
\text{ for all }\ t\geq\tau,$

  * (iii)

For every $\/\mathrm{d}\/_{\mathcal{W}}$-bounded set $B\subset\mathcal{W}$,

$\lim_{\tau\to-\infty}\delta(T(t,\tau,\omega)B,\mathscr{A}_{\infty}(\theta_{t}\omega))=0,\
\text{ for all }\ t\geq\tau,$

  * (iv)

$\mathscr{A}_{\infty}(\omega)$ is measurable with respect to the
$\mathbb{P}$-completion of $\mathscr{F}$.

  * (v)

$\mathscr{A}_{\infty}(\omega)$ is the minimal
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-closed set with property
$\mathrm{(iii)}$, that is, if $\widehat{\mathscr{A}}_{\infty}(\omega)$ is an
another $\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-closed set with
the property $\mathrm{(iii)}$, then
$\mathscr{A}_{\infty}(\omega)\subset\widehat{\mathscr{A}}_{\infty}(\omega)$.

###  2.2 Weak asymptotic autonomy

In this subsection, our aim is to establish an abstract theory for the _weak
asymptotic autonomy_ of pullback stochastic weak attractors. In applications
to stochastic evolution equations perturbed by white noise, the sample space
$\Omega$ in the probability space $(\Omega,\mathscr{F},\mathbb{P})$ is

$\displaystyle\Omega=\\{\omega\in\mathrm{C}(\mathbb{R};\mathrm{Y}):\omega(0)=0\\}=:\mathrm{C}_{0}(\mathbb{R};\mathrm{Y}),$
(2.2)

where $\mathrm{Y}$ is some Banach space. In this case, there exists a group
$\\{\theta_{t}\\}_{t\in\mathbb{R}}$ of measure preserving transformations of
$(\Omega,\mathscr{F},\mathbb{P})$ which is given by ([1, 23])

$\displaystyle(\theta_{t}\omega)(s)=\omega(t+s)-\omega(t),\ \ \ \text{ for all
}\ \ s,t\in\mathbb{R}.$ (2.3)

In fact, we are proposing an abstract theory on those probability spaces where
the existence of a group $\\{\theta_{t}\\}_{t\in\mathbb{R}}$ of measure
preserving transformations is known.

######  Remark 2.8.

Note that if the existence of a group $\\{\theta_{t}\\}_{t\in\mathbb{R}}$ of
measure preserving transformations is known, it implies the existence of an
$\Omega_{t}$ in Theorem 2.5. In this case, one can consider $\Omega_{t}$ as
the set $\theta_{t}\Omega$.

Let $\\{\mathscr{A}(t,\omega)\\}_{t\in\mathbb{R},\omega\in\Omega}$ and
$\mathcal{B}(t,\omega)$ be a minimal pullback stochastic weak attractor and
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact absorbing set at
time $t$, respectively, associated with the non-autonomous stochastic
dynamical system $\\{S(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$ on
$\mathcal{W}$. Also, let
$\\{\mathscr{A}_{\infty}(\omega)\\}_{\omega\in\Omega}$ and
$\mathcal{B}_{\infty}(\omega)$ be a minimal stochastic weak attractor and
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact absorbing set,
respectively, associated with the autonomous stochastic dynamical system
$\\{T(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$ on $\mathcal{W}$. We show
that $\\{\mathscr{A}(t,\omega)\\}_{t\in\mathbb{R},\omega\in\Omega}$ upper
semicontinuously forward converges to
$\\{\mathscr{A}_{\infty}(\omega)\\}_{\omega\in\Omega}$ with respect to the
metric $\delta$.

######  Remark 2.9.

For the sample space $\Omega=\mathrm{C}_{0}(\mathbb{R};\mathrm{Y})$, as the
existence of a group $\\{\theta_{t}\\}_{t\in\mathbb{R}}$ of measure preserving
transformations of $(\Omega,\mathscr{F},\mathbb{P})$ is known, it is
remarkable that the autonomous stochastic dynamical system
$\\{T(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$ over
$(\Omega,\mathscr{F},\mathbb{P})$ satisfies Hypothesis 2.6.

The following theorem provides the abstract result for the weak asymptotic
autonomy of pullback stochastic weak attractors.

######  Theorem 2.10.

Let $\\{\mathscr{A}(\tau,\omega)\\}_{\tau\in\mathbb{R},\omega\in\Omega}$ and
$\\{\mathscr{A}_{\infty}(\omega)\\}_{\omega\in\Omega}$ are
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact. Suppose that the
following properties hold.

  * (i)

$\overline{\bigcup\limits_{t\geq\tau}\mathscr{A}(t,\omega)}^{\delta}$ is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact.

  * (ii)

The mappings $S(\cdot,\cdot,\cdot)$ and $T(\cdot,\cdot,\cdot)$ satisfy,

$\displaystyle\lim\limits_{\tau\to+\infty}\delta(S(t+\tau,\tau,\omega)\boldsymbol{w}_{\tau},T(t+\tau,\tau,\omega)\boldsymbol{w}_{0})=0,\
\text{ for all }t\geq 0\text{ and }\omega\in\widetilde{\Omega},$ (2.4)

whenever $\\{\boldsymbol{w}_{\tau}\\}_{\tau\in\mathbb{R}}$ is a
$d_{\mathcal{W}}$-bounded sequence, $\boldsymbol{w}_{0}\in\mathcal{W}$ and
$\lim\limits_{\tau\to+\infty}\delta(\boldsymbol{w}_{\tau},\boldsymbol{w}_{0})=0,$
where $\widetilde{\Omega}\subset\Omega$ is a $\theta_{t}$-invariant full
measure set.

  * (iii)

There exists a $t_{0}>0$ such that
$\mathcal{B}_{t_{0}}(\omega):=\overline{\bigcup\limits_{t\geq
t_{0}}\mathcal{B}(t,\omega)}^{\delta}$ is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact set.

Then, the weak asymptotic autonomy follows for $\mathscr{A}$, that is,

$\displaystyle\lim_{t\to+\infty}\mathrm{dist}_{\mathcal{W}}^{\delta}(\mathscr{A}(t,\omega),\mathscr{A}_{\infty}(\omega))=0,\
\ \mathbb{P}\text{-a.e. }\omega\in\Omega,$ (2.5)

where, $\mathrm{dist}_{\mathcal{W}}^{\delta}(\cdot,\cdot)$ denotes the
Hausdorff semi-distance between two non-empty subsets of the Banach space
$\mathcal{W}$ with respect to the metric $\delta$, that is, for non-empty sets
$A,B\subset\mathcal{W}$

$\mathrm{dist}_{\mathcal{W}}^{\delta}(A,B)=\sup_{a\in A}\inf_{b\in
B}\delta(a,b).$

###### Proof.

In order to prove the convergence given in (2.5), it is enough to show that
$\mathbb{P}(\Omega_{0})=1$, where

$\Omega_{0}=\mathopen{}\mathclose{{}\left\\{\omega\in\Omega:\lim\limits_{t\to+\infty}\mathrm{dist}_{\mathcal{W}}^{\delta}\mathopen{}\mathclose{{}\left(\mathscr{A}(t,\omega),\mathscr{A}_{\infty}(\omega)}\right)=0}\right\\}.$

Let us assume that $\mathbb{P}(\Omega_{0})<1,$ then
$\mathbb{P}(\Omega\backslash\Omega_{0})>0$. Let
$\Omega_{1}=(\Omega\backslash\Omega_{0})\cap\widetilde{\Omega}$, where
$\widetilde{\Omega}$ is the $\theta_{t}$-invariant full measure set given in
(2.4). Then $\mathbb{P}(\Omega_{1})>0$ and
$\theta_{s}\Omega_{1}\subset\widetilde{\Omega}$, for all $s\in\mathbb{R}$.

Let $\omega\in\Omega_{1}$ be fixed. Since $\omega\notin\Omega_{0}$, we can
find $\varepsilon>0$ and $t_{n}\geq 2t_{0}>0$ (WLOG) with
$t_{n}\uparrow\infty$ such that

$\displaystyle\mathrm{dist}_{\mathcal{W}}^{\delta}\mathopen{}\mathclose{{}\left(\mathscr{A}(t_{n},\omega),\mathscr{A}_{\infty}(\omega)}\right)\geq
3\varepsilon,\ \ \text{ for all }\ n\in\mathbb{N}.$

In view of $\/\mathrm{d}\/_{\mathcal{W}}$-boundedness and $\delta$-compactness
of $\mathscr{A}(t_{n},\omega)$, there exists
$\boldsymbol{w}_{n}\in\mathscr{A}(t_{n},\omega)$ such that

$\displaystyle\delta\mathopen{}\mathclose{{}\left(\boldsymbol{w}_{n},\mathscr{A}_{\infty}(\omega)}\right)=\mathrm{dist}_{\mathcal{W}}^{\delta}\mathopen{}\mathclose{{}\left(\mathscr{A}(t_{n},\omega),\mathscr{A}_{\infty}(\omega)}\right)\geq
3\varepsilon,\ \ \text{ for all }n\in\mathbb{N}.$ (2.6)

By the invariance property of pullback stochastic weak attractors, we also
have that there exists
$\boldsymbol{x}_{n}\in\mathscr{A}\mathopen{}\mathclose{{}\left(\frac{t_{n}}{2},\theta_{-\frac{t_{n}}{2}}\omega}\right)$
such that
$S(t_{n},\frac{t_{n}}{2},\omega)\boldsymbol{x}_{n}=\boldsymbol{w}_{n}.$ Now by
condition $(\mathrm{i})$, there exists an element
$\boldsymbol{x}\in\overline{\bigcup\limits_{t_{n}\geq
2t_{0}}\mathscr{A}\mathopen{}\mathclose{{}\left(\frac{t_{n}}{2},\theta_{-\frac{t_{n}}{2}}\omega}\right)}^{\delta}$
with

$\displaystyle\lim\limits_{n\to+\infty}\delta(\boldsymbol{x}_{n},\boldsymbol{x})=0.$
(2.7)

From (2.7) and condition $(\mathrm{ii})$, we get

$\displaystyle\lim_{n\to+\infty}\delta\mathopen{}\mathclose{{}\left(S\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x}_{n},T\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x}}\right)=0,$

which confirms the existence of an $N_{1}\in\mathbb{N}$ such that

$\displaystyle\delta\mathopen{}\mathclose{{}\left(S\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x}_{n},T\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x}}\right)\leq\varepsilon,\
\ \ \text{ for all }\ n\geq N_{1}.$ (2.8)

By the invariance property of $\mathscr{A}$ and the absorption property of
absorbing set $\mathcal{B}$, there exists a $\tau<0$ such that

$\displaystyle\mathscr{A}\mathopen{}\mathclose{{}\left(\frac{t_{n}}{2},\theta_{-\frac{t_{n}}{2}}\omega}\right)\
=\
S\mathopen{}\mathclose{{}\left(\frac{t_{n}}{2},\tau,\omega}\right)\mathscr{A}\mathopen{}\mathclose{{}\left(\tau,\theta_{\tau-
t_{n}}\omega}\right)\ \subset\
\mathcal{B}\mathopen{}\mathclose{{}\left(\frac{t_{n}}{2},\theta_{-\frac{t_{n}}{2}}\omega}\right),$

which gives

$\displaystyle\boldsymbol{x}\in\overline{\bigcup\limits_{t_{n}\geq
2t_{0}}\mathscr{A}\mathopen{}\mathclose{{}\left(\frac{t_{n}}{2},\theta_{-\frac{t_{n}}{2}}\omega}\right)}^{\delta}\
\subset\ \overline{\bigcup\limits_{t_{n}\geq
2t_{0}}\mathcal{B}\mathopen{}\mathclose{{}\left(\frac{t_{n}}{2},\theta_{-\frac{t_{n}}{2}}\omega}\right)}^{\delta}=\mathcal{B}_{t_{0}}\mathopen{}\mathclose{{}\left(\theta_{-\frac{t_{n}}{2}}\omega}\right).$
(2.9)

By condition $(\mathrm{iii})$, (2.9) and the attracting property of
$\mathscr{A}_{\infty}$, there exists $N_{2}\in\mathbb{N}$ such that

$\displaystyle\delta\mathopen{}\mathclose{{}\left(T\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x},\mathscr{A}_{\infty}(\omega)}\right)\leq\varepsilon,\
\ \ \text{ for all }\ n\geq N_{2}.$ (2.10)

Combining (2.8) and (2.10), we deduce for all $n\geq N:=\max\\{N_{1},N_{2}\\}$
that

$\displaystyle\mathrm{dist}_{\mathcal{W}}^{\delta}\mathopen{}\mathclose{{}\left(\mathscr{A}(t_{n},\omega),\mathscr{A}_{\infty}(\omega)}\right)$
$\displaystyle=\delta\mathopen{}\mathclose{{}\left(\boldsymbol{w}_{n},\mathscr{A}_{\infty}(\omega)}\right)$
$\displaystyle\leq\delta\mathopen{}\mathclose{{}\left(S\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x}_{n},T\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x}}\right)+\delta\mathopen{}\mathclose{{}\left(T\mathopen{}\mathclose{{}\left(t_{n},\frac{t_{n}}{2},\omega}\right)\boldsymbol{x},\mathscr{A}_{\infty}(\omega)}\right)$
$\displaystyle\leq 2\varepsilon,$

which contradicts (2.6). ∎

##  3 Global Solvability of 2D stochastic Euler equations driven by
multiplicative noise

We first provide the necessary function spaces needed for the further
analysis. Then, we define a transformation which helps to convert our
stochastic system into an equivalent pathwise deterministic system. Finally,
we prove the existence and uniqueness of the 2D stochastic Euler system (1.2).

###  3.1 Function spaces

For $p\in[1,\infty]$, the scalar as well as vector valued Lebesgue spaces will
be denoted by
$\mathrm{L}^{p}(\mathcal{O})=\mathrm{L}^{p}(\mathcal{O};\mathbb{R})$ and
$\mathbb{L}^{p}(\mathcal{O})=\mathrm{L}^{p}(\mathcal{O};\mathbb{R}^{2})$,
respectively. For $s>0$ and $p\in[1,\infty]$, the usual scalar and vector
valued Sobolev spaces will be represented by
$\mathrm{W}^{s,p}(\mathcal{O})=\mathrm{W}^{s,p}(\mathcal{O};\mathbb{R})$ and
$\mathbb{W}^{s,p}(\mathcal{O})=\mathrm{W}^{s,p}(\mathcal{O};\mathbb{R}^{2})$,
respectively. For $p=2$, the Hilbertian Sobolev spaces will be denoted by
$\mathrm{H}^{s}(\mathcal{O})=\mathrm{W}^{s,2}(\mathcal{O})$ and
$\mathbb{H}^{s}(\mathcal{O})=\mathbb{W}^{s,2}(\mathcal{O})$. Let us define

$\displaystyle\mathbb{H}$
$\displaystyle:=\\{\boldsymbol{u}\in\mathbb{L}^{2}(\mathcal{O}):\nabla\cdot\boldsymbol{u}=0\
\text{ and }\
\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\mathcal{O}}=0\\},$
$\displaystyle\mathbb{V}$
$\displaystyle:=\\{\boldsymbol{u}\in\mathbb{H}^{1}(\mathcal{O}):\nabla\cdot\boldsymbol{u}=0\
\text{ and }\
\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\mathcal{O}}=0\\},$

where $\boldsymbol{n}$ is the unit outward normal to the boundary
$\partial\mathcal{O}$, and
$\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\mathcal{O}}$ should be
understood in the sense of trace in $\mathbb{H}^{-1/2}(\partial\mathcal{O})$
(cf. [58, Chapter 1, Theorem 1.2]). The spaces $\mathbb{H}$ and $\mathbb{V}$
are endowed with the norms

$\displaystyle\|\boldsymbol{u}\|_{\mathbb{H}}^{2}=\int_{\mathcal{O}}|\boldsymbol{u}(x)|^{2}\/\mathrm{d}\/x\
\text{ and }\
\|\boldsymbol{u}\|_{\mathbb{V}}^{2}=\int_{\mathcal{O}}\mathopen{}\mathclose{{}\left[|\boldsymbol{u}(x)|^{2}+|\nabla\boldsymbol{u}(x)|^{2}}\right]\/\mathrm{d}\/x,$

respectively. Let $(\cdot,\cdot)$ denote the inner product in the Hilbert
space $\mathbb{H}$ and $\langle\cdot,\cdot\rangle$ represent the induced
duality between the spaces $\mathbb{V}$ and its dual $\mathbb{V}^{\prime}$.
Note that $\mathbb{H}$ can be identified with its dual $\mathbb{H}^{\prime}$.
Furthermore, we have the Gelfand triple
$\mathbb{V}\hookrightarrow\mathbb{H}\cong\mathbb{H}^{\prime}\hookrightarrow\mathbb{V}^{\prime}$
with dense and continuous embedding, and the embedding
$\mathbb{V}\hookrightarrow\mathbb{H}$ is compact.

###  3.2 Equivalent pathwise deterministic system

In the system (1.2), $\mathrm{W}(t,\omega)$ is the standard scalar Wiener
process on the probability space $(\Omega,\mathscr{F},\mathbb{P}),$ where

$\Omega=\\{\omega\in\mathrm{C}(\mathbb{R};\mathbb{R}):\omega(0)=0\\}=:\mathrm{C}_{0}(\mathbb{R};\mathbb{R}),$

endowed with the compact-open topology given by the complete metric

$\displaystyle
d_{\Omega}(\omega,\omega^{\prime}):=\sum_{m=1}^{\infty}\frac{1}{2^{m}}\frac{\|\omega-\omega^{\prime}\|_{m}}{1+\|\omega-\omega^{\prime}\|_{m}},\
\text{ where }\ \|\omega-\omega^{\prime}\|_{m}:=\sup_{-m\leq t\leq
m}|\omega(t)-\omega^{\prime}(t)|,$

and $\mathscr{F}$ is the Borel sigma-algebra induced by the compact-open
topology of $(\Omega,d_{\Omega}),$ $\mathbb{P}$ is the two-sided Wiener
measure on $(\Omega,\mathscr{F})$. Also, define
$\\{\theta_{t}\\}_{t\in\mathbb{R}}$ by

$\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),\ \ \ t\in\mathbb{R}\
\text{ and }\ \omega\in\Omega.$

Hence, $(\Omega,\mathscr{F},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$ is
a metric dynamical system.

Let us now consider

$\displaystyle
y(\theta_{t}\omega)=\int_{-\infty}^{t}e^{-(t-\xi)}\/\mathrm{d}\/\mathrm{W}(\xi),\
\ \omega\in\Omega,$ (3.1)

which is the stationary solution of the one dimensional Ornstein-Uhlenbeck
equation

$\displaystyle\/\mathrm{d}\/y(\theta_{t}\omega)+y(\theta_{t}\omega)\/\mathrm{d}\/t=\/\mathrm{d}\/\mathrm{W}(t).$
(3.2)

It is known from [30] that there exists a $\theta$-invariant subset
$\widetilde{\Omega}\subset\Omega$ of full measure such that
$y(\theta_{t}\omega)$ is continuous in $t$ for every
$\omega\in\widetilde{\Omega},$ and

$\displaystyle\lim_{t\to\pm\infty}\frac{\omega(t)}{|t|}=\lim_{t\to\pm\infty}\frac{|y(\theta_{t}\omega)|}{|t|}$
$\displaystyle=\lim_{t\to\pm\infty}\frac{1}{t}\int_{0}^{t}y(\theta_{\xi}\omega)\/\mathrm{d}\/\xi=\lim_{t\to\infty}e^{-\delta
t}|y(\theta_{-t}\omega)|=0,$ (3.3)

for all $\delta>0$, where $\Gamma$ is the Gamma function. For further analysis
of this work, we do not distinguish between $\widetilde{\Omega}$ and $\Omega$.

Define a new variable $\boldsymbol{v}$ by

$\displaystyle\boldsymbol{v}(t;\tau,\omega,\boldsymbol{v}_{\tau})=e^{-\sigma
y(\theta_{t}\omega)}\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau})\ \ \
\text{ with }\ \ \ \boldsymbol{v}_{\tau}=e^{-\sigma
y(\theta_{\tau}\omega)}\boldsymbol{u}_{\tau},$ (3.4)

where $\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau})$ and
$y(\theta_{t}\omega)$ are the solutions of (1.2) and (3.2), respectively. Then
$\boldsymbol{v}(\cdot):=\boldsymbol{v}(\cdot;\tau,\omega,\boldsymbol{v}_{\tau})$
satisfies:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\boldsymbol{v}}{\/\mathrm{d}\/t}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right]\boldsymbol{v}+e^{\sigma
y(\theta_{t}\omega)}(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+e^{-\sigma
y(\theta_{t}\omega)}\nabla p&=e^{-\sigma
y(\theta_{t}\omega)}\boldsymbol{f},\hskip 14.22636pt\text{ in }\
\mathcal{O}\times(\tau,\infty),\\\ \nabla\cdot\boldsymbol{v}&=0,\hskip
48.36967pt\text{ in }\ \mathcal{O}\times(\tau,\infty),\\\
\boldsymbol{v}\cdot\boldsymbol{n}&=0,\hskip 48.36967pt\text{ on }\
\partial\mathcal{O}\times(\tau,\infty),\\\
\boldsymbol{v}|_{t=\tau}&=\boldsymbol{v}_{\tau},\hskip 42.67912pt\text{ in
}\mathcal{O},\end{aligned}}\right.$ (3.5)

in the distributional sense, where $\sigma>0$. It can be easily seen from
(3.5) that in $\mathbb{V}^{\prime}$

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\boldsymbol{v}}{\/\mathrm{d}\/t}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right]\boldsymbol{v}+e^{\sigma
y(\theta_{t}\omega)}(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}&=e^{-\sigma
y(\theta_{t}\omega)}\boldsymbol{f},\ \ \ \ \ t>\tau,\\\
\boldsymbol{v}|_{t=\tau}&=\boldsymbol{v}_{\tau}.\end{aligned}}\right.$ (3.6)

Next, we define the vorticity

$\varrho:=\nabla\land\boldsymbol{v}=\frac{\partial v_{2}}{\partial
x_{1}}-\frac{\partial v_{1}}{\partial x_{2}}.$

Then the vorticity equation corresponding to (3.5) is given by

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\varrho}{\/\mathrm{d}\/t}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right]\varrho+e^{\sigma
y(\theta_{t}\omega)}(\boldsymbol{v}\cdot\nabla\varrho)&=e^{-\sigma
y(\theta_{t}\omega)}\nabla\land\boldsymbol{f},\ \ \ \ \ t>\tau,\\\
\varrho|_{t=\tau}&=\nabla\land\boldsymbol{v}_{\tau},\end{aligned}}\right.$
(3.7)

in $\mathrm{L}^{2}(\mathcal{O})$, since for the 2D flows the vortex stretching
term is zero.

######  Remark 3.1.

One can use an another transformation to change the system (1.2) into an
equivalent pathwise deterministic system. For a given $t\in\mathbb{R}$ and
$\omega\in\Omega$, let $\hat{z}(t,\omega)=e^{-\sigma\omega(t)}$. Then,
$\hat{z}$ satisfies the equation

$\displaystyle\/\mathrm{d}\/\hat{z}=\frac{\sigma^{2}}{2}\hat{z}\/\mathrm{d}\/t-\sigma\hat{z}\/\mathrm{d}\/\mathrm{W}.$
(3.8)

Let $\boldsymbol{v}$ be a new random variable given by

$\displaystyle\boldsymbol{v}(t;\tau,\omega,\boldsymbol{v}_{\tau})=\hat{z}(t,\omega)\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau})\
\ \ \text{ with }\ \ \
\boldsymbol{v}_{\tau}=\hat{z}(\tau,\omega)\boldsymbol{u}_{\tau},$ (3.9)

where $\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau})$ and
$\hat{z}(t,\omega)$ are the solutions of (1.2) and (3.8), respectively. Then
the new random variable
$\boldsymbol{v}(\cdot;\tau,\omega,\boldsymbol{v}_{\tau})$ satisfies the
following system:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\boldsymbol{v}}{\/\mathrm{d}\/t}+\frac{\sigma^{2}}{2}\boldsymbol{v}+\frac{1}{\hat{z}(t,\omega)}\mathrm{B}\big{(}\boldsymbol{v}\big{)}+\hat{z}(t,\omega)\nabla
p&=\hat{z}(t,\omega)\boldsymbol{f},\quad t>\tau,\\\
\boldsymbol{v}|_{t=\tau}&=\boldsymbol{v}_{\tau},\qquad\quad
x\in\mathcal{O},\end{aligned}}\right.$ (3.10)

which is a pathwise deterministic system and is equivalent to the system
(1.2). Even though the system (3.10) is easier to handle than the system
(3.5), there is a technical difficulty in using the transformation (3.9) in
the context of attractors. Since $\omega(t)$ is identified with
$\mathrm{W}(t,\omega)$, the change of variable given by (3.9) involves the
Wiener process explicitly. Therefore, $\hat{z}(t,\omega)$ is not a stationary
process and the change of variable given by (3.9) is not stationary. This fact
creates troubles if one uses the change of variable given by (3.9) in order to
have conjugated random dynamical systems. Since $z(t,\omega)=e^{-\sigma
y(\theta_{t}\omega)}$ is stationary, the change of variable is given by a
homeomorphism (also called conjugation) which transforms one random dynamical
system into an another equivalent one (cf. [31, Proposition B.9]). For this
reason, it is appropriate to use the change of variable mentioned in (3.4). We
also point out here that the authors in [33] used the transformation given in
(3.8) to study the global (or local) existence of smooth pathwise solutions of
2D (or 3D) Euler equations. But they are not investigating any qualitative
properties of the solutions (cf. [55] also). The authors in [43] used the
transformation given in (3.4) to study the random dynamics of 2D SNSE on the
whole space.

Here, the solvability of stochastic Euler system (1.2) has been proved by a
vanishing viscosity method. Let us consider the following Navier-Stokes type
equations:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\boldsymbol{v}_{\nu}}{\/\mathrm{d}\/t}-\nu\Delta\boldsymbol{v}_{\nu}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right]\boldsymbol{v}_{\nu}&+e^{\sigma
y(\theta_{t}\omega)}(\boldsymbol{v}_{\nu}\cdot\nabla)\boldsymbol{v}_{\nu}\\\
+e^{-\sigma y(\theta_{t}\omega)}\nabla p_{\nu}&=e^{-\sigma
y(\theta_{t}\omega)}\boldsymbol{f},\text{ in }\
\mathcal{O}\times(\tau,\infty),\\\ \nabla\cdot\boldsymbol{v}_{\nu}&=0,\ \ \
\text{ in }\ \mathcal{O}\times(\tau,\infty),\\\
\nabla\land\boldsymbol{v}_{\nu}&=0,\ \ \ \text{ on }\
\partial\mathcal{O}\times(\tau,\infty),\\\
\boldsymbol{v}_{\nu}\cdot\boldsymbol{n}&=0,\ \ \ \text{ on }\
\partial\mathcal{O}\times(\tau,\infty),\\\
\boldsymbol{v}_{\nu}|_{t=\tau}&=\boldsymbol{v}_{\tau},\ \ \text{ in
}\mathcal{O},\end{aligned}}\right.$ (3.11)

in the distributional sense, where $\sigma>0$, $0<\nu\leq\nu_{0}$ with
$\nu_{0}$ sufficiently small (will be specified later) and
$\nabla\land\boldsymbol{v}_{\nu}=\frac{\partial v_{\nu,2}}{\partial
x_{1}}-\frac{\partial v_{\nu,1}}{\partial x_{2}}.$

###  3.3 Bilinear form and linear operator

Let the space $\mathbb{H}_{\mathrm{div}}(\mathcal{O})$ be defined by

$\mathbb{H}_{\mathrm{div}}(\mathcal{O}):=\\{\boldsymbol{u}_{\nu}\in\mathbb{L}^{2}(\mathcal{O}):\nabla\cdot\boldsymbol{u}_{\nu}\in\mathrm{L}^{2}(\mathcal{O})\\},$

endowed with the inner product
$(\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu})_{\mathbb{H}_{\mathrm{div}}}=(\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu})+(\mathrm{div\
}\boldsymbol{u}_{\nu},\mathrm{div\ }\boldsymbol{v}_{\nu})$. Let us define the
bilinear form $a(\cdot,\cdot):\mathbb{V}\times\mathbb{V}\to\mathbb{R}$ by

$\displaystyle a(\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu})$
$\displaystyle:=\nu\int_{\mathcal{O}}\nabla\boldsymbol{u}_{\nu}(x):\nabla\boldsymbol{v}_{\nu}(x)\/\mathrm{d}\/x-\nu\int_{\partial\mathcal{O}}\kappa(\rho)\gamma_{0}(\boldsymbol{u}_{\nu}(\rho))\cdot\gamma_{0}(\boldsymbol{v}_{\nu}(\rho))\/\mathrm{d}\/\rho$
$\displaystyle\qquad+\frac{\sigma^{2}}{2}\int_{\mathcal{O}}\boldsymbol{u}_{\nu}(x)\cdot\boldsymbol{v}_{\nu}(x)\/\mathrm{d}\/x,$
(3.12)

for all $\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu}\in\mathbb{V}$, where
$\kappa(\cdot)\in\mathrm{L}^{\infty}(\partial\mathcal{O})$ is the curvature
and
$\gamma_{0}\in\mathcal{L}(\mathbb{H}_{\mathrm{div}}(\mathcal{O});\mathrm{H}^{-\frac{1}{2}}(\partial\mathcal{O}))\cap\mathcal{L}(\mathbb{H}^{s}(\mathcal{O});\mathrm{H}^{s-\frac{1}{2}}(\partial\mathcal{O})),$
for $s>0$ is the trace operator (cf. [2, 8, 46]). We claim that the bilinear
form (3.3) is well defined. Indeed, consider

$\displaystyle|a(\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu})|$
$\displaystyle\leq\nu\|\nabla\boldsymbol{u}_{\nu}\|_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\nu}\|_{\mathbb{H}}+C\nu\|\gamma_{0}(\boldsymbol{u}_{\nu})\|_{\mathrm{H}^{-\frac{1}{2}}(\partial\mathcal{O})}\|\gamma_{0}(\boldsymbol{v}_{\nu})\|_{\mathrm{H}^{\frac{1}{2}}(\partial\mathcal{O})}+\frac{\sigma^{2}}{2}\|\boldsymbol{u}_{\nu}\|_{\mathbb{H}}\|\boldsymbol{v}_{\nu}\|_{\mathbb{H}}$
$\displaystyle\leq\max\mathopen{}\mathclose{{}\left\\{\nu,\frac{\sigma^{2}}{2}}\right\\}\|\boldsymbol{u}_{\nu}\|_{\mathbb{V}}\|\boldsymbol{v}_{\nu}\|_{\mathbb{V}}+C\nu\|\boldsymbol{u}_{\nu}\|_{\mathbb{H}}\|\boldsymbol{v}_{\nu}\|_{\mathbb{V}},$
(3.13)

which is finite for all
$\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu}\in\mathbb{V}$, where we have used
the Trace theorems [58, Theorem 1.2, p.7] and [53, Theorem 3, p.206].
Moreover, by the definition of $a(\cdot,\cdot)$, we have

$\displaystyle a(\boldsymbol{u}_{\nu},\boldsymbol{u}_{\nu})$
$\displaystyle=\nu\int_{\mathcal{O}}|\nabla\boldsymbol{u}_{\nu}(x)|^{2}\/\mathrm{d}\/x+\frac{\sigma^{2}}{2}\int_{\mathcal{O}}|\boldsymbol{u}_{\nu}(x)|^{2}\/\mathrm{d}\/x-\nu\int_{\partial\mathcal{O}}\kappa(\rho)|\gamma_{0}(\boldsymbol{u}_{\nu}(\rho))|^{2}\/\mathrm{d}\/\rho.$
(3.14)

By using the Trace theorem, and interpolation, Hölder’s and Young’s
inequalities, we obtain

$\displaystyle\mathopen{}\mathclose{{}\left|\int_{\partial\mathcal{O}}\kappa(\rho)|\gamma_{0}(\boldsymbol{u}_{\nu}(\rho))|^{2}\/\mathrm{d}\/\rho}\right|$
$\displaystyle\leq
C\|\gamma_{0}(\boldsymbol{u}_{\nu})\|_{\mathrm{L}^{2}(\partial\mathcal{O})}^{2}\leq
C\|\boldsymbol{u}_{\nu}\|_{\mathbb{H}^{\frac{1}{2}}(\mathcal{O})}^{2}\leq
C\|\boldsymbol{u}_{\nu}\|_{\mathbb{V}}\|\boldsymbol{u}_{\nu}\|_{\mathbb{H}}$
$\displaystyle\leq\frac{1}{2}\|\boldsymbol{u}_{\nu}\|_{\mathbb{V}}^{2}+C_{\frac{1}{2}}\|\boldsymbol{u}_{\nu}\|_{\mathbb{H}}^{2}.$
(3.15)

Therefore, from (3.14), we immediately have

$\displaystyle
a(\boldsymbol{u}_{\nu},\boldsymbol{u}_{\nu})\geq\frac{\min\mathopen{}\mathclose{{}\left\\{\nu,\sigma^{2}}\right\\}}{2}\|\boldsymbol{u}_{\nu}\|_{\mathbb{V}}^{2}-C\nu\|\boldsymbol{u}_{\nu}\|_{\mathbb{H}}^{2},$
(3.16)

which implies that $a(\cdot,\cdot)$ is $\mathbb{V}$-coercive (cf. [34, page
352]). We set

$\mathrm{D}(\mathrm{A}_{\nu,\sigma})=\big{\\{}\boldsymbol{v}_{\nu}\in\mathbb{V}\cap\mathbb{H}^{2}(\mathcal{O}):(\nabla\land\boldsymbol{v}_{\nu})|_{\partial\mathcal{O}}=0\big{\\}},$

and define the linear operator
$\mathrm{A}_{\nu,\sigma}:\mathrm{D}(\mathrm{A}_{\nu,\sigma})\to\mathbb{H}$, as

$\displaystyle\langle\mathrm{A}_{\nu,\sigma}\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu}\rangle=a(\boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu}),\
\text{ for all }\ \boldsymbol{u}_{\nu},\boldsymbol{v}_{\nu}\in\mathbb{V}.$
(3.17)

There is also an extension
$\mathrm{A}_{\nu,\sigma}:\mathbb{V}\to\mathbb{V}^{\prime}$ of the operator
$\mathrm{A}_{\nu,\sigma}\boldsymbol{u}_{\nu}=-\nu\Delta\boldsymbol{u}_{\nu}+\frac{\sigma^{2}}{2}\boldsymbol{u}_{\nu}$
by an application of the Lax-Milgram lemma.

###  3.4 Trilinear form and bilinear operator

Let us define the _trilinear form_
$b(\cdot,\cdot,\cdot):\mathbb{V}\times\mathbb{V}\times\mathbb{V}\to\mathbb{R}$
by

$b(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})=\int_{\mathcal{O}}(\boldsymbol{u}(x)\cdot\nabla)\boldsymbol{v}(x)\cdot\boldsymbol{w}(x)\/\mathrm{d}\/x=\sum_{i,j=1}^{2}\int_{\mathcal{O}}\boldsymbol{u}_{i}(x)\frac{\partial\boldsymbol{v}_{j}(x)}{\partial
x_{i}}\boldsymbol{w}_{j}(x)\/\mathrm{d}\/x.$

If $\boldsymbol{u},\boldsymbol{v}$ are such that the linear map
$b(\boldsymbol{u},\boldsymbol{v},\cdot)$ is continuous on $\mathbb{V}$, the
corresponding element of $\mathbb{V}^{\prime}$ is denoted by
$\mathrm{B}(\boldsymbol{u},\boldsymbol{v})$. We also denote
$\mathrm{B}(\boldsymbol{v})=\mathrm{B}(\boldsymbol{v},\boldsymbol{v})=(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}$.
An integration by parts implies (cf. [60])

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\langle\mathrm{B}(\boldsymbol{u},\boldsymbol{v}),\boldsymbol{v}\rangle&=0,\
\text{ for all }\ \boldsymbol{u},\boldsymbol{v}\in\mathbb{V},\\\
\langle\mathrm{B}(\boldsymbol{u},\boldsymbol{v}),\boldsymbol{w}\rangle&=-\langle\mathrm{B}(\boldsymbol{u},\boldsymbol{w}),\boldsymbol{v}\rangle,\
\text{ for all }\
\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in\mathbb{V}.\end{aligned}}\right.$
(3.18)

###  3.5 Abstract formulation and existence

We can rewrite the system (1.2) in the following abstract form in
$\mathbb{V}^{\prime}$:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\boldsymbol{v}_{\nu}}{\/\mathrm{d}\/t}+\mathrm{A}_{\nu,\sigma}\boldsymbol{v}_{\nu}+e^{\sigma
y(\theta_{t}\omega)}\mathrm{B}(\boldsymbol{v}_{\nu})&=e^{-\sigma
y(\theta_{t}\omega)}\boldsymbol{f}+\sigma
y(\theta_{t}\omega)\boldsymbol{v}_{\nu},\ \ \ \ \ t>\tau,\\\
\boldsymbol{v}_{\nu}|_{t=\tau}&=\boldsymbol{v}_{\tau},\end{aligned}}\right.$
(3.19)

where $\sigma>0$, $0<\nu\leq\nu_{0}$.

Let us now show that there exists a global in time solution to the system
(1.2) in the following sense:

######  Definition 3.2.

A stochastic process
$\boldsymbol{u}(t):=\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau})$ is a
global weak (analytic) solution of the system (1.2) for $t\geq\tau$ and
$\omega\in\Omega$ if for $\mathbb{P}$-a.e. $\omega\in\Omega$

$\displaystyle\boldsymbol{u}(\cdot)\in\mathrm{C}([\tau,\infty);\mathbb{H})\cap\mathrm{L}^{\infty}_{\mathrm{loc}}(\tau,\infty;\mathbb{V})$

and

$\displaystyle(\boldsymbol{u}(t),\boldsymbol{\psi})+\int_{\tau}^{t}b(\boldsymbol{u}(\xi),\boldsymbol{u}(\xi),\boldsymbol{\psi})\/\mathrm{d}\/\xi=(\boldsymbol{u}_{\tau},\boldsymbol{\psi})+\int_{\tau}^{t}(\boldsymbol{f}(\xi),\boldsymbol{\psi})\/\mathrm{d}\/\xi+\sigma\int_{\tau}^{t}(\boldsymbol{u}(\xi),\boldsymbol{\psi})\/\mathrm{d}\/\mathrm{W}(\xi),$

for every $t\geq\tau$ and for every $\boldsymbol{\psi}\in\mathbb{V}$.

In order to prove the existence of a solution to the system (1.2), it is
enough to show that there exists a solution to the system (3.5).

######  Theorem 3.3.

Suppose that $\boldsymbol{u}_{\tau}\in\mathbb{V}$,
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$
and
$\nabla\land\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{2}(\mathcal{O}))$.
Then, there exists at least one global weak (analytic) solution
$\boldsymbol{u}$ of the system (1.2). In addition, $\boldsymbol{u}(\cdot)$
satisfies

$\displaystyle\|\boldsymbol{u}(t)\|^{2}_{\mathbb{V}}$ $\displaystyle\leq
Ce^{2\sigma
y(\theta_{t}\omega)}\bigg{[}\|\boldsymbol{u}_{\tau}\|_{\mathbb{V}}^{2}e^{-2\sigma
y(\theta_{\tau}\omega)}e^{\int_{\tau}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}$
$\displaystyle\quad+\int_{\tau}^{t}e^{\int_{\xi}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}e^{-2\sigma
y(\theta_{\xi}\omega)}\big{\\{}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi\bigg{]},$
(3.20)

for all $t\geq\tau$.

###### Proof.

Let $\\{e_{1},e_{2},\ldots,e_{n},\ldots\\}\subset\mathrm{D}(\mathrm{A})$ be a
complete orthonormal system in $\mathbb{H}$ and let $\mathbb{H}_{n}$ be the
$n$-dimensional subspace of $\mathbb{H}$ spanned by
$\\{e_{1},e_{2},\ldots,e_{n}\\}$. Let us denote the projection of the space
$\mathbb{V}^{\prime}$ into $\mathbb{H}_{n}$ by $\mathrm{P}_{n}$, that is,
$\mathrm{P}_{n}\boldsymbol{x}=\sum\limits_{j=1}^{n}\langle\boldsymbol{x},e_{j}\rangle
e_{j}$. As we know that every element $\boldsymbol{x}\in\mathbb{H}$ induces a
functional $\boldsymbol{x}^{\prime}\in\mathbb{H}$ by the formula
$\langle\boldsymbol{x}^{\prime},\boldsymbol{y}\rangle=(\boldsymbol{x},\boldsymbol{y}),\;\boldsymbol{y}\in\mathbb{V}$,
then $\mathrm{P}_{n}\big{|}_{\mathbb{H}}$, the orthogonal projection of
$\mathbb{H}$ onto $\mathbb{H}_{n}$ is given by
$\mathrm{P}_{n}\boldsymbol{x}=\sum\limits_{j=1}^{n}(\boldsymbol{x},e_{j})e_{j}$.
Clearly, $\mathrm{P}_{n}$ is the orthogonal projection from $\mathbb{H}$ onto
$\mathbb{H}_{n}$. Let us consider the following finite-dimensional system:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\boldsymbol{v}_{n,\nu}}{\/\mathrm{d}\/t}+\mathrm{A}_{\nu,\sigma}\boldsymbol{v}_{n,\nu}+e^{\sigma
y(\theta_{t}\omega)}\mathrm{P}_{n}\mathrm{B}(\boldsymbol{v}_{n,\nu})&=e^{-\sigma
y(\theta_{t}\omega)}\mathrm{P}_{n}\boldsymbol{f}+\sigma
y(\theta_{t}\omega)\boldsymbol{v}_{n,\nu},\ \ \ \ \ t>\tau,\\\
\boldsymbol{v}_{n,\nu}|_{t=\tau}&=\mathrm{P}_{n}\boldsymbol{v}_{\tau},\end{aligned}}\right.$
(3.21)

where

$\displaystyle\mathrm{P}_{n}\boldsymbol{v}_{\tau}=\sum_{i=1}^{n}(\boldsymbol{v}_{\tau},e_{i})e_{i}.$

The existence of a local solution to the finite-dimensional system (3.21)
follows by an application of Carathéodory’s existence theorem. Taking the
inner product by $\boldsymbol{v}_{n,\nu}(\cdot)$ in the first equation of
(3.21), and making use of (3.3) and (3.18), we find

$\displaystyle\frac{1}{2}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}+\nu\|\nabla\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right]\|\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\quad=\nu\int_{\partial\mathcal{O}}\kappa(\rho)|\gamma_{0}(\boldsymbol{v}_{n,\nu}(t,\rho))|^{2}\/\mathrm{d}\/\rho+e^{-\sigma
y(\theta_{t}\omega)}(\boldsymbol{f}(t),\boldsymbol{v}_{n,\nu}(t)),$ (3.22)

for a.e. $t\geq\tau$. Similar to (3.3), we get

$\displaystyle\int_{\partial\mathcal{O}}\kappa(\rho)|\gamma_{0}(\boldsymbol{v}_{n,\nu}(\rho))|^{2}\/\mathrm{d}\/\rho\leq
C\|\boldsymbol{v}_{n,\nu}\|_{\mathbb{H}}\|\boldsymbol{v}_{n,\nu}\|_{\mathbb{V}}\leq\frac{1}{2}\|\nabla\boldsymbol{v}_{n,\nu}\|^{2}_{\mathbb{H}}+C_{\frac{1}{2}}\|\boldsymbol{v}_{n,\nu}\|^{2}_{\mathbb{H}},$
(3.23)

where $C_{\frac{1}{2}}>0$ is a constant. An application of the Cauchy-Schwarz
and Young’s inequalities yields

$\displaystyle e^{-\sigma
y(\theta_{t}\omega)}|(\boldsymbol{f},\boldsymbol{v}_{n,\nu})|\leq\frac{\sigma^{2}}{4}\|\boldsymbol{v}_{n,\nu}\|^{2}_{\mathbb{H}}+\frac{e^{-2\sigma
y(\theta_{t}\omega)}}{\sigma^{2}}\|\boldsymbol{f}\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}.$
(3.24)

Combining (3.5)-(3.24), we obtain (for $0<\nu\leq\nu_{0}$)

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}+\nu\|\nabla\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-2\nu
C_{\frac{1}{2}}-2\sigma
y(\theta_{t}\omega)}\right]\|\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq\frac{2e^{-2\sigma
y(\theta_{t}\omega)}}{\sigma^{2}}\|\boldsymbol{f}(t)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})},$
(3.25)

for a.e. $t\geq\tau$. Making use of Gronwall’s inequality, we find

$\displaystyle\|\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq\mathopen{}\mathclose{{}\left[\|\boldsymbol{v}_{n,\nu}(\tau)\|_{\mathbb{H}}^{2}e^{\int_{\tau}^{t}\mathopen{}\mathclose{{}\left[-C^{*}(\nu,\sigma)+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}+\frac{2}{\sigma^{2}}\int_{\tau}^{t}e^{\int_{\xi}^{t}\mathopen{}\mathclose{{}\left[-C^{*}(\nu,\sigma)+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}z^{2}(\xi,\omega)\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}\/\mathrm{d}\/\xi}\right],$
(3.26)

for all $t\geq\tau$, $0<\nu\leq\nu_{0}$ and
$C^{*}(\nu,\sigma)=\frac{\sigma^{2}}{2}-2\nu C_{\frac{1}{2}}>0$ (we choose
$\nu_{0}$ such that $\frac{\sigma^{2}}{2}-2\nu C_{\frac{1}{2}}>0,$ for all
$0<\nu\leq\nu_{0}$). Since $y(\cdot)$ is continuous,
$\|\boldsymbol{v}_{n,\nu}(\tau)\|_{\mathbb{H}}\leq\|\boldsymbol{v}_{\tau}\|_{\mathbb{H}}$
and
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$,
we infer from (3.5) that

$\displaystyle\sup_{t\in[\tau,\tau+T]}\|\boldsymbol{v}_{n,\nu}(t)\|^{2}_{\mathbb{H}}\leq
C<\infty,$ (3.27)

where the constant $C$ is independent of $n$. Integrating (3.5) over
$(\tau,\tau+T)$ and using (3.27), we attain
$\\{\boldsymbol{v}_{n,\nu}\\}_{n\in\mathbb{N}}\subset\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V}).$

Now, for $\boldsymbol{\psi}(\cdot)\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})$,
we consider

$\displaystyle\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle\frac{\/\mathrm{d}\/\boldsymbol{v}_{n,\nu}}{\/\mathrm{d}\/t},\boldsymbol{\psi}}\right\rangle}\right|$
$\displaystyle=\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle\mathrm{A}_{\nu,\sigma}\boldsymbol{v}_{n,\nu}+e^{\sigma
y(\theta_{t}\omega)}\mathrm{B}(\boldsymbol{v}_{n,\nu})-e^{-\sigma
y(\theta_{t}\omega)}\boldsymbol{f}+y(\theta_{t}\omega)\boldsymbol{v}_{n,\nu},\boldsymbol{\psi}}\right\rangle}\right|$
$\displaystyle\leq\nu|(\nabla\boldsymbol{v}_{n,\nu},\nabla\boldsymbol{\psi})|+\mathopen{}\mathclose{{}\left|\int_{\partial\mathcal{O}}\kappa(y)\gamma_{0}(\boldsymbol{v}_{n,\nu}(t,y))\cdot\gamma_{0}(\boldsymbol{\psi}(t,y))\/\mathrm{d}\/y}\right|+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}+\sigma|y(\theta_{t}\omega)|}\right]$
$\displaystyle\quad\times|(\boldsymbol{v}_{n,\nu},\boldsymbol{\psi})|+e^{\sigma
y(\theta_{t}\omega)}|b(\boldsymbol{v}_{n,\nu},\boldsymbol{v}_{n,\nu},\boldsymbol{\psi})|+e^{-\sigma
y(\theta_{t}\omega)}|(\boldsymbol{f},\boldsymbol{\psi})|$ $\displaystyle\leq
C\mathopen{}\mathclose{{}\left[1+|y(\theta_{t}\omega)|}\right]\big{(}\|\boldsymbol{v}_{\nu,\tau}\|_{\mathbb{V}}+|y(\theta_{t}\omega)|\|\boldsymbol{v}_{n,\nu}\|_{\mathbb{H}}+e^{\sigma
y(\theta_{t}\omega)}\|\boldsymbol{v}_{n,\nu}\|_{\mathbb{H}}\|\boldsymbol{v}_{n,\nu}\|_{\mathbb{V}}$
$\displaystyle\quad+e^{-\sigma
y(\theta_{t}\omega)}\|\boldsymbol{f}\|_{\mathbb{L}^{2}(\mathcal{O})}\big{)}\|\boldsymbol{\psi}\|_{\mathbb{V}},$
(3.28)

which immediately gives
$\\{\frac{\/\mathrm{d}\/\boldsymbol{v}_{n,\nu}}{\/\mathrm{d}\/t}\\}_{n\in\mathbb{N}}\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V}^{\prime})$,
where we have used the fact
$\boldsymbol{v}_{n,\nu}\in\mathrm{L}^{\infty}(\tau,\tau+T;\mathbb{H})\cap\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})$.
Hence by the Aubin-Lions compactness lemma (since
$\boldsymbol{v}_{n,\nu}\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})$ and
$\frac{\/\mathrm{d}\/\boldsymbol{v}_{n,\nu}}{\/\mathrm{d}\/t}\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V}^{\prime})$)
and Banach-Alaoglu theorem (since
$\boldsymbol{v}_{n,\nu}\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})$), there
exists a subsequence (still denoted by the same)
$\boldsymbol{v}_{\nu}\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})$ such that (as
$n\to+\infty$)

$\displaystyle\boldsymbol{v}_{n,\nu}\to$ $\displaystyle\ \boldsymbol{v}_{\nu}\
\text{ in }\ \mathrm{L}^{2}(\tau,\tau+T;\mathbb{H})\ \ \text{ strongly},$
(3.29) $\displaystyle\boldsymbol{v}_{n,\nu}\xrightharpoonup{w}$
$\displaystyle\ \boldsymbol{v}_{\nu}\ \text{ in }\
\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})\ \ \text{ weakly}.$ (3.30)

With the help of the convergences (3.29)-(3.30) (see [58]), one can pass the
limit $n\to+\infty$ in (3.21) and obtain that $\boldsymbol{v}_{\nu}$ satisfies
(3.19) in the distributional sense. Moreover, we also deduce

$\displaystyle\boldsymbol{v}_{\nu}\in\mathrm{C}([\tau,\tau+T];\mathbb{H})\cap\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V}),\
\text{ for all }\ T>0.$

Our next aim is to provide uniform estimates in $\nu$ for the
$\mathbb{V}$-norm of $\boldsymbol{v}_{\nu}$. Let us denote by
$\varrho_{\nu}=\nabla\land\boldsymbol{v}_{\nu}$. Then, (3.11) gives

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\varrho_{\nu}}{\/\mathrm{d}\/t}-\nu\Delta\varrho_{\nu}+\frac{\sigma^{2}}{2}\varrho_{\nu}+e^{\sigma
y(\theta_{t}\omega)}(\boldsymbol{v}_{\nu}\cdot\nabla\varrho_{\nu})&=e^{-\sigma
y(\theta_{t}\omega)}\nabla\land\boldsymbol{f}+\sigma
y(\theta_{t}\omega)\varrho_{\nu},\ \ \ t>\tau,\\\
\varrho_{\nu}|_{t=\tau}&=\nabla\land\boldsymbol{v}_{\tau}.\end{aligned}}\right.$
(3.31)

Taking the inner product with $\varrho_{\nu}$ in the first equation of (3.31),
we obtain

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\varrho_{\nu}(t)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-2\sigma
y(\theta_{t}\omega)}\right]\|\varrho_{\nu}(t)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}+2\nu\|\nabla\varrho_{\nu}(t)\|_{\mathbb{L}^{2}}^{2}$
$\displaystyle\leq\frac{2e^{-2\sigma
y(\theta_{t}\omega)}}{\sigma^{2}}\|\nabla\land\boldsymbol{f}(t)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2},\
\text{ for a.e. }t\geq\tau,$

which implies

$\displaystyle\|\varrho_{\nu}(t)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}$
$\displaystyle\leq\mathopen{}\mathclose{{}\left[\|\varrho_{\nu}(\tau)\|^{2}_{\mathrm{L}^{2}(\mathcal{O})}e^{\int_{\tau}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}+\frac{2}{\sigma^{2}}\int_{\tau}^{t}e^{\int_{\xi}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}e^{-2\sigma
y(\theta_{\xi}\omega)}\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\/\mathrm{d}\/\xi}\right],$
(3.32)

for all $t\geq\tau$. Since
$\Delta\boldsymbol{v}_{\nu}=\nabla(\nabla\cdot\boldsymbol{v}_{\nu})-\nabla\wedge(\nabla\wedge\boldsymbol{v}_{\nu})$,
divergence free condition $\nabla\cdot\boldsymbol{v}_{\nu}=0$ assures that
$\boldsymbol{v}_{\nu}$ satisfies the following elliptic problem:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\Delta\boldsymbol{v}_{\nu}&=-\nabla^{\perp}\varrho_{\nu},\ \ \text{ on }\
\mathcal{O},\\\ \nabla\land\boldsymbol{v}_{\nu}|_{\partial\mathcal{O}}&=0,\\\
\boldsymbol{v}_{\nu}\cdot\boldsymbol{n}|_{\partial\mathcal{O}}&=0,\end{aligned}}\right.$
(3.33)

where $\nabla^{\perp}=(\frac{\partial}{\partial x_{2}},\
-\frac{\partial}{\partial x_{1}})$. Multiplying the first equation of (3.33)
with $\boldsymbol{v}_{\nu}$ and integrating over $\mathcal{O}$, we find

$\displaystyle(\Delta\boldsymbol{v}_{\nu},\boldsymbol{v}_{\nu})=-(\nabla^{\perp}\varrho_{\nu},\boldsymbol{v}_{\nu}).$

Making use of integration by parts and performing calculations similar to
(3.3), we arrive at

$\displaystyle\|\nabla\boldsymbol{v}_{\nu}\|^{2}_{\mathbb{H}}$
$\displaystyle=2\int_{\partial\mathcal{O}}\kappa(\rho)|\gamma_{0}(\boldsymbol{v}_{\nu}(\rho))|^{2}\/\mathrm{d}\/\rho+2\|\varrho_{\nu}\|^{2}_{\mathrm{L}^{2}(\mathcal{O})}-\|\nabla\boldsymbol{v}_{\nu}\|^{2}_{\mathbb{H}}$
$\displaystyle\leq
C\|\boldsymbol{v}_{\nu}\|_{\mathbb{H}}\|\boldsymbol{v}_{\nu}\|_{\mathbb{V}}+2\|\varrho_{\nu}\|^{2}_{\mathrm{L}^{2}(\mathcal{O})}-\|\nabla\boldsymbol{v}_{\nu}\|^{2}_{\mathbb{H}}$
$\displaystyle\leq
C(\|\boldsymbol{v}_{\nu}\|^{2}_{\mathbb{H}}+\|\varrho_{\nu}\|^{2}_{\mathrm{L}^{2}(\mathcal{O})}).$
(3.34)

Combining (3.5) and (3.5)-(3.5), we arrive at

$\displaystyle\|\boldsymbol{v}_{\nu}(t)\|^{2}_{\mathbb{V}}$ $\displaystyle\leq
C\bigg{[}\|\boldsymbol{v}_{\tau}\|_{\mathbb{V}}^{2}e^{\int_{\tau}^{t}\mathopen{}\mathclose{{}\left[-C^{*}(\nu,\sigma)+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}$
$\displaystyle\quad+\frac{2}{\sigma^{2}}\int_{\tau}^{t}e^{\int_{\xi}^{t}\mathopen{}\mathclose{{}\left[-C^{*}(\nu,\sigma)+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}e^{-2\sigma
y(\theta_{\xi}\omega)}\bigg{[}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\bigg{]}\/\mathrm{d}\/\xi\bigg{]},$
(3.35)

for all $t\geq\tau$. Due to $C^{*}(\nu,\sigma)>0$, the inequality (3.5)
implies that $\\{\boldsymbol{v}_{\nu}(\cdot)\\}_{0<\nu\leq\nu_{0}}$ is
uniformly bounded (that is, the bound is independent of $\nu$) in
$\mathrm{L}^{\infty}(\tau,\tau+T;\mathbb{V})$. A similar calculation as in
(3.5) gives
$\\{\frac{\/\mathrm{d}\/\boldsymbol{v}_{\nu}}{\/\mathrm{d}\/t}\\}_{0<\nu\leq\nu_{0}}$
is uniformly bounded in $\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V}^{\prime})$.
Once again, by the Aubin-Lions compactness lemma, there exists a subsequence
(still denoted by the same) and
$\boldsymbol{v}\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})$ with
$\frac{\/\mathrm{d}\/\boldsymbol{v}}{\/\mathrm{d}\/t}\in\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V}^{\prime})$
such that (as $\nu\to 0$)

$\displaystyle\boldsymbol{v}_{\nu}\to$ $\displaystyle\ \boldsymbol{v}\ \text{
in }\ \mathrm{L}^{2}(\tau,\tau+T;\mathbb{H})\ \ \text{ strongly},$ (3.36)
$\displaystyle\boldsymbol{v}_{\nu}\xrightharpoonup{w}$ $\displaystyle\
\boldsymbol{v}\ \text{ in }\ \mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})\ \ \text{
weakly}.$ (3.37)

Again, with the help of convergences (3.36)-(3.37), one can pass the limit
$\nu\to 0$ in (3.19) and obtain that $\boldsymbol{v}(\cdot)$ satisfies (3.5)
in the distributional sense and (3.5) holds for $\boldsymbol{v}(\cdot)$ also.
Since $z(\cdot,\omega)$ is continuous, the transformation (3.4) confirms that
$\boldsymbol{u}(\cdot)$ is a global weak (analytic) solution of the system
(1.2) in the sense of Definition 3.2. In addition, the transformation (3.4)
and the estimate (3.5) imply that $\boldsymbol{u}(\cdot)$ satisfies (3.3),
which completes the proof. ∎

###  3.6 Uniqueness

In Subsection 3.5, we proved that for the initial data
$\boldsymbol{u}_{\tau}\in\mathbb{V}$, and external forcing
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$
and
$\nabla\land\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{2}(\mathcal{O}))$,
there exists a solution for the system (1.2) in the sense of Definition 3.2.
In this subsection, we show that if the initial data and external forcing have
more regularity, that is, $\boldsymbol{u}_{\tau}\in\mathbb{V}$,
$\nabla\land\boldsymbol{u}_{\tau}\in\mathrm{L}^{\infty}(\mathcal{O})$,
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$
and
$\nabla\land\boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$,
the solution of the system (1.2) obtained in Theorem 3.3 is unique. In order
to prove the uniqueness, the following two lemmas play a crucial role.

######  Lemma 3.4 ([47, Theorem 8.5, part (ii)]).

For every $\boldsymbol{v}\in\mathbb{H}^{1}(\mathcal{O})$, there exists a
constant $C>0$ which is independent of $p$, such that

$\displaystyle\|\boldsymbol{v}\|_{\mathbb{L}^{p}(\mathcal{O})}\leq
Cp^{\frac{1}{2}}\|\boldsymbol{v}\|_{\mathbb{H}^{1}(\mathcal{O})},\ \ \ \text{
for all }\ p\in[2,\infty).$ (3.38)

Now, since $\mathrm{div\ }\boldsymbol{v}_{\nu}=0$, we can find a stream
function $\boldsymbol{\psi}_{\nu}$ such that
$\boldsymbol{v}_{\nu}=\nabla^{\perp}\boldsymbol{\psi}_{\nu},$ and
$\boldsymbol{\psi}_{\nu}$ satisfies

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
-\Delta\boldsymbol{\psi}_{\nu}&=\varrho_{\nu},\ \ \text{ on }\ \mathcal{O},\\\
\boldsymbol{\psi}_{\nu}|_{\partial\mathcal{O}}&=0.\end{aligned}}\right.$

Let $\mathcal{G}_{\mathcal{O}}$ be the Green function with homogeneous
boundary conditions. Then, $\boldsymbol{\psi}_{\nu}$ satisfies

$\displaystyle\boldsymbol{\psi}_{\nu}(x)=\int_{\mathcal{O}}\mathcal{G}_{\mathcal{O}}(x,\rho)\varrho_{\nu}(\rho)\/\mathrm{d}\/\rho\
\ \text{ which implies }\ \
\boldsymbol{v}_{\nu}(x)=\int_{\mathcal{O}}\nabla^{\perp}\mathcal{G}_{\mathcal{O}}(x,\rho)\varrho_{\nu}(\rho)\/\mathrm{d}\/\rho.$
(3.39)

It is known that the Green function $\mathcal{G}_{\mathcal{O}}$ and its
derivatives satisfies

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
|\mathcal{G}_{\mathcal{O}}(x,z)|&\leq C\big{[}\log(|x-z|)+1\big{]},\\\
|\nabla^{\perp}\mathcal{G}_{\mathcal{O}}(x,z)|&\leq C|x-z|^{-1},\\\
\bigg{|}\sum_{i=1}^{2}\frac{\partial}{\partial
x_{i}}\nabla^{\perp}\mathcal{G}_{\mathcal{O}}(x,z)\bigg{|}&\leq
C|x-z|^{-2}.\end{aligned}}\right.$ (3.40)

By (3.39)-(3.40) and the Calderón-Zygmund theorem on singular integral (cf.
[56]), we infer that

$\displaystyle\|\nabla\boldsymbol{v}_{\nu}\|_{\mathbb{L}^{p}(\mathcal{O})}\leq
C_{p}\|\varrho_{\nu}\|_{\mathbb{L}^{p}(\mathcal{O})}\ \ \ \text{ for }\
p\in(1,\infty).$ (3.41)

Furthermore, for $p>1$ the constant $C_{p}$ appearing in (3.41) can be
rewritten as

$\displaystyle C_{p}\leq C\frac{p^{2}}{p-1},$

where $C$ is a constant independent of $p$. This fact follows from the
constant which is calculated carefully in Marcinkiewicz interpolation
inequality (cf. [32]). Based on the above discussion, we state the following
lemma which can also be found in the works [69, Lemma 4.1] and [70, Lemma
1.2].

######  Lemma 3.5.

For a function $\boldsymbol{v}$ satisfing the system (3.33), we have

$\displaystyle\|\nabla\boldsymbol{v}\|_{\mathbb{L}^{p}(\mathcal{O})}\leq\frac{Cp^{2}}{p-1}\|\nabla\land\boldsymbol{v}\|_{\mathrm{L}^{p}(\mathcal{O})},\
\ \text{for all}\ \ \nabla\land\boldsymbol{v}\in\mathrm{L}^{p}(\mathcal{O}),$
(3.42)

for any $p>1$, where the constant $C>0$ depends on $\mathcal{O}$ but not on
$p$.

######  Theorem 3.6.

Suppose that $\boldsymbol{u}_{\tau}\in\mathbb{V}$,
$\nabla\land\boldsymbol{u}_{\tau}\in\mathrm{L}^{\infty}(\mathcal{O})$,
$\boldsymbol{f}\in\mathrm{L}^{2}_{\mathrm{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathcal{O}))$
and
$\nabla\land\boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$.
Then, the solution of the system (1.2) obtained in Theorem 3.3 is unique. In
addition, the solution
$\boldsymbol{u}\in\mathrm{L}^{\infty}(\tau,\tau+T;{\mathbb{W}}^{1,p}(\mathcal{O}))\cap\mathrm{L}^{\infty}(\tau,\tau+T;\mathbb{L}^{\infty}(\mathcal{O}))$
for $p\in[2,\infty)$ and
$\nabla\land\boldsymbol{u}\in\mathrm{L}^{\infty}(\tau,\tau+T;\mathbb{L}^{\infty}(\mathcal{O}))$.
Moreover, $\nabla\land\boldsymbol{u}(\cdot)$ satisfies

$\displaystyle\|\nabla\land\boldsymbol{u}(t)\|_{\mathbb{L}^{\infty}(\mathcal{O})}$
$\displaystyle\leq e^{\sigma
y(\theta_{t}\omega)}\bigg{[}e^{\int_{\tau}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r-\sigma
y(\theta_{\tau}\omega)}\|\nabla\land\boldsymbol{u}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})}$
$\displaystyle\quad+\int_{\tau}^{t}e^{\int_{\xi}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r-\sigma
y(\theta_{\xi}\omega)}\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\/\mathrm{d}\/\xi\bigg{]},$
(3.43)

for a.e. $t\geq\tau$.

###### Proof.

The proof is divided into the following two steps: Step I: Claim:
$\boldsymbol{v}_{\nu},\
\boldsymbol{v}\in\mathrm{L}^{\infty}(\tau,\tau+T;{\mathbb{W}}^{1,p}(\mathcal{O}))\cap\mathrm{L}^{\infty}(\tau,\tau+T;\mathbb{L}^{\infty}(\mathcal{O}))$
for $p\in[2,\infty)$.

It implies from (3.38) and (3.5) that

$\displaystyle\|\boldsymbol{v}_{\nu}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;{\mathbb{L}}^{p}(\mathcal{O}))}$
$\displaystyle\leq
p^{\frac{1}{2}}C(\omega)\|\boldsymbol{v}_{\nu}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathbb{V})}$
$\displaystyle\leq
p^{\frac{1}{2}}C(T,\omega,\|\boldsymbol{v}_{\tau}\|_{\mathbb{V}},\|\boldsymbol{f}\|_{\mathrm{L}^{2}(\tau,\tau+T;\mathbb{L}^{2}(\mathcal{O}))},\|\nabla\land\boldsymbol{f}\|_{\mathrm{L}^{2}(\tau,\tau+T;\mathrm{L}^{2}(\mathcal{O}))}),$
(3.44)

for $p\in[2,\infty)$. Taking the inner product of the first equation of (3.31)
with $|\varrho_{\nu}|^{p-2}\varrho_{\nu}$, for $p\in[2,\infty)$, we obtain

$\displaystyle\frac{1}{p}$
$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\varrho_{\nu}(t)\|^{p}_{\mathrm{L}^{p}(\mathcal{O})}+p\nu\||\nabla\varrho_{\nu}(t)||\varrho_{\nu}(t)|^{\frac{p-2}{2}}\|^{2}_{\mathrm{L}^{2}(\mathcal{O})}+\frac{\sigma^{2}}{2}\|\varrho_{\nu}(t)\|^{p}_{\mathrm{L}^{p}(\mathcal{O})}$
$\displaystyle=e^{-\sigma
y(\theta_{t}\omega)}\int_{\mathcal{O}}(\nabla\land\boldsymbol{f}(t,x))|\varrho_{\nu}(t,x)|^{p-2}\varrho_{\nu}(t,x)\/\mathrm{d}\/x+\sigma
y(\theta_{t}\omega)\|\varrho_{\nu}(t)\|^{p}_{\mathrm{L}^{p}(\mathcal{O})}$
$\displaystyle\leq e^{-\sigma
y(\theta_{t}\omega)}\|\nabla\land\boldsymbol{f}(t)\|_{\mathrm{L}^{p}(\mathcal{O})}\|\varrho_{\nu}(t)\|_{\mathrm{L}^{p}(\mathcal{O})}^{p-1}+\sigma|y(\theta_{t}\omega)|\|\varrho_{\nu}(t)\|^{p}_{\mathrm{L}^{p}(\mathcal{O})}$
$\displaystyle\leq Ce^{-p\sigma
y(\theta_{t}\omega)}\|\nabla\land\boldsymbol{f}(t)\|^{p}_{\mathrm{L}^{p}(\mathcal{O})}+(1+\sigma|y(\theta_{t}\omega)|)\|\varrho_{\nu}(t)\|^{p}_{\mathrm{L}^{p}(\mathcal{O})},$
(3.45)

for a.e. $t\geq\tau$, which implies (due to Gronwall’s inequality)

$\displaystyle\sup_{t\in[\tau,\tau+T]}\|\varrho_{\nu}(t)\|_{\mathrm{L}^{p}(\mathcal{O})}^{p}$
$\displaystyle\leq
p\bigg{[}\frac{1}{p}\|\nabla\land\boldsymbol{v}_{\tau}\|_{\mathrm{L}^{p}(\mathcal{O})}^{p}+C\int_{\tau}^{\tau+T}e^{-p\sigma
y(\theta_{\xi}\omega)}\|\nabla\land\boldsymbol{f}(\xi)\|^{p}_{\mathrm{L}^{p}(\mathcal{O})}\/\mathrm{d}\/\xi\bigg{]}e^{p\int_{\tau}^{\tau+T}(1+\sigma|y(\theta_{r}\omega)|)\/\mathrm{d}\/r}.$

Therefore, we infer

$\displaystyle\sup_{t\in[\tau,\tau+T]}\|\varrho_{\nu}(t)\|_{\mathrm{L}^{p}(\mathcal{O})}\leq
p^{\frac{1}{p}}C(T,\omega,\mathcal{O},\|\nabla\land\boldsymbol{u}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})},\|\nabla\land\boldsymbol{f}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathrm{L}^{\infty}(\mathcal{O}))}).$

In view of Lemma 3.5, we deduce

$\displaystyle\|\nabla\boldsymbol{v}_{\nu}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;{\mathbb{L}}^{p}(\mathcal{O}))}\leq\frac{p^{\frac{1}{p}}p^{2}}{p-1}C(T,\omega,\mathcal{O},\|\nabla\land\boldsymbol{u}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})},\|\nabla\land\boldsymbol{f}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathrm{L}^{\infty}(\mathcal{O}))}),$
(3.46)

for $p\in[2,\infty)$. Since the bounds in (3.6) and (3.46) are independent of
$\nu$, one can obtain by passing limit $\nu\to 0$ that

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\|\boldsymbol{v}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;{\mathbb{L}}^{p}(\mathcal{O}))}&\leq
p^{\frac{1}{2}}C(T,\omega,\|\boldsymbol{u}_{\tau}\|_{\mathbb{V}},\|\boldsymbol{f}\|_{\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})}),\\\
\|\nabla\boldsymbol{v}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;{\mathbb{L}}^{p}(\mathcal{O}))}&\leq\frac{p^{\frac{1}{p}}p^{2}}{p-1}C(T,\omega,\mathcal{O},\|\nabla\land\boldsymbol{u}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})},\|\nabla\land\boldsymbol{f}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathrm{L}^{\infty}(\mathcal{O}))}),\end{aligned}}\right.$
(3.47)

for $p\in[2,\infty)$. Moreover, Sobolev’s inequality provides
$\boldsymbol{v}_{\nu},\boldsymbol{v}\in\mathrm{L}^{\infty}(\tau,\tau+T;\mathbb{L}^{\infty}(\mathcal{O}))$.
Let us now apply the maximum principle (see [29, Subsection 7.1.4]) for
$\varrho_{\nu}$, which is a solution of (3.31), to deduce

$\displaystyle\|\varrho_{\nu}(t)\|_{\mathrm{L}^{\infty}(\mathcal{O})}$
$\displaystyle\leq
e^{\int_{\tau}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}\|\varrho_{\nu}(\tau)\|_{\mathrm{L}^{\infty}(\mathcal{O})}+\int_{\tau}^{t}e^{\int_{\xi}^{t}\mathopen{}\mathclose{{}\left[-\frac{\sigma^{2}}{2}+2\sigma
y(\theta_{r}\omega)}\right]\/\mathrm{d}\/r}e^{-\sigma
y(\theta_{\xi}\omega)}\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\/\mathrm{d}\/\xi,$
(3.48)

for all $t\geq\tau$, which immediately gives

$\displaystyle\|\varrho_{\nu}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathrm{L}^{\infty}(\mathcal{O}))}\leq
C(T,\omega)\big{[}\|\nabla\land\boldsymbol{v}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathrm{L}^{\infty}(\mathcal{O}))}\big{]}.$
(3.49)

Therefore, from (3.4) and (3.6), one can easily deduce (3.6). Step II: The
solution of the system (1.2) obtained in Theorem 3.3 is unique. Let
$\boldsymbol{v}_{1}(\cdot)$ and $\boldsymbol{v}_{2}(\cdot)$ be two solutions
of the system (3.5) with
$\boldsymbol{v}_{1}(\tau)=\boldsymbol{v}_{2}(\tau)=\boldsymbol{v}_{\tau}$.
Then
$\boldsymbol{v}(\cdot):=\boldsymbol{v}_{1}(\cdot)-\boldsymbol{v}_{2}(\cdot)$
satisfies for a.e. $t\geq\tau$ and $\boldsymbol{\phi}\in\mathbb{V}$

$\displaystyle\mathopen{}\mathclose{{}\left\langle\frac{\/\mathrm{d}\/\boldsymbol{v}}{\/\mathrm{d}\/t},\boldsymbol{\phi}}\right\rangle+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-y(\theta_{t}\omega)}\right](\boldsymbol{v},\boldsymbol{\phi})+e^{\sigma
y(\theta_{t}\omega)}\langle(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}_{1},\boldsymbol{\phi}\rangle+e^{\sigma
y(\theta_{t}\omega)}\langle(\boldsymbol{v}_{2}\cdot\nabla)\boldsymbol{v},\boldsymbol{\phi}\rangle=0.$
(3.50)

Taking $\boldsymbol{\phi}=\boldsymbol{v}$, we attain (for $p>3$)

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{v}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle=-\mathopen{}\mathclose{{}\left[\sigma^{2}-2\sigma
y(\theta_{t}\omega)}\right]\|\boldsymbol{v}(t)\|^{2}_{\mathbb{H}}-2e^{\sigma
y(\theta_{t}\omega)}\langle(\boldsymbol{v}(t)\cdot\nabla)\boldsymbol{v}_{1}(t),\boldsymbol{v}(t)\rangle$
$\displaystyle\leq 2e^{\sigma
y(\theta_{t}\omega)}\|\nabla\boldsymbol{v}_{1}(t)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\boldsymbol{v}(t)\|^{2}_{\mathbb{L}^{\frac{2p}{p-1}}(\mathcal{O})}+2\sigma|y(\theta_{t}\omega)|\|\boldsymbol{v}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq 2e^{\sigma
y(\theta_{t}\omega)}\|\nabla\boldsymbol{v}_{1}(t)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\boldsymbol{v}(t)\|^{\frac{2}{p-2}}_{\mathbb{L}^{p}(\mathcal{O})}\|\boldsymbol{v}(t)\|^{\frac{2(p-3)}{p-2}}_{\mathbb{H}}+2\sigma|y(\theta_{t}\omega)|\|\boldsymbol{v}(t)\|^{2}_{\mathbb{H}},$
(3.51)

for a.e. $t\geq\tau$. We also have

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{v}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle=\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\mathopen{}\mathclose{{}\left[\|\boldsymbol{v}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}}}\right]^{p-2}=(p-2)\|\boldsymbol{v}(t)\|_{\mathbb{H}}^{\frac{2(p-3)}{p-2}}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{v}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}},$
(3.52)

for a.e. $t\geq\tau$, which implies from (3.6) and
$\|\boldsymbol{v}\|_{\mathbb{H}}^{\frac{2(p-3)}{p-2}}\geq 0$ that

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{v}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}}\leq\frac{2e^{\sigma
y(\theta_{t}\omega)}}{p-2}\|\nabla\boldsymbol{v}_{1}(t)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\boldsymbol{v}(t)\|^{\frac{2}{p-2}}_{\mathbb{L}^{p}(\mathcal{O})}+\frac{2\sigma|y(\theta_{t}\omega)|}{p-2}\|\boldsymbol{v}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}},$
(3.53)

for a.e. $t\geq\tau$. Now, applying Gronwall’s inequality in (3.53) from
$\tau$ to $t$ with $t\geq\tau$, we infer

$\displaystyle\|\boldsymbol{v}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq\mathopen{}\mathclose{{}\left[\frac{2}{p-2}\int_{\tau}^{t}e^{\sigma
y(\theta_{\xi}\omega)}\|\nabla\boldsymbol{v}_{1}(\xi)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\boldsymbol{v}(\xi)\|^{\frac{2}{p-2}}_{\mathbb{L}^{p}(\mathcal{O})}\/\mathrm{d}\/\xi}\right]^{p-2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{\xi}\omega)|\/\mathrm{d}\/\xi}$
$\displaystyle\leq[C(t-\tau)]^{p-2}(Cp)^{2}\frac{1}{p^{\frac{2}{p}}}\mathopen{}\mathclose{{}\left[\frac{p^{2}}{(p-1)(p-2)}}\right]^{p-2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{\xi}\omega)|\/\mathrm{d}\/\xi},$
(3.54)

where we have used the bounds obtained in (3.47) and the constant $C$
appearing in (3.6)

$C=C(T,\omega,\mathcal{O},\|\boldsymbol{u}_{\tau}\|_{\mathbb{V}},\|\boldsymbol{f}\|_{\mathrm{L}^{2}(\tau,\tau+T;\mathbb{L}^{2}(\mathcal{O}))},\|\nabla\land\boldsymbol{u}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})},\|\nabla\land\boldsymbol{f}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathrm{L}^{\infty}(\mathcal{O}))})>0,$

is independent of $p$. Let us now take $t$ sufficiently small such that
$C(t-\tau)<1$. Further, it provides us that
$\lim\limits_{p\to+\infty}[C(t-\tau)]^{p-2}(Cp)^{2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{\xi}\omega)|\/\mathrm{d}\/\xi}=0,$
while
$\lim\limits_{p\to+\infty}p^{-\frac{2}{p}}\mathopen{}\mathclose{{}\left[\frac{p^{2}}{(p-1)(p-2)}}\right]^{p-2}=e^{3}$.
Hence $\boldsymbol{v}(t)=0$ if $C(t-\tau)<1$. We repeat the same arguments a
finite number of times to cover the whole interval $[\tau,\tau+T]$, so that
the uniqueness is proved. ∎

######  Remark 3.7.

Using Moser estimates, Kato-Ponce commutator estimates (see [50]), and Beale-
Kato-Majda type logarithmic Sobolev inequality (see [33, Lemma 2.1] and [33,
Section 9]), the existence of a global smooth solution for the system (1.2)
(for $\boldsymbol{f}=\mathbf{0}$) has been established in [33, Theorem 4.6,
part(1)] for the initial data

$\boldsymbol{u}_{\tau}\in\mathbb{X}_{m,p}:=\mathopen{}\mathclose{{}\left\\{\boldsymbol{u}\in\mathbb{W}^{m,p}(\mathcal{O}):\nabla\cdot\boldsymbol{u}=0\
\text{ and }\
\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\mathcal{O}}=0}\right\\},$

where $m>2/p+1$, $p\in[2,\infty)$. The existence of global smooth solution is
obtained by showing that
$\sup\limits_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{W}^{1,\infty}}<\infty$,
$\mathbb{P}$-a.s. Therefore, if we take our initial data
$\boldsymbol{u}_{\tau}\in\mathbb{X}_{m,p},$ for $m>2/p+1$ with
$\nabla\land\boldsymbol{u}_{\tau}\in\mathrm{L}^{\infty}(\mathcal{O})$, then
the unique weak solution of the system (1.2) obtained in Theorems 3.3 and 3.6
is smooth satisfying
$\boldsymbol{u}(\cdot)\in\mathrm{C}([\tau,\infty);\mathbb{X}_{m,p})$,
$\mathbb{P}$-a.s.

If
$\boldsymbol{u}\in\mathrm{L}^{\infty}(0,T;\mathbb{W}^{1,\infty}(\mathcal{O}))$,
$\mathbb{P}$-a.s., then one can easily obtain the continuous dependence on the
data also. In fact an exponential stability result for $\sigma>0$. For
instance, an estimate similar to (3.6) yields

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\boldsymbol{v}(t)\|_{\mathbb{H}}^{2}+\sigma^{2}\|\boldsymbol{v}(t)\|_{\mathbb{H}}^{2}$
$\displaystyle=-2e^{\sigma
y(\theta_{t}\omega)}\langle(\boldsymbol{v}(t)\cdot\nabla)\boldsymbol{v}_{1}(t),\boldsymbol{v}(t)\rangle+2\sigma
y(\theta_{t}\omega)\|\boldsymbol{v}(t)\|_{\mathbb{H}}^{2}$ $\displaystyle\leq
2\bigg{[}e^{\sigma
y(\theta_{t}\omega)}\|\nabla\boldsymbol{v}_{1}(t)\|_{\mathbb{L}^{\infty}}+\sigma|y(\theta_{t}\omega)|\bigg{]}\|\boldsymbol{v}(t)\|_{\mathbb{H}}^{2},$
(3.55)

for a.e. $t\in[\tau,\tau+T]$. Therefore variation of constants formula gives

$\displaystyle\|\boldsymbol{v}(t)\|_{\mathbb{H}}^{2}\leq
e^{-\sigma^{2}(t-\tau)}\|\boldsymbol{v}_{\tau}\|_{\mathbb{H}}^{2}\exp\mathopen{}\mathclose{{}\left(2\mathopen{}\mathclose{{}\left[\int_{\tau}^{t}e^{\sigma
y(\theta_{\xi}\omega)}\|\nabla\boldsymbol{v}_{1}(\xi)\|_{\mathbb{L}^{\infty}}+\sigma|y(\theta_{\xi}\omega)|}\right]\mathrm{d}\xi}\right),\
\mathbb{P}\text{-a.s.,}$ (3.56)

for all $t\in[\tau,\tau+T]$, where
$\boldsymbol{v}_{\tau}=\boldsymbol{v}_{1}(\tau)-\boldsymbol{v}_{2}(\tau)$.

##  4 Pullback stochastic weak attractor and stochastic weak attractor

In this section, we demonstrate the existence of pullback stochastic weak
attractors (stochastic weak attractors) for 2D non-autonomous (autonomous)
stochastic EE driven by a linear multiplicative white noise. We apply the
abstract theory (Theorems 2.5 and 2.7) developed in [6] to prove the main
result of this section.

###  4.1 Existence of pullback stochastic weak attractor

Let $\delta$, $\/\mathrm{d}\/$ and $\/\mathrm{d}\/_{\infty}$ be the metrics
induced by the norms on $\mathbb{H}$, $\mathbb{V}$ and
$\mathrm{L}^{\infty}(\mathcal{O})$, respectively. We denote by

$\displaystyle\mathcal{W}=\\{\boldsymbol{v}\in\mathbb{V}:\nabla\land\boldsymbol{v}\in\mathrm{L}^{\infty}(\mathcal{O})\\}.$
(4.1)

Also, let $\/\mathrm{d}\/_{\mathcal{W}}$ be the metric on $\mathcal{W}$ which
is defined by

$\displaystyle\/\mathrm{d}\/_{\mathcal{W}}(\boldsymbol{u},\boldsymbol{v})=\/\mathrm{d}\/(\boldsymbol{u},\boldsymbol{v})+\/\mathrm{d}\/_{\infty}(\nabla\land\boldsymbol{u},\nabla\land\boldsymbol{v})=\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb{V}}+\|\nabla\land\boldsymbol{u}-\nabla\land\boldsymbol{v}\|_{\mathrm{L}^{\infty}(\mathcal{O})}.$

Theorems 3.3 and 3.6 allow us to define a family
$\\{S(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$ by
$S(t,\tau,\omega):\mathcal{W}\to\mathcal{W}$ such that

$\displaystyle
S(t,\tau,\omega)\boldsymbol{u}_{\tau}:=\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau}).$
(4.2)

The following lemma proves that the family
$\\{S(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$ satisfies Hypothesis 2.1.

######  Lemma 4.1.

The family of mappings $\\{S(t,\tau,\omega)\\}_{t\geq\tau,\omega\in\Omega}$
associated to the system (1.2) satisfies Hypothesis 2.1.

###### Proof.

The evolution property holds due to the uniqueness of the solution in
$\mathcal{W}$ (see Theorem 3.6). Now, fix $\omega\in\Omega$ and take
$t\geq\tau$. Let $\\{\boldsymbol{w}^{n}_{\tau}\\}_{n\in\mathbb{N}}$ be a
sequence which is $\/\mathrm{d}\/_{\mathcal{W}}$-bounded and
$\delta$-convergent to $\boldsymbol{w}_{\tau}$ in $\mathcal{W}$, that is,
there exists a positive number $K$ such that

$\displaystyle\|\boldsymbol{w}_{\tau}^{n}\|_{\mathcal{W}}\leq K,\ \text{ for
all }\ n\in\mathbb{N}\ \text{ and }\
\lim_{n\to+\infty}\|\boldsymbol{w}_{\tau}^{n}-\boldsymbol{w}_{\tau}\|_{\mathbb{H}}=0.$
(4.3)

Let $\boldsymbol{w}^{n}(\cdot)$ and $\boldsymbol{w}(\cdot)$ be the unique
solutions to the system (1.2) corresponding to the initial data
$\boldsymbol{w}^{n}_{\tau}$ and $\boldsymbol{w}_{\tau}$, respectively. Then
$\mathscr{W}_{n}(\cdot):=\boldsymbol{w}^{n}(\cdot)-\boldsymbol{w}(\cdot)$
satisfies for a.e. $t\in[\tau,\tau+T]$ and $\boldsymbol{\phi}\in\mathbb{V}$

$\displaystyle\mathopen{}\mathclose{{}\left\langle\frac{\/\mathrm{d}\/\mathscr{W}_{n}}{\/\mathrm{d}\/t},\boldsymbol{\phi}}\right\rangle+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right](\mathscr{W}_{n},\boldsymbol{\phi})=-e^{\sigma
y(\theta_{t}\omega)}\langle(\mathscr{W}_{n}\cdot\nabla)\boldsymbol{w},\boldsymbol{\phi}\rangle-e^{\sigma
y(\theta_{t}\omega)}\langle(\boldsymbol{w}^{n}\cdot\nabla)\mathscr{W}_{n},\boldsymbol{\phi}\rangle.$

Taking $\boldsymbol{\phi}=\mathscr{W}_{n}$ and arguing similarly as in the
proof of Theorem 3.6 (see (3.53)), we obtain (for $p>3$)

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\mathscr{W}_{n}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}}$
$\displaystyle\leq\frac{2e^{\sigma
y(\theta_{t}\omega)}}{p-2}\|\nabla\boldsymbol{w}(t)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\mathscr{W}_{n}(t)\|^{\frac{2}{p-2}}_{\mathbb{L}^{p}(\mathcal{O})}+\frac{2\sigma|y(\theta_{t}\omega)|}{p-2}\|\mathscr{W}_{n}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}}$
$\displaystyle\leq\frac{p^{2}Ce^{\sigma
y(\theta_{t}\omega)}}{(p-1)(p-2)}\|\nabla\land\boldsymbol{w}(t)\|_{\mathrm{L}^{p}(\mathcal{O})}\mathopen{}\mathclose{{}\left[p^{\frac{1}{2}}C\|\mathscr{W}_{n}(t)\|_{\mathbb{V}}}\right]^{\frac{2}{p-2}}+\frac{2\sigma|y(\theta_{t}\omega)|}{p-2}\|\mathscr{W}_{n}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}}$
$\displaystyle\leq\frac{p^{2}C(pC)^{\frac{1}{p-2}}e^{\sigma
y(\theta_{t}\omega)}}{(p-1)(p-2)}\|\nabla\land\boldsymbol{w}(t)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\bigg{[}\|\boldsymbol{w}^{n}(t)\|_{\mathbb{V}}+\|\boldsymbol{w}(t)\|_{\mathbb{V}}\bigg{]}^{\frac{2}{p-2}}+\frac{2\sigma|y(\theta_{t}\omega)|}{p-2}\|\mathscr{W}_{n}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}}$
$\displaystyle\leq\frac{p^{2}C_{T}(pC_{K,T})^{\frac{1}{p-2}}e^{\sigma
y(\theta_{t}\omega)}}{(p-1)(p-2)}+\frac{2\sigma|y(\theta_{t}\omega)|}{p-2}\|\mathscr{W}_{n}(t)\|^{\frac{2}{p-2}}_{\mathbb{H}},$
(4.4)

for a.e. $t\geq\tau$, where we have used inequalities (3.5), (3.38) and
(3.42), and the fact that $\boldsymbol{w}^{n},\boldsymbol{w}\in\mathcal{W}$.
Also, the constants $C_{T}$ and $C_{K,T}$ are independent of $p$. Now,
integration of (4.1) from $\tau$ to $t$ with $t\geq\tau$ yields

$\displaystyle\|\mathscr{W}_{n}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq\bigg{[}\|\boldsymbol{w}^{n}_{\tau}-\boldsymbol{w}_{\tau}\|^{\frac{2}{p-2}}_{\mathbb{H}}+\frac{p^{2}C_{T}(pC_{K,T})^{\frac{1}{p-2}}}{(p-1)(p-2)}\int_{\tau}^{t}e^{\sigma
y(\theta_{\xi}\omega)}\/\mathrm{d}\/\xi\bigg{]}^{p-2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{\xi}\omega)|\/\mathrm{d}\/\xi}.$
(4.5)

Further,
$\lim\limits_{n\to+\infty}\|\boldsymbol{w}_{\tau}^{n}-\boldsymbol{w}_{\tau}\|_{\mathbb{H}}=0$
provides for all $t\geq\tau$

$\displaystyle\lim_{n\to+\infty}\|\mathscr{W}_{n}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq pC_{K,T}\bigg{[}C_{T}\int_{\tau}^{t}e^{\sigma
y(\theta_{\xi}\omega)}\/\mathrm{d}\/\xi\bigg{]}^{p-2}\mathopen{}\mathclose{{}\left[\frac{p^{2}}{(p-1)(p-2)}}\right]^{p-2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{\xi}\omega)|\/\mathrm{d}\/\xi},$
(4.6)

where the constants $C_{K,T}$ and $C_{T}$ appearing in (4.6) are independent
of $p$. Let us now take $t$ sufficiently small such that
$C_{T}\int_{\tau}^{t}e^{\sigma y(\theta_{\xi}\omega)}\/\mathrm{d}\/\xi<1$. It
also implies that

$\lim\limits_{p\to+\infty}pC_{K,T}\mathopen{}\mathclose{{}\left[C_{T}\int_{\tau}^{t}e^{\sigma
y(\theta_{\xi}\omega)}\/\mathrm{d}\/\xi}\right]^{p-2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{\xi}\omega)|\/\mathrm{d}\/\xi}=0\
\ \ \text{ while }\ \
\lim\limits_{p\to+\infty}\mathopen{}\mathclose{{}\left[\frac{p^{2}}{(p-1)(p-2)}}\right]^{p-2}=e^{3}.$

Hence
$\lim\limits_{n\to+\infty}\|\boldsymbol{w}^{n}(t)-\boldsymbol{w}(t)\|_{\mathbb{H}}=0,$
for all $t\in(\tau,\tau+t_{0})$ such that
$C_{T}\int_{\tau}^{\tau+t_{0}}e^{\sigma
y(\theta_{\xi}\omega)}\/\mathrm{d}\/\xi<1$. We repeat the same arguments a
finite number of times to cover the whole interval $[\tau,\tau+T]$. Hence part
(ii) of Hypothesis 2.1 is satisfied. ∎

Our next aim is to show that the mapping $(\tau,\omega)\mapsto
S(t,\tau,\omega)\boldsymbol{w}$ satisfies Hypothesis 2.2. The following lemma
shows that for all $t\in\mathbb{R}$ and $\boldsymbol{w}\in\mathcal{W}$, the
mapping $(\tau,\omega)\mapsto S(t,\tau,\omega)\boldsymbol{w}$ is measurable
from $((-\infty,t],\mathscr{B}((-\infty,t])$ to
$(\mathcal{W},\mathscr{B}(\mathbb{H})\cap\mathcal{W})$.

######  Lemma 4.2.

For a given $\boldsymbol{u}^{*}_{\tau}\in\mathcal{W}$, $t\in\mathbb{R}$ and
$\omega\in\Omega$, the $\mathcal{W}$-valued mapping $\tau\mapsto
S(t,\tau,\omega)\boldsymbol{u}^{*}_{\tau}$ is continuous from $(-\infty,t]$ to
$\mathbb{H}$.

###### Proof.

First we show that, for a given $\boldsymbol{u}^{*}_{\tau}\in\mathcal{W}$,
$t\in\mathbb{R}$ and $\omega\in\Omega$, the $\mathcal{W}$-valued mapping
$\tau\mapsto S(t,\tau,\omega)\boldsymbol{u}^{*}_{\tau}$ is right continuous
from $(-\infty,t]$ to $\mathbb{H}$. Since

$S(t,\tau,\omega)\boldsymbol{u}^{*}_{\tau}=\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}^{*}_{\tau})=\boldsymbol{v}(t;\tau,\omega,\boldsymbol{v}^{*}_{\tau})e^{y(\theta_{t}\omega)},$

it is enough to prove that
$\boldsymbol{v}(t;\cdot,\omega,\boldsymbol{v}^{*}_{\tau})$ with
$\boldsymbol{v}^{*}_{\tau}=\boldsymbol{u}^{*}_{\tau}e^{y(\theta_{\cdot}\omega)}$
is right continuous from $(-\infty,t]$ to $\mathbb{H}$. Now fix
$\tau^{*}\in\mathbb{R}$, and take $\boldsymbol{v}^{*}_{\tau}\in\mathcal{W}$
and $\omega\in\Omega$. It is enough to show that for any given
$\varepsilon>0$, there exists a real number
$\varepsilon^{*}=\varepsilon^{*}(\varepsilon,\tau^{*},\omega,\boldsymbol{v}^{*}_{\tau})>0$
such that

$\displaystyle\|\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})-\boldsymbol{v}^{*}_{\tau}\|_{\mathbb{H}}<\varepsilon\
\text{ whenever }\ \tau\in(\tau^{*},\tau^{*}+\varepsilon^{*}),\
\xi\in(\tau,\tau^{*}+\varepsilon^{*}),$ (4.7)

where $\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})$ is the
solution of the system (1.2) with the initial data $\boldsymbol{v}^{*}_{\tau}$
and initial time $\tau$. We have

$\displaystyle\|\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})-\boldsymbol{v}^{*}_{\tau}\|^{2}_{\mathbb{H}}$
$\displaystyle=\|\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}-\|\boldsymbol{v}^{*}_{\tau}\|^{2}_{\mathbb{H}}-2(\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})-\boldsymbol{v}^{*}_{\tau},\boldsymbol{v}^{*}_{\tau})$
$\displaystyle=\int_{\tau}^{\xi}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/\zeta}\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}\/\mathrm{d}\/\zeta-2(\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})-\boldsymbol{v}^{*}_{\tau},\boldsymbol{v}^{*}_{\tau}).$
(4.8)

From (3.6), we have

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/\zeta}\|\boldsymbol{v}(\zeta)\|^{2}_{\mathbb{H}}+\frac{\sigma^{2}}{2}\|\boldsymbol{v}(\zeta)\|^{2}_{\mathbb{H}}\leq\frac{2e^{-2\sigma
y(\theta_{\zeta}\omega)}}{\sigma^{2}}\|\boldsymbol{f}(\zeta)\|^{2}_{\mathbb{H}}+2\sigma
y(\theta_{\zeta}\omega)\|\boldsymbol{v}(\zeta)\|^{2}_{\mathbb{H}},$ (4.9)

which implies

$\displaystyle\sup_{\tau\in[\tau^{*}-1,\tau^{*}+1]}\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}$
$\displaystyle\leq\sup_{\tau\in[\tau^{*}-1,\tau^{*}+1]}\bigg{[}\|\boldsymbol{v}^{*}_{\tau}\|^{2}_{\mathbb{H}}+\frac{2}{\sigma^{2}}\int_{\tau}^{\zeta}e^{-2\sigma
y(\theta_{r}\omega)}\|\boldsymbol{f}(r)\|^{2}_{\mathbb{H}}\/\mathrm{d}\/r\bigg{]}e^{2\sigma\int_{\tau}^{\zeta}|y(\theta_{r}\omega)|\/\mathrm{d}\/r}$
$\displaystyle\leq\mathopen{}\mathclose{{}\left[\|\boldsymbol{v}^{*}_{\tau}\|^{2}_{\mathbb{H}}+\frac{2}{\sigma^{2}}\int_{\tau^{*}-1}^{\zeta}e^{-2\sigma
y(\theta_{r}\omega)}\|\boldsymbol{f}(r)\|^{2}_{\mathbb{H}}\/\mathrm{d}\/r}\right]e^{2\sigma\int_{\tau^{*}-1}^{\zeta}|y(\theta_{r}\omega)|\/\mathrm{d}\/r},$
(4.10)

which is finite and the right hand side is independent of $\tau$. From (4.9),
we infer

$\displaystyle\int_{\tau}^{\xi}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/\zeta}\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}\/\mathrm{d}\/\zeta\leq\frac{2}{\sigma^{2}}\int_{\tau}^{\xi}z^{2}(\zeta,\omega)\|\boldsymbol{f}(\zeta)\|^{2}_{\mathbb{H}}\/\mathrm{d}\/\zeta+2\sigma\int_{\tau}^{\xi}|y(\theta_{\zeta}\omega)|\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}\/\mathrm{d}\/\zeta.$
(4.11)

It follows from (4.1)-(4.11) that there exists a real number
$\varepsilon^{*}_{1}=\varepsilon^{*}_{1}(\varepsilon,\tau^{*},\omega,\boldsymbol{v}^{*}_{\tau})>0$
such that

$\displaystyle\int_{\tau}^{\xi}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/\zeta}\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}\/\mathrm{d}\/\zeta\leq\frac{\varepsilon^{2}}{2},\
\text{ whenever }\ \tau\in(\tau^{*},\tau^{*}+\varepsilon^{*}_{1}),\
\xi\in(\tau,\tau^{*}+\varepsilon^{*}_{1}),$ (4.12)

by the absolute continuity of the Lebesgue integral. Now, using the first
equation of the system (3.6), we have

$\displaystyle|(\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})-\boldsymbol{v}^{*}_{\tau},{\boldsymbol{v}^{*}_{\tau}})|$
$\displaystyle=\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle\int^{\xi}_{\tau}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/\zeta}\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\/\mathrm{d}\/\zeta,{\boldsymbol{v}^{*}_{\tau}}}\right\rangle}\right|$
$\displaystyle\leq\|{\boldsymbol{v}^{*}_{\tau}}\|_{\mathbb{H}}\bigg{(}\int^{\xi}_{\tau}\bigg{\|}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/\zeta}\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\bigg{\|}^{2}_{\mathbb{H}}\/\mathrm{d}\/\zeta\bigg{)}^{1/2}(\tau-\xi)^{1/2}$
$\displaystyle\leq
C\|{\boldsymbol{v}^{*}_{\tau}}\|_{\mathbb{H}}\bigg{(}\int^{\xi}_{\tau}\bigg{[}(1+|y(\theta_{\zeta}\omega)|)\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}+e^{2\sigma
y(\theta_{\zeta}\omega)}\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{L}^{4}(\mathcal{O})}$
$\displaystyle\qquad\times\|\nabla\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{L}^{4}(\mathcal{O})}+e^{-2\sigma
y(\theta_{\zeta}\omega)}\|\boldsymbol{f}(\zeta)\|^{2}_{\mathbb{H}}\bigg{]}\/\mathrm{d}\/\zeta\bigg{)}^{1/2}(\tau-\xi)^{1/2}$
$\displaystyle\leq
C\|{\boldsymbol{v}^{*}_{\tau}}\|_{\mathbb{H}}\bigg{(}\int^{\xi}_{\tau}\bigg{[}(1+|y(\theta_{\zeta}\omega)|)\|\boldsymbol{v}(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathbb{H}}+e^{2\sigma
y(\theta_{\zeta}\omega)}\|\varrho(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathrm{L}^{2}(\mathcal{O})}$
$\displaystyle\qquad\times\|\varrho(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|^{2}_{\mathrm{L}^{4}(\mathcal{O})}+e^{-2\sigma
y(\theta_{\zeta}\omega)}\|\boldsymbol{f}(\zeta)\|^{2}_{\mathbb{H}}\bigg{]}\/\mathrm{d}\/\zeta\bigg{)}^{1/2}(\tau-\xi)^{1/2},$
(4.13)

where we have used (3.38) and (3.42) in the final inequality. Taking the inner
product of first equation of (3.7) with $|\varrho|^{k-2}\varrho_{\nu}$ with
$k\in\\{2,4\\}$, we obtain (see (3.6))

$\displaystyle\frac{1}{k}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/\zeta}\|\varrho(\zeta)\|^{k}_{\mathrm{L}^{k}(\mathcal{O})}+\frac{\sigma^{2}}{2}\|\varrho(\zeta)\|^{k}_{\mathrm{L}^{k}(\mathcal{O})}$
$\displaystyle\leq Ce^{-k\sigma
y(\theta_{\zeta}\omega)}\|\nabla\land\boldsymbol{f}(\zeta)\|^{k}_{\mathrm{L}^{k}(\mathcal{O})}+(1+\sigma|y(\theta_{\zeta}\omega)|\|\varrho(\zeta)\|^{k}_{\mathrm{L}^{k}(\mathcal{O})},$

for a.e. $\zeta\in[0,T]$, which implies by Gronwall’s inequality

$\displaystyle\sup_{\tau\in[\tau^{*}-1,\tau^{*}+1]}\|\varrho(\zeta;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\|_{\mathrm{L}^{k}(\mathcal{O})}^{k}$
$\displaystyle\leq
C\bigg{[}\|\nabla\land\boldsymbol{v}^{*}_{\tau}\|_{\mathrm{L}^{k}(\mathcal{O})}^{k}+\|\nabla\land\boldsymbol{f}\|^{k}_{\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{k}(\mathcal{O}))}\int_{\tau^{*}-1}^{\zeta}e^{-k\sigma
y(\theta_{r}\omega)}\/\mathrm{d}\/r\bigg{]}e^{\int_{\tau^{*}-1}^{\zeta}(1+\sigma|y(\theta_{r}\omega)|)\/\mathrm{d}\/r},$
(4.14)

which is finite and the right hand side is independent of $\tau$. From the
fact that
$\boldsymbol{v}(\cdot;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\in\mathrm{C}([\tau,+\infty);\mathbb{H})$,
$\varrho(\cdot;\tau,\omega,\boldsymbol{v}^{*}_{\tau})\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\tau,+\infty;\mathrm{L}^{p}(\mathcal{O}))$
for $p\in[2,\infty)$, continuity of $y(\cdot)$, (4.1) and (4.1), it follows
from (4.1) that for $\varepsilon>0$ same as in (4.7), we can find a positive
real number
$\varepsilon^{*}_{2}=\varepsilon^{*}_{2}(\varepsilon,\tau^{*},\omega,\boldsymbol{v}^{*}_{\tau})$
such that

$\displaystyle|(\boldsymbol{v}(\xi;\tau,\omega,\boldsymbol{v}^{*}_{\tau})-\boldsymbol{v}^{*}_{\tau},{\boldsymbol{v}^{*}_{\tau}})|\leq\frac{\varepsilon^{2}}{2},\
\text{ whenever }\ \tau\in(\tau^{*},\tau^{*}+\varepsilon^{*}_{2}),\
\xi\in(\tau,\tau^{*}+\varepsilon^{*}_{2}).$ (4.15)

Taking $\varepsilon^{*}=\min\\{\varepsilon^{*}_{1},\varepsilon^{*}_{2}\\}$, we
obtain (4.7) by combining (4.1), (4.12) and (4.15).

Further, one can prove that, for given
$\boldsymbol{u}^{*}_{\tau}\in\mathcal{W}$, $t\in\mathbb{R}$ and
$\omega\in\Omega$, the $\mathcal{W}$-valued mapping $\tau\mapsto
S(t,\tau,\omega)\boldsymbol{u}^{*}_{\tau}$ is left continuous from
$(-\infty,t]$ to $\mathbb{H}$ following the similar steps as above. This
completes the proof. ∎

Since, $\omega(\cdot)$ has sub-exponential growth (cf. [14, Lemma 11]),
$\Omega$ can be written as $\Omega=\bigcup\limits_{N\in\mathbb{N}}\Omega_{N}$,
where

$\displaystyle\Omega_{N}:=\\{\omega\in\Omega:|\omega(t)|\leq Ne^{|t|},\text{
for all }t\in\mathbb{R}\\},\text{ for all }\ N\in\mathbb{N}.$

Moreover, for each $N\in\mathbb{N}$, $(\Omega_{N},d_{\Omega_{N}})$ is a polish
space (cf. [14, Lemma 17]).

######  Lemma 4.3.

For each $N\in\mathbb{N}$, suppose $\omega_{k},\omega_{0}\in\Omega_{N}$ such
that $d_{\Omega}(\omega_{k},\omega_{0})\to 0$ as $k\to+\infty$. Then, for each
$\tau\in\mathbb{R}$, $T\in\mathbb{R}^{+}$ and $a\in\mathbb{R}$ ,

$\displaystyle\sup_{t\in[\tau,\tau+T]}$
$\displaystyle\bigg{[}|y(\theta_{t}\omega_{k})-y(\theta_{t}\omega_{0})|+|e^{ay(\theta_{t}\omega_{k})}-e^{ay(\theta_{t}\omega_{0})}|\bigg{]}\to
0\ \text{ as }\ k\to+\infty,$ (4.16)
$\displaystyle\sup_{k\in\mathbb{N}}\sup_{t\in[\tau,\tau+T]}$
$\displaystyle|y(\theta_{t}\omega_{k})|\leq C(\tau,T,\omega_{0}).$ (4.17)

###### Proof.

The proof is straightforward (cf. [25, Corollary 22] and [48, Lemma 2.5]). ∎

The following lemma proves that for all $t\in\mathbb{R}$ and
$\boldsymbol{w}\in\mathcal{W}$, the mapping $(\tau,\omega)\mapsto
S(t,\tau,\omega)\boldsymbol{w}$ is measurable from $(\Omega,\mathscr{F})$ to
$(\mathcal{W},\mathscr{B}(\mathcal{H})\cap\mathcal{W})$.

######  Lemma 4.4.

Suppose that $\tau\in\mathbb{R}$, $t\geq\tau$ and
$\boldsymbol{u}_{\tau}\in\mathcal{W}$. For each $N\in\mathbb{N}$, the
$\mathcal{W}$-valued mapping $\omega\mapsto
S(t,\tau,\omega)\boldsymbol{u}_{\tau}$ is continuous from
$(\Omega_{N},d_{\Omega_{N}})$ to $\mathbb{H}$.

###### Proof.

Let
$\mathbf{V}^{k}(\cdot):=\boldsymbol{v}^{k}(\cdot)-\boldsymbol{v}^{0}(\cdot),$
where
$\boldsymbol{v}^{k}(\cdot):=\boldsymbol{v}(\cdot;\tau,\omega_{k},\boldsymbol{v}_{\tau})$
and
$\boldsymbol{v}^{0}(\cdot):=\boldsymbol{v}(\cdot;\tau,\omega_{0},\boldsymbol{v}_{\tau})$.
Then, $\mathbf{V}^{k}(\cdot)$ satisfies:

$\displaystyle\frac{\/\mathrm{d}\/\mathbf{V}^{k}}{\/\mathrm{d}\/r}+\frac{\sigma^{2}}{2}\mathbf{V}^{k}$
$\displaystyle=-e^{\sigma
y(\theta_{r}\omega_{0})}\big{[}\mathrm{B}(\boldsymbol{v}^{k},\mathbf{V}^{k})+\mathrm{B}(\mathbf{V}^{k},\boldsymbol{v}^{0})\big{]}-\mathopen{}\mathclose{{}\left[e^{\sigma
y(\theta_{r}\omega_{k})}-e^{\sigma
y(\theta_{r}\omega_{0})}}\right]\big{[}\mathrm{B}(\boldsymbol{v}^{k},\mathbf{V}^{k})+\mathrm{B}(\boldsymbol{v}^{k},\boldsymbol{v}^{0})\big{]}$
$\displaystyle\quad+[e^{-\sigma y(\theta_{r}\omega_{k})}-e^{-\sigma
y(\theta_{r}\omega_{0})}]\boldsymbol{f}+\sigma\big{[}y(\theta_{r}\omega_{k})-y(\theta_{r}\omega_{0})\big{]}\boldsymbol{v}^{0}+\sigma
y(\theta_{r}\omega_{k})\mathbf{V}^{k},$

in $\mathbb{H}$. By taking the inner product with $\mathbf{V}^{k}(\cdot)$ in
the above equation, we obtain (for $p>3$)

$\displaystyle\frac{1}{2}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/r}\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}+\frac{\sigma^{2}}{2}\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}$
$\displaystyle=-e^{\sigma
y(\theta_{r}\omega_{0})}b(\mathbf{V}^{k}(r),\boldsymbol{v}^{0}(r),\mathbf{V}^{k}(r))-\mathopen{}\mathclose{{}\left[e^{\sigma
y(\theta_{r}\omega_{k})}-e^{\sigma
y(\theta_{r}\omega_{0})}}\right]b(\boldsymbol{v}^{k}(r),\boldsymbol{v}^{0}(r),\mathbf{V}^{k}(r))$
$\displaystyle\quad+[e^{-\sigma y(\theta_{r}\omega_{k})}-e^{-\sigma
y(\theta_{r}\omega_{0})}](\boldsymbol{f}(r),\mathbf{V}^{k}(r))+\sigma\big{[}y(\theta_{r}\omega_{k})-y(\theta_{r}\omega_{0})\big{]}(\boldsymbol{v}^{0}(r),\mathbf{V}^{k}(r))$
$\displaystyle\quad+\sigma
y(\theta_{r}\omega_{k})\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq e^{\sigma
y(\theta_{r}\omega_{0})}\|\nabla\boldsymbol{v}^{0}(r)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{L}^{\frac{2p}{p-1}}(\mathcal{O})}+|e^{-\sigma
y(\theta_{r}\omega_{k})}-e^{-\sigma
y(\theta_{r}\omega_{0})}|\|\boldsymbol{f}(r)\|_{\mathbb{H}}\|\mathbf{V}^{k}(r)\|_{\mathbb{H}}$
$\displaystyle\quad+\mathopen{}\mathclose{{}\left|e^{\sigma
y(\theta_{r}\omega_{k})}-e^{\sigma
y(\theta_{r}\omega_{0})}}\right|\|\boldsymbol{v}^{k}(r)\|_{\mathbb{L}^{4}(\mathcal{O})}\|\nabla\boldsymbol{v}^{0}(r)\|_{\mathbb{L}^{4}(\mathcal{O})}\|\mathbf{V}^{k}(r)\|_{\mathbb{H}}$
$\displaystyle\quad+\sigma|y(\theta_{r}\omega_{k})-y(\theta_{r}\omega_{0})|\|\boldsymbol{v}^{0}(r)\|_{\mathbb{H}}\|\mathbf{V}^{k}(r)\|_{\mathbb{H}}+\sigma|y(\theta_{r}\omega_{k})|\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq e^{\sigma
y(\theta_{r}\omega_{0})}\|\nabla\boldsymbol{v}^{0}(r)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\mathbf{V}^{k}(r)\|^{\frac{2}{p-2}}_{\mathbb{L}^{p}(\mathcal{O})}\|\mathbf{V}^{k}(r)\|^{\frac{2(p-3)}{p-2}}_{\mathbb{H}}+C|e^{-\sigma
y(\theta_{r}\omega_{k})}-e^{-\sigma
y(\theta_{r}\omega_{0})}|^{2}\|\boldsymbol{f}(r)\|^{2}_{\mathbb{H}}$
$\displaystyle\quad+C\mathopen{}\mathclose{{}\left|e^{\sigma
y(\theta_{r}\omega_{k})}-e^{\sigma
y(\theta_{r}\omega_{0})}}\right|^{2}\|\boldsymbol{v}^{k}(r)\|^{2}_{\mathbb{L}^{4}(\mathcal{O})}\|\nabla\boldsymbol{v}^{0}(r)\|^{2}_{\mathbb{L}^{4}(\mathcal{O})}+C|y(\theta_{r}\omega_{k})-y(\theta_{r}\omega_{0})|^{2}\|\boldsymbol{v}^{0}(r)\|^{2}_{\mathbb{H}}$
$\displaystyle\quad+\sigma|y(\theta_{r}\omega_{k})|\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}+\frac{\sigma^{2}}{4}\|\mathbf{V}^{k}(r)\|_{\mathbb{H}}^{2}$
$\displaystyle\leq(Cp)^{\frac{1}{p-2}}\frac{Cp^{\frac{1}{p}}p^{2}}{p-1}\|\mathbf{V}^{k}(r)\|^{\frac{2(p-3)}{p-2}}_{\mathbb{H}}+C\mathopen{}\mathclose{{}\left|e^{\sigma
y(\theta_{r}\omega_{k})}-e^{\sigma
y(\theta_{r}\omega_{0})}}\right|^{2}+C|y(\theta_{r}\omega_{k})-y(\theta_{r}\omega_{0})|^{2}$
$\displaystyle\quad+C|e^{-\sigma y(\theta_{r}\omega_{k})}-e^{-\sigma
y(\theta_{r}\omega_{0})}|^{2}\|\boldsymbol{f}(r)\|^{2}_{\mathbb{H}}+\sigma|y(\theta_{r}\omega_{k})|\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}+\frac{\sigma^{2}}{4}\|\mathbf{V}^{k}(r)\|_{\mathbb{H}}^{2},$

for a.e. $r\in[\tau,\tau+T]$, where we have used the bounds from (3.47) and
(4.16), and the constant

$\displaystyle
C=C(T,\omega_{0},\mathcal{O},\|\boldsymbol{u}_{\tau}\|_{\mathbb{V}},\|\boldsymbol{f}\|_{\mathrm{L}^{2}(\tau,\tau+T;\mathbb{V})},\|\nabla\land\boldsymbol{u}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})},\|\nabla\land\boldsymbol{f}\|_{\mathrm{L}^{\infty}(\tau,\tau+T;\mathrm{L}^{\infty}(\mathcal{O}))})>0,$

is independent of $p$ and $k$. Taking integration from $\tau$ to $t>\tau$, we
obtain

$\displaystyle\|\mathbf{V}^{k}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\leq|e^{-2\sigma y(\theta_{\tau}\omega_{k})}-e^{-2\sigma
y(\theta_{\tau}\omega_{0})}|\|\boldsymbol{u}_{\tau}\|^{2}_{\mathbb{H}}+C\int_{\tau}^{t}|e^{-\sigma
y(\theta_{r}\omega_{k})}-e^{-\sigma
y(\theta_{r}\omega_{0})}|^{2}\|\boldsymbol{f}(r)\|^{2}_{\mathbb{H}}\/\mathrm{d}\/r$
$\displaystyle\quad+C\int_{\tau}^{t}\mathopen{}\mathclose{{}\left|e^{\sigma
y(\theta_{r}\omega_{k})}-e^{\sigma
y(\theta_{r}\omega_{0})}}\right|^{2}\/\mathrm{d}\/r+(Cp)^{\frac{1}{p-2}}\frac{Cp^{\frac{1}{p}}p^{2}}{p-1}\int_{\tau}^{t}\|\mathbf{V}^{k}(r)\|^{\frac{2(p-3)}{p-2}}_{\mathbb{H}}\/\mathrm{d}\/r$
$\displaystyle\quad+2\sigma\int_{\tau}^{t}|y(\theta_{r}\omega)|\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}\/\mathrm{d}\/r$
(4.18)
$\displaystyle=:M(k)+(Cp)^{\frac{1}{p-2}}\frac{Cp^{\frac{1}{p}}p^{2}}{p-1}\int_{\tau}^{t}\|\mathbf{V}^{k}(r)\|^{\frac{2(p-3)}{p-2}}_{\mathbb{H}}\/\mathrm{d}\/r+2\sigma\int_{\tau}^{t}|y(\theta_{r}\omega)|\|\mathbf{V}^{k}(r)\|^{2}_{\mathbb{H}}\/\mathrm{d}\/r,$

where $M(k)$ is the first three terms appearing in the right hand side of
(4.18). By applying [27, Theorem 21] to (4.18), we arrive at

$\displaystyle\|\mathbf{V}^{k}(t)\|^{2}_{\mathbb{H}}\leq\bigg{[}\\{M(k)\\}^{\frac{1}{p-2}}+\frac{(Cp)^{\frac{1}{p-2}}Cp^{\frac{1}{p}}p^{2}}{(p-2)(p-1)}(t-\tau)\bigg{]}^{p-2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{r}\omega)|\/\mathrm{d}\/r},$

for all $t\in[\tau,\tau+T]$. Now, using the convergence in (4.16), we find

$\displaystyle\lim_{k\to+\infty}\|\mathbf{V}^{k}(t)\|^{2}_{\mathbb{H}}\leq(Cp)^{2}\\{C(t-\tau)\\}^{p-2}\frac{1}{p^{\frac{2}{p}}}\bigg{[}\frac{p^{2}}{(p-2)(p-1)}\bigg{]}^{p-2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{r}\omega)|\/\mathrm{d}\/r}.$

Let us now take $t$ sufficiently small such that $C(t-\tau)<1$. Further, it
provides us that
$\lim\limits_{p\to+\infty}[C(t-\tau)]^{p-2}(Cp)^{2}e^{2\sigma\int_{\tau}^{t}|y(\theta_{r}\omega)|\/\mathrm{d}\/r}=0,$
whereas
$\lim\limits_{p\to+\infty}p^{-\frac{2}{p}}\mathopen{}\mathclose{{}\left[\frac{p^{2}}{(p-1)(p-2)}}\right]^{p-2}=e^{3}$.
Hence

$\lim\limits_{k\to+\infty}\|\mathbf{V}^{k}(t)\|^{2}_{\mathbb{H}}=0\ \ \mbox{
if }\ C(t-\tau)<1.$

We repeat the same arguments a finite number of times to cover the interval
$[\tau,\tau+T]$, which completes the proof. ∎

######  Lemma 4.5.

Suppose that Hypothesis 1.2 holds. Then, there exists a
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded absorbing set for each $t\in\mathbb{R}$
which is $\delta$-compact.

###### Proof.

From (3.3) and (3.6), we infer

$\displaystyle\|\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau})\|^{2}_{\mathbb{V}}$
$\displaystyle\leq
Ce^{-\frac{\sigma^{2}}{2}t+2\sigma\int_{0}^{t}y(\theta_{r}\omega)\/\mathrm{d}\/r+2\sigma
y(\theta_{t}\omega)}\bigg{[}e^{\frac{\sigma^{2}}{2}\tau+2\sigma\int_{\tau}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-2\sigma
y(\theta_{\tau}\omega)}\|\boldsymbol{u}_{\tau}\|_{\mathbb{V}}^{2}$
$\displaystyle\quad+\int_{\tau}^{t}e^{\frac{\sigma^{2}}{2}\xi+2\sigma\int_{\xi}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-2\sigma
y(\theta_{\xi}\omega)}\big{\\{}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi\bigg{]},$
(4.19)

and

$\displaystyle\|\nabla\land\boldsymbol{u}(t;\tau,\omega,\boldsymbol{u}_{\tau})\|_{\mathbb{L}^{\infty}(\mathcal{O})}$
$\displaystyle\leq
e^{-\frac{\sigma^{2}}{2}t+2\sigma\int_{0}^{t}y(\theta_{r}\omega)\/\mathrm{d}\/r+\sigma
y(\theta_{t}\omega)}\bigg{[}e^{\frac{\sigma^{2}}{2}\tau+2\sigma\int_{\tau}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-\sigma
y(\theta_{\tau}\omega)}\|\nabla\land\boldsymbol{u}_{\tau}\|_{\mathrm{L}^{\infty}(\mathcal{O})}$
$\displaystyle\quad+\int_{\tau}^{t}e^{\frac{\sigma^{2}}{2}\xi+2\sigma\int_{\xi}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-\sigma
y(\theta_{\xi}\omega)}\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\/\mathrm{d}\/\xi\bigg{]}.$
(4.20)

From (3.3), we obtain the existence of $R_{1},R_{2},R_{3},R_{4}<0$ such that

$\displaystyle-2\sigma y(\theta_{\xi}\omega)$
$\displaystyle\leq-\mathopen{}\mathclose{{}\left(\frac{\sigma^{2}}{4}-\frac{\delta_{1}}{2}}\right)\xi,\
\ \ \ \ \ \ \mbox{for all $\xi\leq R_{1},$}$ $\displaystyle-\sigma
y(\theta_{\xi}\omega)$
$\displaystyle\leq-\mathopen{}\mathclose{{}\left(\frac{\sigma^{2}}{4}-\frac{\delta_{2}}{2}}\right)\xi,\
\ \ \ \ \ \ \mbox{for all $\xi\leq R_{2},$}$
$\displaystyle\frac{\sigma^{2}}{2}\xi+2\sigma\int^{0}_{\xi}y(\theta_{r}\omega)\/\mathrm{d}\/r$
$\displaystyle\leq\mathopen{}\mathclose{{}\left(\frac{\sigma^{2}}{4}+\frac{\delta_{1}}{2}}\right)\xi,\
\ \ \ \ \ \ \ \mbox{for all $\xi\leq R_{3},$}$
$\displaystyle\frac{\sigma^{2}}{2}\xi+2\sigma\int^{0}_{\xi}y(\theta_{r}\omega)\/\mathrm{d}\/r$
$\displaystyle\leq\mathopen{}\mathclose{{}\left(\frac{\sigma^{2}}{4}+\frac{\delta_{2}}{2}}\right)\xi,\
\ \ \ \ \ \ \ \mbox{for all $\xi\leq R_{4},$}$

where $\sigma$ is the constant appearing in (1.2), and $\delta_{1}$ and
$\delta_{2}$ are the constants appearing in (1.3). Therefore, for all $\xi\leq
R:=\min\\{R_{1},R_{2},R_{3},R_{4}\\}$,

$\displaystyle\frac{\sigma^{2}}{2}\xi+2\sigma\int^{0}_{\xi}y(\theta_{r}\omega)\/\mathrm{d}\/r-2\sigma
y(\theta_{\xi}\omega)$ $\displaystyle\leq\delta_{1}\xi,$ (4.21)

and

$\displaystyle\frac{\sigma^{2}}{2}\xi+2\sigma\int^{0}_{\xi}y(\theta_{r}\omega)\/\mathrm{d}\/r-\sigma
y(\theta_{\xi}\omega)$ $\displaystyle\leq\delta_{2}\xi.$ (4.22)

From (4.21)-(4.22) and Hypothesis 1.2, we infer that the right hand sides of
(4.1) and (4.1) are bounded as $\tau\to-\infty$. Therefore, for given bounded
subset $B\subset\mathcal{W}$, $t\in\mathbb{R}$ and $\omega\in\Omega$, there
exist a $\/\mathrm{d}\/_{\mathcal{W}}$-bounded set
$\mathcal{B}(t,\omega)\subset\mathcal{W}$, and a time $\tau_{0}(B)$ depending
on $B$ such that

$\displaystyle S(t,\tau,\omega)B\subset\mathcal{B}(t,\omega),\ \ \text{ for
all }\ \tau\leq\tau_{0}.$

Since the embedding $\mathbb{V}\subset\mathbb{H}$ is compact, we deduce that
there exists a $\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact
random ball which absorbs $\/\mathrm{d}\/_{\mathcal{W}}$-bounded sets of
$\mathcal{W},$ for all $t\in\mathbb{R}$ and $\omega\in\Omega$, and this
completes the proof. ∎

The following theorem demonstrates the main result of this section which is a
direct consequence of the abstract result (Theorem 2.5).

######  Theorem 4.6.

Suppose that Hypotheses 1.2, 2.1 and 2.2 hold. Then, there exists a stochastic
weak attractor $\mathscr{A}(t,\omega)$ for the system (1.2) for all
$t\in\mathbb{R}$ and $\omega\in\Omega$, which attracts bounded sets from
$-\infty$.

###### Proof.

Lemma 4.1 ensures that Hypothesis 2.1 is satisfied, Lemmas 4.2 and 4.4 verify
Hypothesis 2.2, and Lemma 4.5 proves the existence of
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact random absorbing
set. Therefore, by Theorem 2.5, there exists a stochastic weak attractor for
the system (1.2) for all $t\in\mathbb{R}$ and $\omega\in\Omega$, which
attracts $\/\mathrm{d}\/_{\mathcal{W}}$-bounded sets from $-\infty$. Hence the
proof is completed. ∎

###  4.2 Existence of stochastic weak attractor

Let us now consider the following autonomous 2D stochastic EE in
$\mathcal{O}$:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\/\mathrm{d}\/\bar{\boldsymbol{u}}+\mathopen{}\mathclose{{}\left[(\bar{\boldsymbol{u}}\cdot\nabla)\bar{\boldsymbol{u}}+\nabla\bar{p}}\right]\/\mathrm{d}\/t&=\boldsymbol{f}_{\infty}\/\mathrm{d}\/t+\sigma\bar{\boldsymbol{u}}\/\mathrm{d}\/\mathrm{W},\
\text{ in }\ \mathcal{O}\times(\tau,\infty),\\\
\nabla\cdot\bar{\boldsymbol{u}}&=0,\hskip 73.97716pt\text{ in }\
\mathcal{O}\times[0,\infty),\\\
\bar{\boldsymbol{u}}\cdot\boldsymbol{n}&=0,\hskip 73.97716pt\text{ on }\
\partial\mathcal{O}\times[0,\infty),\\\
\bar{\boldsymbol{u}}|_{t=0}&=\bar{\boldsymbol{u}}_{0},\hskip 68.2866pt\text{
in }\ \mathcal{O},\\\ \end{aligned}}\right.$ (4.23)

where $\boldsymbol{f}_{\infty}$ is a time-independent external forcing. Define
a new variable $\bar{\boldsymbol{v}}$ by

$\displaystyle\bar{\boldsymbol{v}}(t;\omega,\boldsymbol{v}_{0})=e^{-\sigma
y(\theta_{t}\omega)}\bar{\boldsymbol{u}}(t;\omega,\boldsymbol{u}_{0})\ \ \
\text{ with }\ \ \ \bar{\boldsymbol{v}}_{0}=e^{-\sigma
y(\omega)}\bar{\boldsymbol{u}}_{0},$ (4.24)

where $\bar{\boldsymbol{u}}(t;\omega,\boldsymbol{u}_{0})$ and
$y(\theta_{t}\omega)$ are the solutions of (4.23) and (3.2), respectively.
Then
$\bar{\boldsymbol{v}}(\cdot):=\bar{\boldsymbol{v}}(\cdot;\omega,\boldsymbol{v}_{0})$
satisfies:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\/\mathrm{d}\/\bar{\boldsymbol{v}}}{\/\mathrm{d}\/t}+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right]\bar{\boldsymbol{v}}+e^{\sigma
y(\theta_{t}\omega)}(\bar{\boldsymbol{v}}\cdot\nabla)\bar{\boldsymbol{v}}+e^{-\sigma
y(\theta_{t}\omega)}\nabla\bar{p}&=e^{-\sigma
y(\theta_{t}\omega)}\boldsymbol{f}_{\infty},\hskip 14.22636pt\text{ in }\
\mathcal{O}\times(\tau,\infty),\\\ \nabla\cdot\bar{\boldsymbol{v}}&=0,\hskip
48.36967pt\text{ in }\ \mathcal{O}\times(\tau,\infty),\\\
\bar{\boldsymbol{v}}\cdot\boldsymbol{n}&=0,\hskip 48.36967pt\text{ on }\
\partial\mathcal{O}\times(\tau,\infty),\\\
\bar{\boldsymbol{v}}|_{t=0}&=\bar{\boldsymbol{v}}_{0},\hskip 42.67912pt\text{
in }\mathcal{O},\end{aligned}}\right.$ (4.25)

in the distributional sense, where $\sigma>0$. We have shown in Theorem 4.6
that there exists a pullback stochastic weak attractor $\mathscr{A}(t,\omega)$
for the system (1.2) with time-dependent forcing $\boldsymbol{f}$. For the
time-independent forcing
$\boldsymbol{f}_{\infty}\in\mathbb{L}^{2}(\mathcal{O})$ and
$\nabla\land\boldsymbol{f}_{\infty}\in\mathrm{L}^{\infty}(\mathcal{O}),$ the
Hypothesis 1.2 is automatically satisfied for
$\delta_{1},\delta_{2}\in(0,\frac{\sigma^{2}}{2})$ . Since the sample space is
the two-sided Wiener space $\mathrm{C}_{0}(\mathbb{R};\mathbb{R})$, Hypothesis
2.6 is satisfied. In view of the abstract result stated in Theorem 2.7, the
proof of the following Theorem is analogues to the proof of Theorem 4.6.
Therefore, we are only state the theorem and not repeat the proof here.

######  Theorem 4.7.

Suppose that Hypotheses 2.1 and 2.2 hold. Let the time-independent forcing
$\boldsymbol{f}_{\infty}$ be such that
$\boldsymbol{f}_{\infty}\in\mathbb{L}^{2}(\mathcal{O})$ and
$\nabla\land\boldsymbol{f}_{\infty}\in\mathrm{L}^{\infty}(\mathcal{O})$. Then,
there exists a stochastic weak attractor $\mathscr{A}_{\infty}(\omega)$ for
the system (4.23) for all $\omega\in\Omega$, which attracts bounded sets from
$-\infty$.

##  5 Weak asymptotic autonomy of pullback stochastic weak attractors

In this section, we apply the abstract results established in Section 2.2 to
the 2D non-autonomous stochastic Euler system (1.2). In Section 4, we proved
the existence of a minimal pullback stochastic weak attractor (Theorem 2.5)
for the system (1.2) and the existence of a minimal stochastic weak attractor
(Theorem 2.7) for the system (4.23). In order to apply Theorem 2.10, we have
to show that
$\\{\mathscr{A}(\tau,\omega)\\}_{\tau\in\mathbb{R},\omega\in\Omega}$ and
$\\{\mathscr{A}_{\infty}(\omega)\\}_{\omega\in\Omega}$ satisfy the all three
conditions stated in Theorem 2.10.

The following lemma provides us that
$\\{\mathscr{A}(\tau,\omega)\\}_{\tau\in\mathbb{R},\omega\in\Omega}$ satisfies
condition $(\mathrm{i})$ of Theorem 2.10.

######  Lemma 5.1.

$\overline{\bigcup\limits_{t\geq\tau}\mathscr{A}(t,\omega)}^{\delta}$ is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact.

###### Proof.

Let $\\{\boldsymbol{w}_{n}\\}_{n=1}^{\infty}$ be an arbitrary sequence
extracted from $\bigcup\limits_{t\geq\tau}\mathscr{A}(t,\omega)$. Then, there
exists a sequence $t_{n}\geq\tau$ such that
$\boldsymbol{w}_{n}\in\mathscr{A}(t_{n},\omega)$ for each $n\in\mathbb{N}$.
Now, for the sequence $s_{n}\to+\infty$, by the invariance property of
$\mathscr{A},$ we have $\boldsymbol{w}_{n}\in
S(t_{n},t_{n}-s_{n},\omega)\mathscr{A}(t_{n}-s_{n},\theta_{-s_{n}}\omega)$. It
implies that we can find
$\boldsymbol{w}_{0,n}\in\mathscr{A}(t_{n}-s_{n},\theta_{-s_{n}}\omega)$ such
that $\boldsymbol{w}_{n}=S(t_{n},t_{n}-s_{n},\omega)\boldsymbol{w}_{0,n}$. By
the property of absorbing set, there exists a number $N_{0}\in\mathbb{N}$ such
that
$\boldsymbol{w}_{0,n}\in\mathscr{A}(t_{n}-s_{n},\theta_{-s_{n}}\omega)\subseteq\mathcal{B}(t_{n}-s_{n},\theta_{-s_{n}}\omega)$
for all $n\geq N_{0}$. Since the absorbing set is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact, the sequence
$\\{\boldsymbol{w}_{0,n}\\}_{n=1}^{\infty}$ is
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded. It implies that
$\\{S(t_{n},t_{n}-s_{n},\omega)\boldsymbol{w}_{0,n}\\}_{n=1}^{\infty}$ or
$\\{\boldsymbol{w}_{n}\\}_{n=1}^{\infty}$ is also a
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded sequence. We infer from
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded sequence
$\\{\boldsymbol{w}_{n}\\}_{n=1}^{\infty}$ that the sequences
$\\{\boldsymbol{w}_{n}\\}_{n=1}^{\infty}$ and
$\\{\nabla\land\boldsymbol{w}_{n}\\}_{n=1}^{\infty}$ are bounded in
$\mathbb{V}$ and $\mathrm{L}^{\infty}(\mathcal{O})$, respectively. In view of
the Banach-Alaoglu theorem, and the uniqueness of weak and weak∗ limits, we
find an element $\boldsymbol{w}\in\mathcal{W}$ such that
$\boldsymbol{w}_{n}\rightharpoonup\boldsymbol{w}$ weakly in $\mathbb{V}$ and
$\nabla\land\boldsymbol{w}_{n}\xrightharpoonup{*}\nabla\land\boldsymbol{w}$
weak∗ in $\mathrm{L}^{\infty}(\mathcal{O})$. Now, the compact embedding
$\mathbb{V}\hookrightarrow\mathbb{H}$ gives that
$\boldsymbol{w}_{n}\to\boldsymbol{w}$ strongly in $\mathbb{H}$. Hence, the
arbitrary sequence $\\{\boldsymbol{w}_{n}\\}_{n=1}^{\infty}$ converges
strongly to an element $\boldsymbol{w}\in\mathcal{W}$ in $\delta$-metric. This
completes the proof. ∎

The following lemma shows that the stochastic dynamical systems
$S(\cdot,\cdot,\cdot)$ and $T(\cdot,\cdot,\cdot)$ satisfy condition
$(\mathrm{ii})$ of the Theorem 2.10.

######  Proposition 5.2.

Suppose that Hypothesis 1.1 is satisfied. Then the solution $\boldsymbol{v}$
of the system (3.5) forward converges to the solution $\bar{\boldsymbol{v}}$
of the system (4.25) in $\delta$-metric, that is,

$\displaystyle\lim_{\tau\to+\infty}\delta(\boldsymbol{v}(T+\tau,\tau,\theta_{-\tau}\omega,\boldsymbol{v}_{\tau}),\bar{\boldsymbol{v}}(T,\omega,\bar{\boldsymbol{v}}_{0}))=0,\
\ \text{ for all }T>0\ \text{ and }\ \omega\in\Omega,$

whenever $\\{\boldsymbol{v}_{\tau}\\}_{\tau\in\mathbb{R}}$ is a
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded sequence,
$\bar{\boldsymbol{v}}_{0}\in\mathcal{W}$ and
$\delta(\boldsymbol{v}_{\tau},\bar{\boldsymbol{v}}_{0})\to 0$ as
$\tau\to+\infty.$

###### Proof.

Let
$\mathscr{V}_{\tau}(t):=\boldsymbol{v}(t+\tau,\tau,\theta_{-\tau}\omega,\boldsymbol{v}_{\tau})-\bar{\boldsymbol{v}}(t,\omega,\bar{\boldsymbol{v}}_{0})$
for $t\geq 0$. Then, $\mathscr{V}_{\tau}(\cdot)$ satisfies for a.e. $t\geq 0$
and $\boldsymbol{\phi}\in\mathbb{V}$

$\displaystyle\mathopen{}\mathclose{{}\left\langle\frac{\/\mathrm{d}\/\mathscr{V}_{\tau}}{\/\mathrm{d}\/t},\boldsymbol{\phi}}\right\rangle+\mathopen{}\mathclose{{}\left[\frac{\sigma^{2}}{2}-\sigma
y(\theta_{t}\omega)}\right](\mathscr{V}_{\tau},\boldsymbol{\phi})$
$\displaystyle=-e^{\sigma
y(\theta_{t}\omega)}\langle(\mathscr{V}_{\tau}\cdot\nabla)\bar{\boldsymbol{v}},\boldsymbol{\phi}\rangle-e^{\sigma
y(\theta_{t}\omega)}\langle(\boldsymbol{v}\cdot\nabla)\mathscr{V}_{\tau},\boldsymbol{\phi}\rangle+e^{-\sigma
y(\theta_{t}\omega)}(\boldsymbol{f}(t+\tau)-\boldsymbol{f}_{\infty},\boldsymbol{\phi}).$

Taking $\boldsymbol{\phi}=\mathscr{V}_{\tau}$, we obtain

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\mathscr{V}_{\tau}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle=-\mathopen{}\mathclose{{}\left[\sigma^{2}-2\sigma
y(\theta_{t}\omega)}\right]\|\mathscr{V}_{\tau}(t)\|^{2}_{\mathbb{H}}-2e^{\sigma
y(\theta_{t}\omega)}\langle(\mathscr{V}_{\tau}(t)\cdot\nabla)\bar{\boldsymbol{v}}(t),\mathscr{V}_{\tau}(t)\rangle$
$\displaystyle\quad+2e^{-\sigma
y(\theta_{t}\omega)}(\boldsymbol{f}(t+\tau)-\boldsymbol{f}_{\infty},\mathscr{V}_{\tau}(t))$
$\displaystyle\leq 2e^{\sigma
y(\theta_{t}\omega)}\|\nabla\bar{\boldsymbol{v}}(t)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\mathscr{V}_{\tau}(t)\|^{2}_{\mathbb{L}^{\frac{2p}{p-1}}(\mathcal{O})}+2\sigma|y(\theta_{t}\omega)|\|\mathscr{V}_{\tau}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\quad+\frac{e^{-2\sigma