\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 246, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/246\hfil Random attractors in $H^1$]
{Random attractors in $H^1$ for stochastic two dimensional micropolar
fluid flows with spatial-valued noises}

\author[W. Zhao \hfil EJDE-2014/246\hfilneg]
{Wenqiang Zhao}  % in alphabetical order

\address{Wenqiang Zhao \newline
School Of Mathematics and Statistics, Chongqing Technology and Business
University,  Chongqing 400067, China}
\email{gshzhao@sina.com}

\thanks{Submitted March 26, 2014. Published November 21, 2014.}
\subjclass[2000]{60H15, 35R60, 35B40, 35B41}
\keywords{Random dynamical system; stochastic micropolar fluid flows;
\hfill\break\indent random attractor; additive noises; Wiener process}

\begin{abstract}
 This work studies the long-time behavior of two-dimensional micropolar fluid
 flows perturbed by the generalized time derivative of the infinite
 dimensional Wiener processes. Based on the \emph{omega-limit compactness}
 argument as well as some new estimates of solutions, it is proved that
 the generated random dynamical system admits an $H^1$-random attractor
 which is compact in $H^1$ space and attracts all tempered random subsets
 of $L^2$ space in $H^1$ topology.  We also give a general abstract result
 which shows that the continuity condition  and absorption of the
 associated random dynamical system in  $H^1$ space is not necessary for
 the existence of random attractor in $H^1$ space.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

 The micropolar fluid model is a qualitative generalization of the
well-known Navier-Stokes model in the sense that it takes into account the
microstructure of fluid \cite{Luk1}. It was introduced by Eringen
\cite{Erin} as an important model to describe a class of
non-Newtonian fluid motion with micro-rotational effects and inertia
involved.

  Let $\mathcal{O}\subset \mathbb{R}^2$ be a smooth bounded domain.
This paper is concerned with the micropolar fluid flows driven by the
time-space additive noises
\begin{equation} \label{e1.1}
 \begin{gathered}
\nabla\cdot v=0\quad \text{on } \mathcal{O}\times\mathbb{R}^+, \\
\frac{dv}{dt}-(\nu+\kappa)\Delta v-2\kappa\nabla\times V+
\nabla\pi+v.\nabla v=f+\dot{W}_1\quad \text{on }
\mathcal{O}\times\mathbb{R}^+, \\
\frac{dV}{dt}-\gamma\Delta V+4\kappa V-2k\nabla\times v+v.\nabla
V= g+\dot{W}_2\quad\text{on }\mathcal{O}\times\mathbb{R}^+,
\end{gathered}
\end{equation}
associated with the hard wall boundary condition
\begin{equation} \label{e1.2}
v=0\quad\text{on }\partial\mathcal{O}\times\mathbb{R}^+,\quad
V=0\quad\text{on }\partial\mathcal{O}\times\mathbb{R}^+,
\end{equation}
and the initial value condition
\begin{equation} \label{e1.3}
v(x,0)=v_0, \quad  V(x,0)=V_0,
\end{equation}
with  a velocity vector field
$v=v(x,t)=(v_1(x,t),v_2(x,t))\in \mathbb{R}^2$, a scalar microrotation field
$V=V(x,t)\in \mathbb{R}$, a scalar pressure $\pi=\pi(x,t)\in\mathbb{R}$. In
equations \eqref{e1.1}, the constants $\nu>0$, $\kappa\geq0$, $\gamma>0$
($\nu$ is usually called the Newtonian viscosity, $\gamma$ and $\kappa$ are
the microrotation viscosity coefficients), and
$f(x)=(f_1(x),f_2(x))$ and $g(x)$ denote the exterior body
force and the moment, respectively. Moreover, $ W_1(t)$ and $
W_2(t)$ are independent two-sided real-valued Wiener processes with
values in appropriate function spaces specified later. In addition,
$\Delta$ is the Laplacian on $\mathcal{O}$ and
$$
\nabla\times v=\frac{\partial v_2}{\partial x_1}-\frac{\partial
v_1}{\partial x_2},\quad
\nabla\cdot  v=\frac{\partial v_1}{\partial
x_1}+\frac{\partial v_2}{\partial x_2},\quad
\nabla\times V=\Big(\frac{\partial V}{\partial x_2}, -\frac{\partial V}{\partial
x_1}\Big).
$$

There is a large volume of literature on the mathematical theory of
the autonomous or non-autonomous micropolar fluid model; see, e.g.,
\cite{Kasz,Luk1,Luk3,Luk4, Zhaoc,  Dongb, Chenj1, Chenj3}.
Especially, for this two dimensional autonomous model,
Lukaszewicz \cite{Luk1} proved the existence of $L^2$-global
attractor in a bounded domain; Dong and Chen \cite{Dongb}
established the existence of $L^2$-global attractor in some
unbounded domains; Chen et al \cite{Chenj1} proved that the
$L^2$-global attractor was compact in the space $H^2$ based on the
notion of the so-called Kuratowski measure of noncompactness of a
bounded set \cite{Zhong}.  As for the non-autonomous model, Zhao et
al \cite{Zhaoc} proved the existence of $H^1$-uniform attractor in
an unbounded Poincar\'{e} domains by utilizing the energy method
originated from \cite{Luk3}; Chen \cite{Cheng} considered the
non-homogeneous micropolar fluid flows and obtained the existence of
$L^2$-pullback attractor in a Lipschitz bounded domain by energy
equation method; Chen et al \cite{Chenj3} and Lukaszewicz and
Tarasi\'{n}ska \cite{Luk4} obtained the existence of $H^1$-pullback
attractors in a bounded domain from a viewpoint of measuring
noncompactness \cite{Zhong}, respectively.

It is well known that the random attractor, which  was initiated by
\cite{Bsc,Hcr2},  is an appropriate notion to describe the long-time
behavior of the solution of stochastic partial differential
equation. The applications cover a wide range of concrete
differential equations; see, recently, \cite{Wang2,Zhao1,Zhao2, Yan1,Bate} and
the references cited there. Such a attractor, which generalizes
non-trivially the global attractors
 well developed (see, e.g., \cite{Tem,Babin}),  is a compact
 invariant random set which attracts every orbit in the state
space. It is uniquely determined by attracting deterministic compact
sets of  phase space \cite{Hcr3}.

 The goal of this article is to prove the existence of
random attractors of the  micropolar fluid
model \eqref{e1.1}--\eqref{e1.3} in $H^1$ space with irregular and spatially valued
noise. On account of the irregularity of solutions in $H^1$ space,
the Sobolev compact imbedding method is unavailable in the proof the
compactness of the random attractor.  To achieve our study, we
utilize the technique developed in  \cite{Zhao2,Zhao3}  to surmount
this obstacle. Specifically, the notion of omega-limits compactness,
which was initiated in \cite{Yan1} and \cite{Kloe} in the framework
of RDS, is successfully employed.  The main advantage of this
technique is that we need not to estimate the solutions in
functional spaces of higher regularity to show the existence of
compact random absorbing set which does not work in this case
\cite{Fla1}.

To solve our problem, we first prove an abstract result. We show that for
the bi-space $(X,Z)$ with a sequence uniqueness (see section 2 Hypothesis A),
 the random attractor on the space $X$ is a random attractor on the space $Y$
only if the RDS $\varphi$ possess omega-limit compactness on the space $Y$.
 The continuity (even quasi-continuity \cite{Yan1}), absorption of the associated
random dynamical systems on $Z$ is not necessary, see Theorem \ref{thm2.3}.
This result is new even in deterministic case, see \cite{Zhong,Yan1}.

The outline of  this article is as follows. Section 2 presents some
basic facts needed for further considerations, including some
notions and an abstract result about random attractors and the
appropriate spaces and operators.
 In section 3 we recall the Ornstein-Uhlenbeck process and its regular
hypothesis and then give the main conclusion of this study.
Section 4 is the proof of  our main conclusion.

\section{Preliminaries}

This section contains some background material which we will use in
further discussion.

\subsection{Random dynamical systems and an abstract result}

In this subsection, we list some appropriate concepts  and tools
from the theory of random dynamical systems (RDSs) and obtain a
abstract result. For more details the readers may refer to \cite{
Arn,Ches, Hcr2}.

 Let $(X, \|\cdot\|_X)$ and $(Z,\|\cdot\|_Z)$ be two completely separable
Banach spaces with Borel
$\sigma$-algebras $\mathcal{B}(X)$ and $\mathcal{B}(Z)$,
respectively.

A random dynamical system on a Banach space $X$ is a family of
measurable mappings $\varphi$ incorporated a metric dynamical
system (MDS) $\theta$, where the metric dynamical system $\theta$ is
a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a group
$\theta_t,t\in \mathbb{R},$ of measure preserving transformations of
$(\Omega,\mathcal{F},\mathbb{P})$, and  the  family of measurable
mappings $\varphi: \mathbb{R}^+\times\Omega\times X\to X;\
(t,\omega,x)\mapsto\varphi(t,\omega)x$ satisfies the cocycle
property
$$
\varphi(0,\omega)=\text{id}, \quad
\varphi(t+s,\omega)=\varphi(t,\theta_s\omega)\circ\varphi(s,\omega),
$$
for all $s,t\in \mathbb{R}^+$. We will denote  this RDS by the
simple notation $\varphi$.  An RDS $\varphi$ is continuous in the
meaning that the mappings $\varphi(t, \omega): X\to X$ are
continuous in $X$ for all $t\in \mathbb{R^+}$ and .

 A random set $D=\{D(\omega)\}_{\omega\in\Omega}$ is a family of
closed subsets of $X$ indexed by $\omega\in \Omega$ such that for
every $x\in X$, the mapping $\omega\mapsto d_X(x,D(\omega))$ is
measurable with respect to $\mathcal{F}$, where for the nonempty
sets $A,B\in 2^X$,
 $$
d_X(A,B)=\sup_{x\in A} \inf_{y\in B}
\|x-y\|_X,
$$
and in particular $d_X(x,B)=d_X(\{x\}, B)$.

A random variable $R\in \mathbb{R}^+$ over a MDS $\theta$ is
tempered if
\begin{equation} \label{e2.1}
\lim_{t\to\pm\infty}\frac{1}{|t|}
\log ^+R(\theta_t\omega)=0,
\end{equation}
for $\mathbb{P}$-a.e.  $\omega\in\Omega$. Note that \eqref{e2.1} is
equivalent to
\[
\lim_{t\to\pm\infty}e^{-\lambda|t|}R(\theta_t\omega)=0\quad 
\text{for any } \lambda>0,
\]
see \cite{Imke,Caraba}. A random set $D=\{D(\omega)\}_{\omega\in\Omega}\in 2^X$
is called tempered  if $R(\omega)=\sup_{x\in D(\omega)}\|x\|_{X}$ is a
tempered random variable.

Let $\mathcal{D}_X$ and  $\mathcal{D}_Z$ denote the collection of
all  tempered random subsets of $X$ and $Z$, respectively. In
addition, we assume  that range  $\varphi (X)\subseteq Z$. In the
following we recall the basic concepts about bi-space random
attractor; see \cite{Babin,Zhao1}.

\begin{definition} \label{def2.1}\rm
 (1) A random set $K_Z\in \mathcal{D}_Z$ is
called an $(X, Z)$-random absorbing set for the RDS $\varphi$ over
a MDS $\theta$ if for every $D\in \mathcal{D}_X$ and
$\mathbb{P}$-a.e.  $\omega\in \Omega$, there exists an
$T=T(D,\omega)>0$ such that $\text{for all}\ t\geq T$,
$$
\varphi (t,\theta_{-t}\omega)D(\theta_{-t}\omega)\subseteq K_Z(\omega).
$$

(2) A compact random set $\mathcal{A}_Z\in 2^Z$ is said to be an
$(X, Z)$-random attractor for the RDS $\varphi$ over a MDS $\theta$
if the invariance property
$$
\varphi(t,\omega)\mathcal{A}_Z(\omega)=\mathcal{A}_Z(\theta_{t}\omega)
$$
 is satisfied for all $ t\geq0$ and $\mathbb{P}$-a.e. $\omega\in \Omega$, and if
 in addition, the pullback attracting property
$$
\lim_{t \to \infty}d_Z(\varphi(t,\theta_{-t}\omega)D(\theta_{-t}\omega),
{\mathcal{A}_Z}(\omega))= 0
$$
holds for every $D\in \mathcal{D}_X$ and
$\mathbb{P}$-a.e. $\omega\in \Omega$.
\end{definition}

\begin{definition} \label{def2.2} \rm
 An RDS $\varphi$ over an MDS $\theta$ is
said to be $(X, Z)$-omega-limit compact if for every $\varepsilon>0$
and $D\in \mathcal{D}_X$, there exists $T=T(\varepsilon, D,
\omega)$ such that for all $t\geq T$,
\[
k\Big(\cup_{t\geq T}\varphi(t,\theta_{-t}\omega)D(\theta_{-t}\omega)\Big)
\leq\varepsilon,\quad \mathbb{P}\text{-a.e. } \omega\in \Omega,
 \]
where  $k(B)$ is the Kuratowski measure of non-compactness of a
bounded subset  $B\subset Z$ defined by
$$
k(B)=\inf\{d>0: B \text{ admits a finite cover by sets of diameter}
\leq d\}.
$$
\end{definition}

\noindent\textbf{Hypothesis (A1).}
  Assume that the bi-space $(X,Z)$ satisfies the
sequence limits uniqueness,  in the sense that for every bounded
sequence $\{x_n\}_n\subset X\cap Z$ such that $x_n\to x$ in
$X$ and $x_n\to y$ in $Z$, respectively, then we have $x=y$. The
nested relation between $X$ and $Z$ is unknown except that
$\varphi (X)\subseteq Z$.


\begin{theorem} \label{thm2.3}
Assume that the bi-space $(X,Z)$ satisfies
{\rm (A1)}, and  $\varphi$ is a continuous RDS on $X$ over
a MDS $\theta$. If there exists an ($X,X$)-random absorbing set
$K$ for $\varphi$ and
 $\varphi$ is ($X,X$)-omega-limit compact, then the random set
$\mathcal{A}_X$,
\begin{equation} \label{e2.2}
\mathcal{A}_X(\omega)=\cap_{s\geq0}\overline{\cup_{t\geq
s}\varphi (t, \theta_{-t}\omega)K(\theta_{-t}\omega)}^{X},\quad
\omega\in \Omega,
\end{equation}
is a unique  ($X,X$)-random attractor for  $\varphi$ in $X$, where
$\overline{B}^{X}$
denotes the closure of $B$ with respect to the $X$-norm.

Furthermore, if $\varphi$ is ($X,Z$)-omega-limit compact then the
random set $\mathcal{A}_Z$,
\begin{equation} \label{e2.3}
\mathcal{A}_Z(\omega)=\cap_{s\geq0}\overline{\cup_{t\geq
s}\varphi (t, \theta_{-t}\omega)K(\theta_{-t}\omega)}^{Z},\quad
\omega\in \Omega,
\end{equation}
is a unique ($X,Z$)-random attractor for $\varphi$. In
addition, $\mathcal{A}_X=\mathcal{A}_Z\in \mathcal{D}_X$.
\end{theorem}

\begin{proof}
The first part  is the same as \cite[Theorem 4.1]{Yan1},
so we omit the proof. We prove the second result.
First, \eqref{e2.3} makes sense by our assumption that
$\varphi(t,\omega)X\subseteq Z$. We show that $\mathcal{A}_Z$ is a
random attractor in the space $Z$, that is, $\mathcal{A}_Z$
satisfies the compact, attracting and invariant property.

 By \cite[Lemma 2.5(v)]{Yan1} and the omega-limit compactness of
$\varphi$ in $Z$,  we have
$$
k\Big(\overline{\cup_{t\geq
T}\varphi(t,\theta_{-t}\omega)K(\theta_{-t}\omega)}^{Z}\Big)=k\Big(\cup_{t\geq
T}\varphi(t,\theta_{-t}\omega)K(\theta_{-t}\omega)\Big)\to0\
\ \text{as}\ T\to\infty.
$$
At the same time,  $\overline{\cup_{t\geq
T}\varphi(t,\theta_{-t}\omega)K(\theta_{-t}\omega)}^Z$ is
norm-closed in $Z$. Then thanks to the nested property of the
Kuratowski measure of non-compactness (see \cite[Lemma 2.5 (iv)]{Yan1}), 
we know that $\mathcal{A}_Z$ is nonempty and compact
as required.

 Furthermore  by a similar argument as in \cite{Hcr2,Zhao1, Yan1}
we can show that $\mathcal{A}_Z$  possesses $(X,Z)$-attracting property.

 By the definitions of formula \eqref{e2.2}--\eqref{e2.3} and the omega-limit
compactness of $\varphi$ in $X$ and $Z$, it
is easy to show that $ \mathcal{A}_X=\mathcal{A}_Z$. Thus
$\mathcal{A}_Z$ is invariant since $\mathcal{A}_X$ is invariant.
\end{proof}

\begin{remark} \label{rmk2.4}\rm
 In applications, one can choose $X=L^2$ and
$Z=L^p$ $(p>2)$ or $H^1$ with bounded or unbounded spatial domain. Note
that  $(L^2,L^p)$ and $(L^2,H^1)$ satisfy (A1).
Therefore, the {\rm (A1)} is not restrictive in concrete
problems. In particular, our Theorem \ref{thm2.3} implies
\cite[Theorem 2.8]{Zhao1}.
\end{remark}

\subsection{Functional settings}

In this subsection, we introduce some spaces and operators stated as
follows.

Let $L^p(\mathcal{O})$ and $H^s(\mathcal{O})$ be the usual Sobolev
spaces. We set $L^2=(L^2(\mathcal{O}))^2\times L^2(\mathcal{O}), $
 endowed with the following  scalar inner
product
$$
(.,.)=(.,.)_{(L^2(\mathcal{O}))^2}+(.,.)_{L^2(\mathcal{O})},
$$
and the norms in $(L^2(\mathcal{O}))^2$, $L^2(\mathcal{O})$ and
$L^2$ are together denoted by the same notation $ |.|$, without any
confusion. We define a functional space $\mathscr{V}$ integrated the
boundary and also the divergence free condition,
$$
\mathscr{V}=\{(v,V)\in (C_0^\infty(\mathcal{O}))^2\times
C_0^\infty(\mathcal{O}):\operatorname{div}v=0\}.
$$
Define $H^1=(H^1(\mathcal{O}))^2\times H^1(\mathcal{O})$, where
$H^1(\mathcal{O})$ is the usual Sobolev space.
 Let $\mathcal{H}$ be the closure of $\mathscr{V}$ with respect to the
$L^2$-norm. The norm in $\mathcal{H}$ is still denoted by $|\cdot|$.
Moreover, we let $\mathcal{V}$ be the closure of $\mathscr{V}$ with
respect to the $H^1$-norm, possessing the equivalent norm in
$\mathcal{V}$ is denoted by  $\|\cdot\|=|\nabla\cdot |$. In addition,
$\mathcal{V}'$ denotes the dual space of $\mathcal{V}$. Then
we have $\mathcal{V}\subset \mathcal{H}\subset \mathcal{V}'$.

 For $\mathbf{V}=(v,V)$, we define the operators
\begin{gather*}
A_1v=-(\nu+\kappa)\Delta v,\quad  A_2V=-\gamma\Delta V,  \\
B_1(v,v)=v.\nabla v, \quad B_2(v,V)=v.\nabla V, \\
A\mathbf{V}=(A_1v,  A_2V),\quad B(v,\mathbf{V})=(B_1(v,v),B_2(v,V)), \\
Lu=(-2\kappa\nabla\times V, -2\kappa\nabla\times v+4\kappa V) ,\quad
F=(f,  g), \quad f\in (L^2(\mathcal{O}))^2, \quad
g\in L^2(\mathcal{O}).
\end{gather*}

It is obvious that the operator $A$ is a positive self-adjoint
 unbounded operator and then $A^{-1}$ is also self-adjoint but
 compact operator  in $\mathcal{H}$, and we can utilize the elementary
 spectral theory in a Hilbert space. We infer that there exists a
compete orthonormal family of
 $\mathcal{H}$, $\{\mathbf{e}_j\}_{j=1}^\infty$ of eigenvectors of $A$.
The corresponding  spectrum of
 $A$ is discrete and denoted by $\{\lambda_j\}_{j=1}^\infty$ which
 are positive, increasing and tend to infinity as
 $j\to\infty$.

 In particular, we also can use the spectrum theory to allow
 us to define the operator $A^s$, the power of $A$. For $s>0$, the
 operator $A^s$ is also a strictly positive and self-adjoint unbounded
 operator in $\mathcal{H}$ with a dense domain $D(A^s)\subset \mathcal{H}$.
 This allows us to introduce the function spaces
\[
D(A^s)=\Big\{\mathbf{V}=\sum_{j=1}^\infty(\mathbf{V},\mathbf{e}_j)\mathbf{e}_j
:\|\mathbf{V}\|^2_{D(A^s)}
=\sum_{j=1}^\infty(\mathbf{V},\mathbf{e}_j)^2\lambda_j^{2s}<+\infty\Big\}.
\]
This norm $\|\cdot\|_{D(A^s)}$ on $D(A^s)$ is equivalent to the usual
norm induced by $H^{2s}$; see Temam \cite{Tem} for details. In
particular, $D(A^0)=\mathcal{H}$ and
$D(A^{1/2})=\mathcal{V}$. Furthermore, we have
\begin{equation} \label{e2.4}
\min\{\nu+\kappa,\gamma\}\|\mathbf{V}\|^2\leq(A\mathbf{V},\mathbf{V})\leq
\lambda_0^{-1/2}|A\mathbf{V}|\,\|\mathbf{V}\|,
\end{equation}
for all $\mathbf{V}=(v,V)\in D(A)$, where $\lambda_0>0$ satisfies the Poincar\'{e}
inequality $\lambda_0|\mathbf{V}|^2\leq\|\mathbf{V}\|^2$.

Based on the orthonormal basis $\{\mathbf{e}_j\}_{j=1}^\infty$ of
eigenfunctions of $A$, we define the $m$-dimensional subspace
$\mathcal{V}_m=\operatorname{span}\{\mathbf{e}_1,\mathbf{e}_2,\dots,
\mathbf{e}_m\}\subset \mathcal{V}$
 and the canonical orthogonal projection
$P_m: \mathcal{V}\mapsto \mathcal{V}_m$ such that for every
$\mathbf{V}\in \mathcal{V}$, $\mathbf{V}$ has a unique decomposition:
$\mathbf{V}=P_m\mathbf{V}+\mathbf{V}_m$, where
\begin{equation} \label{e2.5}
P_m\mathbf{V}=\sum_{j=1}^m(\mathbf{V},\mathbf{e}_j)\mathbf{e}_j\in
\mathcal{V}_m,\quad
\mathbf{V}_m=(I-P_m)\mathbf{V}=\sum_{j=m+1}^\infty(\mathbf{V},\mathbf{e}_j)\mathbf{e}_j\in
\mathcal{V}_m^{\bot};
 \end{equation}
that is, $\mathcal{V}=\mathcal{V}_m\oplus \mathcal{V}_m^{\bot}$.

 According to the above notation, we write \eqref{e1.1}--\eqref{e1.3}
as  the  evolution equation
\begin{equation} \label{e2.6}
d\mathbf{V}+A\mathbf{V}dt+B(v,\mathbf{V})dt+L\mathbf{V}dt =Fdt+dW,\quad 
\mathbf{V}(0)=\mathbf{V}_0\in \mathcal{H},
\end{equation}
where $\mathbf{V}=(v,V)$ and $W=(W_1,W_2)$.

\section{Existence of $(\mathcal{H}, \mathcal{V})$-random attractor for the
generated RDS $\varphi$}


To obtain a priori estimate, we now use the method of
Chueshov and Schmalfu{\ss} \cite{Ches1} to transform the evolution
equation \eqref{e2.6} to a deterministic partial differential equation with
a random parameter without white noise.

  A standard model for a spatially correlated noise is the the
  generalized time derivative of a two-sided \emph{Brownian motion}
  $\omega=\omega(x,t), x\in \mathbb{R}^2$.
Let $\mathcal{H}$ be the separable Hilbert space with norm $|\cdot|$
which is defined in section 2. As usual,
we introduce the spatially valued \emph{Brownian motion} MDS
$\theta=(\Omega, \mathcal{F},\mathbb{P},
(\theta_t)_{t\in\mathbb{R}})$, where
$\Omega=\{\omega\in C_0(\mathbb{R}, \mathcal{H}):\omega(0)=0\}$ with compact open
topology. This topology is metrizable by the complete metric
$$
d(\omega_1,\omega_2)=\sum_{n=1}^\infty\frac{1}{2^n}\frac{d_n(\omega_1,\omega_2)}{1+d_n(\omega_1,\omega_2)},
$$
where $d_n(\omega_1,\omega_2)=\max_{|t|\leq n}|\omega_1-\omega_2|$
for $\omega_1$ and $\omega_2$ in $\Omega$.
$\mathcal{F}=\mathcal{B}(C_0(\mathbb{R}, \mathcal{H}))$ is the
Borel-$\sigma$-algebra
induced by the compact open topology of $\Omega$.
Suppose the Wiener process $\omega$ has covariance operator $Q$.
Let $\mathbb{P}$ be the Wiener measure with respect to $Q$.
The Wiener shift is defined by
$$
\theta_s\omega(t)=\omega(t+s)-\omega(s), \quad \omega\in \Omega,\;
t, s\in \mathbb{R}.
$$
Then the measure $\mathbb{P}$ is ergodic and invariant with respect
to the shift $\theta$. Then $\theta$ is an ergodic MDS.

The associated probability space defines a canonical Wiener process
$W$. We also note that such a Wiener process $W$ generates a
filtration $(\mathcal{F}_t)_{t\in \mathbb{R}}$,
$$
\mathcal{F}_t\equiv\{W(\tau)|\tau\leq t\}\subset \mathcal{F}.
$$

 We  introduce the following
stochastic partial differential equation on $\mathcal{O}$,
\begin{equation} \label{e3.1}
d\mathbf{Z}+A \mathbf{Z}dt=dW.
\end{equation}
 Because $A$ is a positive and self-adjoint operator, there
exists a mild solution to this stochastic equation with the form
$$
\mathbf{Z}(t)=\mathbf{Z}(0)+\int_0^te^{-(t-\tau)A}dW,\ \  \  t>0,
$$
which is called an Ornstein-Uhlenbeck process; see \cite{Prato}. For
the Ornstein-Uhlenbeck process we have the regularity hypothesis;
see also \cite{Ches1}.

\begin{lemma} \label{lem3.1}
Suppose that  the covariance operator $Q$ of the
Wiener process $\omega$ has a finite trace; i.e.,  $Q$ satisfies
\begin{equation} \label{e3.2}
\operatorname{tr}_\mathcal{H}(QA^{2s-1+\delta})
=\operatorname{tr}_\mathcal{H}(A^{s-1/2+\delta/2}QA^{s-1/2+\delta/2})<+\infty,
\end{equation}
for some $s\geq0$ and some (arbitrary small) $\delta>0$, where
$\operatorname{tr}_\mathcal{H}$ denotes the trace of the covariance. Then an
$\mathcal{F}_0$-measurable Gaussian variable $\mathbf{Z}=(z,Z)\in
D(A^s)$ exists, and the process
$(t,\omega)\to\mathbf{Z}(\theta_t\omega)$ is a continuous
stationary solution to the stochastic equation \eqref{e3.1}. Furthermore,
the random variable $\|\mathbf{Z}(\omega)\|^2_{D(A^s)}$ is tempered
and the expectation
$$
\mathbb{E}\|\mathbf{Z}\|^2_{D(A^s)}
=\frac{1}{2}\operatorname{tr}_\mathcal{H}(A^{s-\frac{1}{2}}Q
A^{s-\frac{1}{2}})<+\infty.
$$
\end{lemma}

 Introducing a new variable $\mathbf{U}=\mathbf{V}-\mathbf{Z}(\theta_t\omega)$,
we can rewrite \eqref{e2.6} as the following
 evolution equation with a random parameter $\omega$,
\begin{equation} \label{e3.3}
\begin{gathered}
\frac{d\mathbf{U}}{dt}+A
\mathbf{U}+B(u+z(\theta_t\omega),\mathbf{U}+\mathbf{Z}(\theta_t\omega))
+L(\mathbf{U}+\mathbf{Z}(\theta_t\omega))=F(x) \\
\mathbf{U}(0)=\mathbf{U}_0\in \mathcal{H},
\end{gathered}
\end{equation}
or the following functional form
\begin{align*}
&(\frac{d\mathbf{U}}{dt},\phi)+(A
\mathbf{U},\phi)+(B(u+z(\theta_t\omega),\mathbf{U}
+\mathbf{Z}(\theta_t\omega)),\phi)
+(L(\mathbf{U}+\mathbf{Z}(\theta_t\omega)),\phi)\\
&=(F(x),\phi), \phi\in \mathcal{V}
\end{align*}
 where $\mathbf{U}=(u,U)$, $u=v-z(\theta_t\omega)$,
$U=V-Z(\theta_t\omega)$.

From the analysis above, we obtain the existence of a weak solution for
the  problem \eqref{e3.3}--\eqref{e3.4} by the standard Galerkin
approximation, see, e.g., \cite{Chenj01}.

\begin{lemma} \label{lem3.2}
Let $F=(f, g)\in \mathcal{H}$ and
$\mathbf{U}_0=(u_0,U_0)\in \mathcal{H}$. Then for
$\mathbb{P}$-a.e. $\omega\in \Omega$, the initial problem \eqref{e3.3}--\eqref{e3.4}
possesses a unique solution $\mathbf{U}(t,\omega,\mathbf{U}_0(\omega))$, where
$$
\mathbf{U}=(u,U)\in L^\infty(0,\infty; \mathcal{H})\cap L^2 (0,T;
\mathcal{V})\cap C([0,\infty; \mathcal{H}).
$$
Furthermore, the mapping $\mathbf{U}_0\mapsto \mathbf{U}(t,
\omega,\mathbf{U}_0(\omega))$ from $\mathcal{H}$ to $\mathcal{H}$
is continuous for all $t\geq0$.
\end{lemma}

 By a standard argument on the measurability  we can show that the solution
generates a continuous RDS $\psi$ in the space
$\mathcal{H}$ given by
$\psi(t,\omega)\mathbf{U}_0(\omega)=\mathbf{U}(t,\omega,\mathbf{U}_0(\omega))$.
 Put
$\mathbf{V}(t,\omega,\mathbf{V}_0(\omega))
=\mathbf{U}(t,\omega,\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))
+\mathbf{Z}(\theta_t\omega)$.
Then $\mathbf{V}(t,\omega,\mathbf{V}_0(\omega))$ or briefly
$\mathbf{V}(t)$ is a solution to \eqref{e2.6} with initial value
$\mathbf{V}_0(\omega)\in \mathcal{H}$. Given
$$
\varphi(t,\omega)\mathbf{V}_0(\omega)
=\mathbf{V}(t,\omega,\mathbf{V}_0(\omega))
=\mathbf{U}(t,\omega,\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))
+\mathbf{Z}(\theta_t\omega),
\quad \omega\in \Omega,
$$
then $\varphi$ is also a continuous RDS on  $\mathcal{H}$ for the
original equation \eqref{e2.6}, i.e., system \eqref{e1.1}--\eqref{e1.3}.

The main conclusion of this study reads as follows.

\begin{theorem} \label{thm3.3}
 We suppose that \eqref{e3.2} holds. Set
\begin{equation} \label{e3.4}
M(\omega)=C(\|Z(\omega)\|^2_{H^2}+\|z(\omega)\|^2_{H^1})
-\frac{1}{2}\lambda_0\varsigma,
\end{equation}
where $C$ is a positive constant depending only on the physical coefficients
of the fluid model, $\varsigma=\min\{\nu,\gamma\}$ and $\lambda_0$
is the same as in \eqref{e2.4}. Assume that the mathematical expectation
$\mathbb{E}M<0$.
 Then the RDS $\varphi$ generated by \eqref{e1.1}--\eqref{e1.3} admits
 a unique ($\mathcal{H}, \mathcal{V}$)-random attractor
$\mathcal{A}_\mathcal{V}$. In
addition, $\mathcal{A}_\mathcal{V}= \mathcal{A}_\mathcal{H}$,
where $\mathcal{A}_\mathcal{H}$ is the ($\mathcal{H},
\mathcal{H}$)-random attractor.
\end{theorem}

\section{Proofs of main results}


First we list some basic facts. By using the  Young's
inequality and combination with the divergence free condition, the
operators $A, L$ and $B$ possess the following relationships:
\begin{gather} \label{e4.1}
-(A\mathbf{U},\mathbf{U})-(L\mathbf{U},\mathbf{U})\leq
-\min\{\nu,\gamma\}\|\mathbf{U}\|^2,\quad \forall \mathbf{U}=(u,U)\in
\mathcal{V}, \\
 \label{e4.2}
\begin{aligned}
-(L\mathbf{U},A\mathbf{U})
&=-2\kappa(\nu+\kappa)(\nabla\times U,
\Delta u)-2\kappa\gamma(\nabla\times u, \Delta
U)-4\kappa\gamma\|U\|^2 \\
&\leq \frac{1}{2}|A\mathbf{U}|^2+2\kappa^2\|\mathbf{U}\|^2,\quad
\forall \mathbf{U}=(u,U)\in D(A),
\end{aligned} \\
 \label{e4.3}
\begin{gathered}
B(u, \mathbf{U}, \mathbf{V})=-B(u, \mathbf{V}, \mathbf{U}), \quad
B(u, \mathbf{U}, \mathbf{U})=0,  \\
 \forall (u, \mathbf{U}, \mathbf{V})\in (H^1(\mathcal{O}))^2\times
 \mathcal{V}\times \mathcal{V}.
\end{gathered}
\end{gather}
We recall the Agmon's inequality; see \cite{Tem},
\begin{equation} \label{e4.4}
\|u\|_{L^\infty}\leq c|u|^{1/2}|A_1u|^{1/2}, \quad
\forall\  u\in D(A_1).
\end{equation}


For the projector $P_m$ defined in Section 2,
one can easily show that
\[
|A_1P_m u|^2\leq\lambda_{m+1} |A_1^{1/2}P_m u|^2,\quad \forall
\mathbf{U}=(u,U)\in D(A),
\]
and hence by the classic Brezis-Gallouet's inequality; see
\cite{Brezis, Cao}, we have
\begin{equation} \label{e4.5}
\begin{aligned}
\|P_mu\|_{L^\infty}
&\leq c|A_1^{1/2}P_mu|\Big(1+\log \frac{|A_1P_mu|^2}{\lambda_1|A_1^{1/2}
 P_mu|^2}\Big)^{1/2} \\
&\leq
c\|u\|\Big(1+\log \frac{\lambda_{m+1}}{\lambda_1}\Big)^{1/2},
\quad \forall u\in D(A_1),
\end{aligned}
\end{equation}
where  the letter $c$ in \eqref{e4.4} and \eqref{e4.5} is a deterministic
positive constant  and $\lambda_1$ is the first eigenvalue of the
stokes operator $A$.

\begin{lemma} \label{lem4.1}
There exist positive constants $C$ and $c$ depending only on the physical
coefficients of this model and the domain $\mathcal{O}$ such that
\begin{gather} \label{e4.6}
\frac{d}{dt}|\mathbf{U}|^2+\varsigma\|\mathbf{U}\|^2 \leq
M(\theta_t\omega)|\mathbf{U}|^2+G(\theta_t\omega),\\
\frac{d}{dt}((\nu+\kappa)\|u\|^2+\gamma\|U\|^2)\leq
((\nu+\kappa)\|u\|^2+\gamma\|U\|^2) g(t,\omega)+h(t,\omega) \label{e4.7}
\end{gather}
where $\varsigma=\min\{\nu,\gamma\}$ and
\begin{gather*}
M(\omega)=C(\|z(\omega)\|^2+\|Z(\omega)\|_{H^2}^2)-\frac{1}{2}\lambda_0\varsigma, \\
G(\omega)=c(\|z(\omega)\|^4+|z(\omega)|^2\
\|Z(\omega)\|_{H^2}^2+|z(\omega)|^2+|Z(\omega)|^2+|F|^2), \\
 g(t,\omega)=c(\|u\|^2+\|U\|^2)(|u|^2+|z|^2), \\
h(t,\omega)=4\kappa^2\|\mathbf{U}\|^2+c|u|^2(\|z\|^4+\|Z\|^4)+H(\omega), \\
H(\omega)=c(\|z\|_{H^2}^2+|z|^2\|z\|^4+|z|^2\|Z\|^4+\|z\|^2+\|Z\|^2+|Z|^2+|F|^2).
\end{gather*}
\end{lemma}


\emph{Proof}
  We multiply \eqref{e3.3} by $\mathbf{U}$ and then integrate over
$\mathcal{O}$, along with \eqref{e4.1}
and \eqref{e4.3},  to yield
\begin{equation} \label{e4.8}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}|\mathbf{U}|^2\\
&=-(A\mathbf{U},\mathbf{U})-(L\mathbf{U},\mathbf{U})
-(B(u+z,\mathbf{U}+\mathbf{Z}),
\mathbf{U})-(L\mathbf{Z},\mathbf{U})+(F,\mathbf{U}) \\
&\leq -\varsigma\|\mathbf{U}\|^2-(B(u+z,\mathbf{U}+\mathbf{Z}),
\mathbf{U})-(L\mathbf{Z},\mathbf{U})+(F,\mathbf{U}) \\
&=-\varsigma\|\mathbf{U}\|^2-(B(u+z,\mathbf{Z}),
\mathbf{U})-(L\mathbf{Z},\mathbf{U})+(F,\mathbf{U}) \\
&=-\varsigma\|\mathbf{U}\|^2-(B_1(u+z,u+z), u)-(B_2(u+z,Z),
U)-(L\mathbf{Z},\mathbf{U})+(F,\mathbf{U}),
\end{aligned}
\end{equation}
where
\begin{equation} \label{e4.9}
\begin{aligned}
&-(L\mathbf{Z},\mathbf{U})+(F,\mathbf{U})\\
&=2\kappa(\nabla\times Z,
u)+2\kappa(\nabla\times z, U)-4\kappa(Z,U)+(F,\mathbf{U}) \\
&=2\kappa(\nabla\times u, Z)+2\kappa(\nabla\times U,
z)-4\kappa(Z,U)+(F,\mathbf{U}) \\
&\leq \frac{\varsigma}{8}\|\mathbf{U}\|^2+c(|z|^2+|Z|^2+|F|^2).
\end{aligned}
\end{equation}
By the H\"{o}lder's inequality  and Gagliardo-Nirenberg's
inequality, we have
\begin{gather} \label{e4.10}
\begin{aligned}
(B_1(u+z,u+z), u)
&=(B_1(u,z), u)+(B_1(z,z), u) \\
&\leq c\|u\|_{L^4}\ \| z\|\ \|u\|_{L^4}+c\|z\|_{L^4} \|z\|\ \|u\|_{L^4} \\
&\leq c|u|\ \|z\|\ \|u\|+c \|z\|^2\  \|u\| \\
&\leq \frac{\varsigma}{16} \|u\|^2+\frac{C}{2}|u|^2\ \|z\|^2+c \|z\|^4,
\end{aligned} \\
 \label{e4.11}
\begin{aligned}
(B_2(u+z,Z), U)
&\leq c|u+z|\ \|\nabla Z\|_{L^4}\|U\|_{L^4} \\
&\leq c|u|\ \|Z\|_{H^2}\|U\|+c|z|\ \|Z\|_{H^2}\|U\| \\
&\leq \frac{\varsigma}{16} \|U\|^2+\frac{C}{2}|u|^2\ \|Z\|_{H^2}^2+c
|z|^2\ \|Z\|_{H^2}^2.
\end{aligned}
\end{gather}
Then  from \eqref{e4.8}--\eqref{e4.11} it follows that
\begin{equation} \label{e4.12}
\begin{aligned}
&\frac{d}{dt}|\mathbf{U}|^2+\frac{3}{2}\varsigma\|\mathbf{U}\|^2\\
&\leq C(\|z\|^2+\|Z\|_{H^2}^2)|\mathbf{U}|^2+c(\|z\|^4+|z|^2\
\|Z\|_{H^2}^2+|z|^2+|Z|^2+|F|^2),
\end{aligned}
\end{equation}
where $c$ and $C$ is the positive constants independent of $t, u, U,
z, Z$. Further, the Poincar\'{e}'s inequality implies
\begin{equation} \label{e4.13}
\frac{d}{dt}|\mathbf{U}|^2+\varsigma\|\mathbf{U}\|^2 \leq
M(\theta_t\omega)|\mathbf{U}|^2+G(\theta_t\omega),
\end{equation}
which shows that \eqref{e4.6} holds.

Multiplying \eqref{e3.3} by $A\mathbf{U}$ and then integrating over $\mathcal{O}$,
along with \eqref{e4.2},  gives
\begin{equation} \label{e4.14}
\begin{aligned}
&\frac{d}{dt}((\nu+\kappa)\|u\|^2+\gamma\|U\|^2)\\
&=-2|A\mathbf{U}|^2-2(L\mathbf{U},A\mathbf{U})
-2(B(u+z,\mathbf{U}+\mathbf{Z}),A\mathbf{U})
-2(L\mathbf{Z},A\mathbf{U})+2(F,A\mathbf{U}) \\
&\leq -|A\mathbf{U}|^2+4\kappa^2\|\mathbf{U}\|^2
-2(B(u+z,\mathbf{U}+\mathbf{Z}), A\mathbf{U})
-2(L\mathbf{Z},A\mathbf{U})+2(F,A\mathbf{U}),
\end{aligned}
\end{equation}
where
\begin{equation} \label{e4.15}
\begin{aligned}
&-(L\mathbf{Z},A\mathbf{U})+(F,A\mathbf{U})\\
&=2\kappa(\nu+\kappa)(\nabla\times
Z, \Delta u)+2\kappa\gamma(\nabla\times z, \Delta
U)+4\kappa\gamma(Z,\Delta U)+(F,A\mathbf{U}) \\
&\leq \frac{1}{16}|A\mathbf{U}|^2+c(\|z\|^2+\|Z\|^2+|Z|^2+|F|^2).
\end{aligned}
\end{equation}
Consider that
\begin{equation} \label{e4.16}
(B(u+z,\mathbf{U}+\mathbf{Z}),A\mathbf{U})
=(B_1(u+z,u+z),A_1u)+(B_2(u+z, U+Z),A_2U).
\end{equation}
Then by \eqref{e4.4} we have
\begin{equation} \label{e4.17}
\begin{aligned}
&(B_1(u+z,u+z),A_1u)\\
&\leq \|u+z\|_{L^\infty}\|u+z\|\, |A_1u| \\
&\leq c|u+z|^{1/2}|A_1(u+z)|^{1/2}\|u+z\|\, |A_1u| \\
&\leq \frac{1}{8}|A_1u|^2+c|u+z|\, |A_1(u+z)|\, \|u+z\|^2 \\
&\leq \frac{1}{8}|A_1u|^2+c|u+z|\, \|u+z\|^2\, |A_1u|+c|u+z|\, \|u+z\|^2\ |A_1z| \\
&\leq \frac{1}{8}|A_1u|^2+\frac{1}{8}|A_1u|^2+c|u+z|^2\|u+z\|^4+c|A_1z|^2 \\
&\leq \frac{1}{4}|A_1u|^2+c(\|u\|^4|u|^2+\|u\|^4|z|^2+|u|^2\|z\|^4+|z|^2\|z\|^4
 +|A_1z|^2),
\end{aligned}
\end{equation}

\begin{equation} \label{e4.18}
\begin{aligned}
&(B_2(u+z, U+Z),A_2U)\\
&\leq\|u+z\|_{L^\infty}\|U+Z\|\ |A_2U| \\
&\leq c|u+z|^{1/2}|A_1(u+z)|^{1/2}\ \|U+Z\|\ |A_2U| \\
&\leq \frac{1}{8}|A_2U|^2+c|u+z|\ |A_1(u+z)|\ \|U+Z\|^2 \\
&\leq\frac{1}{8}|A_2U|^2+c|u+z|\  \|U+Z\|^2|A_1u| +c|u+z|\  \|U+Z\|^2|A_1z| \\
&\leq\frac{1}{8}|A_2U|^2+\frac{1}{8}|A_1u|^2+c|u+z|^2\|U+Z\|^4+c|A_1z|^2 \\
&\leq\frac{1}{8}|A_2U|^2+\frac{1}{8}|A_1u|^2+c|u|^2\|U\|^4+c|u|^2\|Z\|^4
 +c|z|^2\|U\|^4 \\
&\quad +c|z|^2\|Z\|^4+c|A_1z|^2.
\end{aligned}
\end{equation}
Therefore, from \eqref{e4.14}--\eqref{e4.18} it follows that
\begin{align*}
&\frac{d}{dt}((\nu+\kappa)\|u\|^2+\gamma\|U\|^2)\\
&\leq c(\|u\|^4+\|U\|^4)(|u|^2+|z|^2)+4\kappa^2\|\mathbf{U}\|^2+c|u|^2(\|z\|^4
 +\|Z\|^4) \\
&\quad +c(|A_1z|^2+|z|^2\|z\|^4+|z|^2\|Z\|^4+\|z\|^2+\|Z\|^2+|Z|^2+|F|^2) \\
&\leq c((\nu+\kappa)\|u\|^2+\gamma\|U\|^2)(\|u\|^2+\|U\|^2)(|u|^2+|z|^2)
 +c|u|^2(\|z\|^4+\|Z\|^4) \\
&\quad +4\kappa^2\|\mathbf{U}\|^2+c(\|z\|_{H^2}^2+|z|^2\|z\|^4
 +|z|^2\|Z\|^4+\|z\|^2+\|Z\|^2+|Z|^2+|F|^2),
\end{align*}
which proves \eqref{e4.7}.


 \begin{lemma} \label{lem4.2}
Assume that the expectation $\mathbb{E}M<0$, where $M$ is in \eqref{e3.4}.
 Let $D\in \mathcal{D}$. Then for
$\mathbb{P}$-a.e. $\omega\in \Omega$, there exist tempered random
radius $R_1(\omega)$ and constant $T=T(D,\omega)>0$ such that for
all $t\geq T$, the solution
$\mathbf{U}(t,\omega,\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))$ of
the initial problem \eqref{e3.3} with $\mathbf{V}_0\in D$ satisfies that
for every $l\in [t,t+1]$,
\begin{equation} \label{e4.19}
|\mathbf{U}(l, \theta_{-t-1}\omega,
\mathbf{V}_0(\theta_{-t-1}\omega)-\mathbf{Z}(\theta_{-t-1}\omega))|
\leq R_1(\omega),
\end{equation}
where $\mathcal{D}$ is the
collection of all tempered random subsets of $\mathcal{H}$.
\end{lemma}


\begin{proof}
 Applying the Gronwall's lemma (see \cite{Tem}) to \eqref{e4.6},
we find that for every $l\geq0$,
\[
|\mathbf{U}(l, \omega,
\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))|^2
\leq e^{\int_0^l{M}(\theta_{\tau}\omega)d\tau}|\mathbf{U}_0(\omega)|^2
+c\int_0^lG(\theta_{s}\omega)e^{\int_s^l{M}(\theta_{\tau}\omega)d\tau}ds,
\]
 in which we replace $\omega$ with
 $\theta_{-t-1}\omega$ and get that for $l\in [t,t+1]$,
\begin{equation} \label{e4.20}
\begin{aligned}
&|\mathbf{U}(l,\theta_{-t-1}\omega,\mathbf{V}_0(\theta_{-t-1}\omega)
 -\mathbf{Z}(\theta_{-t-1}\omega))|^2 \\
&\leq e^{\int_0^l{M}(\theta_{\tau-t-1}\omega)d\tau}|
 \mathbf{U}_0(\theta_{-t-1}\omega)|^2
+ \int_0^lG(\theta_{s-t-1}\omega)e^{\int_s^l{M}(\theta_{\tau-t-1}\omega)d\tau}ds \\
&=e^{\int_{-t-1}^{l-t-1}{M}(\theta_{\tau}\omega)d\tau}
|\mathbf{U}_0(\theta_{-t-1}\omega)|^2
+\int_{-t-1}^{l-t-1}G(\theta_{s}\omega)e^{\int_s^{l-t-1}{M}(\theta_{\tau}\omega)d\tau}ds \\
&\leq 2e^{\int_{-t-1}^{l-t-1}{M}(\theta_{\tau}\omega)d\tau}
(|\mathbf{V}_0(\theta_{-t-1}\omega)|^2+|\mathbf{Z}(\theta_{-t-1}\omega)|^2) \\
&\quad +\int_{-t-1}^{0}G(\theta_{s}\omega)e^{\int_s^{l-t-1}{M}
(\theta_{\tau}\omega)d\tau}ds.
\end{aligned}
\end{equation}
By noting that for $l\in [t, t+1]$, $l-t-1\in [-1,0]$, we deduce
that for $s\in [-t-1, 0]$,
\begin{equation} \label{e4.21}
\begin{aligned}
e^{\int_{s}^{l-t-1}M(\theta_\tau\omega)d\tau}
&=e^{-\frac{1}{2}\lambda_0\varsigma(l-t-1)+\frac{1}{2}\lambda_0\varsigma s}
e^{\int_{s}^{l-t-1}C(\|Z(\theta_\tau\omega)\|^2_{H^2}
 + \|z(\theta_\tau\omega)\|^2)d\tau} \\
&\leq e^{\frac{1}{2}\lambda_0\varsigma
 +\frac{1}{2}\lambda_0\varsigma s}e^{\int_{s}^{0}C(\|Z(\theta_\tau\omega)\|^2_{H^2}
+ \|z(\theta_\tau\omega)\|^2)d\tau} \\
&\leq e^{\frac{1}{2}\lambda_0\varsigma}e^{\int_{s}^{0}M(\theta_\tau\omega)d\tau}.
\end{aligned}
\end{equation}
Then by \eqref{e4.20} and \eqref{e4.21} we obtain that for every $l\in [t,t+1]$,
\begin{align*}
&|\mathbf{U}(l, \theta_{-t-1}\omega,\mathbf{V}_0(\theta_{-t-1}\omega)
-\mathbf{Z}(\theta_{-t-1}\omega))|^2\\
&\leq K \Big(e^{\int_{-t-1}^{0}{M}(\theta_{s}\omega)ds}
|\mathbf{V}_0(\theta_{-t-1}\omega)|^2+|\mathbf{Z}(\theta_{-t-1}\omega)|^2)\\
&\quad +\int_{-t-1}^{0}G(\theta_{s}\omega)e^{\int_s^{0}{M}
 (\theta_{\tau}\omega)d\tau}ds\Big),
\end{align*}
where $K=2e^{\frac{1}{2}\lambda_0\varsigma}$.
By the Birkhoff's ergodic theorem and along with our assumption that
$\mathbb{E}M(\omega)<0$, it yields that
$$
\lim_{t\to\pm\infty}\frac{1}{t+1}\int_{-t-1}^0M(\theta_\tau\omega)d\tau
=\mathbb{E}M(\omega)=\hat{\mu}<0,
$$
which implies
\[
{\int_{-t-1}^0M(\theta_\tau\omega)d\tau}\approx {\hat{\mu}(t+1)}, \quad
\text{as } t\to +\infty.
\]
 Since $G(\theta_s)$ is sub-exponential growth,
\[
\int_{-\infty}^{0}G(\theta_{s}\omega)e^{\int_s^{0}{M}
(\theta_{\tau}\omega)d\tau}ds<+\infty.
\]
Consider that $\|\mathbf{Z}(\omega)\|_{D(A)}$ is tempered. Then
$|\mathbf{Z}(\omega)|$ is tempered, whence by the initial data
$\mathbf{V}_0(\omega)\in D(\omega)$, there exists constant
$T=T(D,\omega)>0$ such that for all $t\geq T $ with  $l\in [t,t+1]$,
\begin{align*}
&|\mathbf{U}(l, \theta_{-t-1}\omega,\mathbf{V}_0(\theta_{-t-1}\omega)
-\mathbf{Z}(\theta_{-t-1}\omega))|^2\\
&\leq R_1(\omega)^2:=K(1+\int_{-\infty}^{0}G(\theta_{s}\omega)
e^{\int_s^{0}{M}(\theta_{\tau}\omega)d\tau}ds).
\end{align*}
We can use the method in \cite[Lemma 4.6]{Caraba} to show that the
random variable $R_1(\omega)$ is tempered. Indeed, for an arbitrary
$\lambda>0$, and let $t<0$. We then have
\begin{equation} \label{e4.22}
\begin{aligned}
&e^{-2\lambda |t|} R_1(\theta_t\omega)^2\\
&=Ke^{\lambda 2t} +Ke^{\lambda t}\int_{-\infty}^{0}G(\theta_{s+t}\omega)e^{\lambda
t+\int_s^{0}{M}(\theta_{\tau+t}\omega)d\tau}ds. \\
&=Ke^{\lambda 2t} +Ke^{\lambda t}\int_{-\infty}^{0}G(\theta_{s+t}\omega)e^{\lambda
t+\int_{s+t}^{0}(M(\theta_{\tau}\omega)-\hat{\mu})d\tau
-\int_{t}^{0}(M(\theta_{\tau}\omega)-\hat{\mu})d\tau-\hat{\mu}s}ds.
\end{aligned}
\end{equation}
Let $0<\varepsilon<\frac{1}{4}\min\{-\hat{\mu},\lambda\}$. Then
using again the sub-exponential growth of  $G(\theta_\tau)$, there
exists $t_1=t_1(\varepsilon,\omega)<0$ such that for all $t< t_1$,
\begin{equation} \label{e4.23}
G(\theta_{s+t})\leq e^{-\varepsilon (s+t)}.
\end{equation}
On the other hand,  there exists $t_2=t_2(\varepsilon,\omega)<0$
such that for all $t< t_2$,
\begin{equation} \label{e4.24}
\int_{s+t}^{0}(M(\theta_{\tau}\omega)-\hat{\mu})d\tau\leq
-\varepsilon (t+s); \quad
-\int_{t}^{0}(M(\theta_{\tau}\omega-\hat{\mu})\leq
-\varepsilon(t+s).
\end{equation}
Put $t_0=\min\{t_1,t_2\}$. Then it follows from \eqref{e4.22}--\eqref{e4.24}
that for all $t< t_0$,
\begin{align*}
 e^{-2\lambda |t|} R_1(\theta_t\omega)^2
&\leq Ke^{\lambda 2t} +Ke^{\lambda t}\int_{-\infty}^{0}e^{\lambda t-3\varepsilon
(t+s)-\hat{\mu}s}ds \\
&\leq Ke^{\lambda 2t} +Ke^{\lambda t}\int_{-\infty}^{0}e^{
\varepsilon (t+s)}ds\to 0
\end{align*}
as $t\to-\infty$. Similarly, we can prove the
convergence for $t\to+\infty$.
\end{proof}


 \begin{lemma} \label{lem4.3}
Assume that the expectation $\mathbb{E}M<0$, where $M$ is in \eqref{e3.4}.
Let $D\in \mathcal{D}$. Then for
$\mathbb{P}$-a.e. $\omega\in \Omega$, there exist random radius
$R_2(\omega)$ and constant $T=T(D,\omega)>0$ such that for all
$t\geq T$, the solution
$\mathbf{U}(t,\omega,\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))$ of
the initial problem \eqref{e3.3} with $\mathbf{V}_0\in D$ satisfies that
for every $l\in [t,t+1]$,
\[
\|\mathbf{U}(l,\theta_{-t-1}\omega,\mathbf{V}_0(\theta_{-t-1}\omega)
-\mathbf{Z}(\theta_{-t-1}\omega))\|
\leq R_2(\omega),
\]
where  $\mathcal{D}$ is the collection of tempered random subsets of
$\mathcal{H}$.
\end{lemma}

\begin{proof}
Integrating \eqref{e4.6} from $t$ to $t+1$  yields
\begin{align*}
&\int_t^{t+1}\|\mathbf{U}(s,\omega,\mathbf{V}_0(\omega)
 -\mathbf{Z}(\omega))\|^2ds \\
&\leq c\int_t^{t+1}M(\theta_s\omega)|\mathbf{U}(s,\omega,\mathbf{V}_0(\omega)
 -\mathbf{Z}(\omega))|^2ds \\
&\quad +c\int_t^{t+1}G(\theta_s\omega)ds+c|\mathbf{U}(t,\omega,\mathbf{V}_0(\omega)
-\mathbf{Z}(\omega))|^2.
\end{align*}
Then by Lemma \ref{lem4.2} we know  that there exists a $T=T(D,\omega)>0$
such that for every $t\geq T$,
\begin{equation} \label{e4.25}
\begin{aligned}
&\int_t^{t+1}\|\mathbf{U}(s,\theta_{-t-1}\omega,
 \mathbf{V}_0(\theta_{-t-1}\omega)-\mathbf{Z}(\theta_{-t-1}\omega))\|^2ds \\
&\leq cR_1(\omega)^2\int_t^{t+1}M(\theta_{s-t-1}\omega)ds
 +c\int_t^{t+1}G(\theta_{s-t-1}\omega)ds+cR_1(\omega)^2 \\
&=c(R_1(\omega)^2\int_{-1}^{0}M(\theta_{s}\omega)ds
 +\int_{-1}^{0}G(\theta_{s}\omega)ds+R_1(\omega)^2):=C_1(\omega).
\end{aligned}
\end{equation}
Using the classic Gronwall's lemma to \eqref{e4.7} on the interval $[s, l]$
with $t\leq s\leq l\leq t+1$, we obtain
\begin{align*}
\|\mathbf{U}(l,\omega,\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))\|^2
&\leq ce^{\int_t^{t+1} g(\tau,\omega)d\tau}(\|\mathbf{U}(s,\omega,
 \mathbf{V}_0(\omega)-\mathbf{Z}(\omega))\|^2\\
&\quad +\int_t^{t+1}h(s,\omega)ds),
\end{align*}
from which it follows that
\begin{equation} \label{e4.26}
\begin{aligned}
&\|\mathbf{U}(l,\theta_{-t-1}\omega,\mathbf{V}_0(\theta_{-t-1}\omega)
 -\mathbf{Z}(\theta_{-t-1}\omega))\| \\
&\leq ce^{\int_t^{t+1} g(\tau,\theta_{-t-1}\omega)d\tau}(\int_t^{t+1}\|\mathbf{U}(s,\theta_{-t-1}\omega,\mathbf{V}_0(\theta_{-t-1}\omega)-\mathbf{Z}(\theta_{-t-1}\omega))\|^2
ds \\
&\quad +\int_t^{t+1}h(s,\theta_{-t-1}\omega)ds).
\end{aligned}
\end{equation}
Note that by Lemma \ref{lem4.1} and association with \eqref{e4.25}, there exists
$T=T(D,\omega)>0$ such that for all $t\geq T$,
\begin{equation} \label{e4.27}
\begin{aligned}
&\int_t^{t+1} g(\tau,\theta_{-t-1}\omega)d\tau\\
&=c\int_t^{t+1}(\|u(s)\|^2+
\|U(s)\|^2)(|u(s)|^2+|z(\theta_{s-t-1}\omega)|^2)ds \\
&\leq c(R_1(\omega)^2+\max_{-1\leq
t\leq0}\{|z(\theta_{t}\omega)|^2\}\int_{t}^{t+1}(\|u(s)\|^2+
\|U(s)\|^2)ds \\
&\leq c(R_1(\omega)^2+\max_{-1\leq
t\leq0}\{|z(\theta_{t}\omega)|^2\})C_1(\omega):=C_2(\omega),
\end{aligned}
\end{equation}
\begin{equation} \label{e4.28}
\begin{aligned}
&\int_t^{t+1}h(s,\theta_{-t-1}\omega)ds\\
&=c\int_t^{t+1}\|\mathbf{U}(s)\|^2ds
+c\int_t^{t+1}|u(s)|^2(\|z(\theta_{s-t-1}\omega)\|^4+\|Z(\theta_{s-t-1}\omega)\|^4)ds\\
&\quad +\int_t^{t+1}H(\theta_{s-t-1}\omega)ds \\
&\leq c\max_{-1\leq
t\leq0}\{\|z(\theta_{t}\omega)\|^4+\|Z(\theta_{t}\omega)\|^4\}(C_1(\omega)
+R_1(\omega)^2)+\int_{-1}^{0}H(\theta_{s}\omega)ds\\
&:=C_3(\omega).
\end{aligned}
\end{equation}
Then by \eqref{e4.26}--\eqref{e4.28} it gives that  for all  $t\geq T$ and
$l\in [t,t+1]$,
\[
\|\mathbf{U}(l,\theta_{-t-1}\omega,\mathbf{V}_0(\theta_{-t}\omega)-\mathbf{Z}(\theta_{-t}\omega))\|^2\leq
ce^{C_2(\omega)}(C_1(\omega)+C_3(\omega)):=R_2(\omega)^2.
\]
This completes the proof.
\end{proof}


 \begin{lemma} \label{lem4.4}
Assume that the expectation $\mathbb{E}M<0$, where $M$ is in \eqref{e3.4}.
 Let $D\in \mathcal{D}$. Then for
$\mathbb{P}$-a.e. $\omega\in \Omega$ and every $\varepsilon>0$, there
are $N=N(\varepsilon,\omega),K=K(\omega)$, and $T=T(\varepsilon,
D,\omega)>0$ such that for all $t\geq T$ and $m\geq N$, the solution
$\mathbf{U}(t,\omega,\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))$ of
the initial problem \eqref{e3.3} with $\mathbf{V}_0\in D$ satisfies that
\begin{gather*}
\|P_m\mathbf{U}(t,\theta_{-t}\omega,\mathbf{V}_0(\theta_{-t}\omega)
-\mathbf{Z}(\theta_{-t}\omega))\|\leq K, \\
\|(I-P_m)\mathbf{U} (t ,\theta_{-t}\omega,\mathbf{V}_0(\theta_{-t}\omega)
-\mathbf{Z}(\theta_{-t}\omega))\|\leq \varepsilon,
\end{gather*}
where $\mathcal{D}$ is the collection
of tempered random subsets of $\mathcal{H}$.
\end{lemma}

\begin{proof}
 Multiplying \eqref{e3.3} by $A\mathbf{U}_m$ and then integrating over
$\mathcal{O}$, we have
\begin{equation} \label{e4.29}
\begin{aligned}
&\frac{d}{dt}((\nu+\kappa)\|u_m\|^2+\gamma\|U_m\|^2)+2|A\mathbf{U}_m|^2\\
&=-2(L\mathbf{U},A\mathbf{U}_m)-2(B(u+z,\mathbf{U}+\mathbf{Z}),A\mathbf{U}_m)
-2(L\mathbf{Z},A\mathbf{U}_m)+2(F,A\mathbf{U}_m),
\end{aligned}
\end{equation}
 where
\begin{gather} \label{e4.30}
\begin{aligned}
&-(L\mathbf{U},A\mathbf{U}_m)\\
&=-2\kappa(\nu+\kappa)(\nabla\times U,
\Delta u_m)-2\kappa\gamma(\nabla\times u, \Delta
U_m)-4\kappa\gamma\|U_m\|^2 \\
&\leq \frac{1}{2}|A\mathbf{U}_m|^2+2\kappa^2\|\mathbf{U}\|^2,
\end{aligned} \\
\label{e4.31}
\begin{aligned}
&-(L\mathbf{Z},A\mathbf{U}_m)\\
&=-2\kappa(\nu+\kappa)(\nabla\times Z,
\Delta u_m)-2\kappa\gamma(\nabla\times z, \Delta
U_m)+4\kappa\gamma(Z,\Delta U_m) \\
&\leq \frac{1}{64}|A\mathbf{U}_m|^2+c(\|z\|^2+\|Z\|^2+|Z|^2),
\end{aligned} \\
 \label{e4.32}
2(F,A\mathbf{U}_m)\leq \frac{1}{32}|A\mathbf{U}_m|^2+c|F|^2.
\end{gather}
Then from \eqref{e4.29}--\eqref{e4.32}, we obtain
\begin{equation} \label{e4.33}
\begin{aligned}
&\frac{d}{dt}((\nu+\kappa)\|u_m\|^2+\gamma\|U_m\|^2)+|A\mathbf{U}_m|^2\\
&\leq -2(B(u+z,\mathbf{U}+\mathbf{Z}),
A\mathbf{U}_m)+\frac{1}{16}|A\mathbf{U}_m|^2 \\
&\quad +c\|\mathbf{U}\|^2+c(\|z\|^2+\|Z\|^2+|Z|^2+|F|^2).
\end{aligned}
\end{equation}
Likewise, we have
\begin{equation} \label{e4.34}
|A\mathbf{U}_m|^2\geq\lambda_{m+1}|A^{1/2}\mathbf{U}_m|^2
=\lambda_{m+1}((\nu+\kappa)\|u_m\|^2+\gamma\|U_m\|^2).
\end{equation}
It remains to estimate the first term on the right hand side of
\eqref{e4.33}. To this end, we rewrite
\begin{equation} \label{e4.35}
\begin{aligned}
&(B(u+z,\mathbf{U}+\mathbf{Z}),A\mathbf{U}_m)\\
&=(B_1(u+z,u+z),A_1u_m)+(B_2(u+z, U+Z),A_2U_m)=I_1+I_2,
\end{aligned}
\end{equation}
where
$$
I_1=(B_1(u+z,u+z),A_1u_m), \quad  I_2=(B_2(u+z, U+Z),A_2U_m).
$$
 To estimate $I_1$, we rewrite it as
\begin{equation} \label{e4.36}
\begin{aligned}
I_1&=(B_1(u,u), A_1u_m)+(B_1(u,z(\theta_t\omega)), A_1u_m) \\
&\quad +(B_1(z(\theta_t\omega),z(\theta_t\omega)),
A_1u_m)+(B_1(z(\theta_t\omega),{v}), A_1u_m),
\end{aligned}
\end{equation}
where by \eqref{e4.4} and \eqref{e4.5},  we calculate that
\begin{gather} \label{e4.37}
\begin{aligned}
&|(B_1(u,u), A_1u_m)|\\
&\leq|(B_1(P_mu,u),A_1u_m)|+|(B_1(u_m,u),A_1u_m)| \\
&\leq \|P_mu\|_{L^\infty}\|u\|\
|A_1u_m|+\|u_m\|_{L^\infty}\|u\|\ |A_1u_m| \\
&\leq c(1+\log\frac{\lambda_{m+1}}{\lambda_1})^{1/2}\|u\|^2\
|A_1u_m|+ c
\|u_m\|^{1/2}|A_1u_m|^{3/2}\|u\| \\
&\leq \frac{1}{32}
|A_1u_m|^2+c(1+\log\frac{\lambda_{m+1}}{\lambda_1})\|u\|^4+c|u|^2\|u\|^4,
\end{aligned} \\
 \label{e4.38}
\begin{aligned}
&|(B_1(u,z(\theta_t\omega)), A_1u_m)|\\
&\leq|(B_1(P_mu,z(\theta_t\omega)),A_1u_m)|+|(B_1(u_m,z(\theta_t\omega)),A_1u_m)| \\
&\leq \|P_mu\|_{L^\infty}\|z(\theta_t\omega)\|\
|A_1u_m|+\|u_m\|_{L^\infty}\|z(\theta_t\omega)\|\ |A_1u_m| \\
 &\leq c(1+\log\frac{\lambda_{m+1}}{\lambda_1})^{1/2}\|u\|\
\|z(\theta_t\omega)\|\ |A_1u_m|+ c
|u_m|^{1/2}|A_1u_m|^{3/2}\|z(\theta_t\omega)\|^2 \\
&\leq \frac{1}{32}
|A_1u_m|^2+c(1+\log\frac{\lambda_{m+1}}{\lambda_1})\|u\|^2\|z(\theta_t\omega)\|^2
+c|u|^2\|z(\theta_t\omega)\|^4,
\end{aligned} \\
 \label{e4.39}
|(B_1(z(\theta_t\omega),z(\theta_t\omega)),
A_1{u_m})|\leq\frac{1}{32}|
A_1{u_m}|^{2}+c\|z(\theta_t\omega)\|_{H^2}^{2}
\|z(\theta_t\omega)\|^2,\\
 \label{e4.40}
 |(B_1(z(\theta_t\omega),u),
A_1{u_m})|\leq\frac{1}{32}|
A_1{u_m}|^{2}+c\|z(\theta_t\omega)\|_{H^2}^{2} \|u\|^2.
\end{gather}
Then it follows from \eqref{e4.35}--\eqref{e4.40} that
\begin{equation} \label{e4.41}
\begin{aligned}
I_1&\leq\frac{1}{8}|
A_1u_m|^{2}+c(1+\log\frac{\lambda_{m+1}}{\lambda_1})(\|u\|^2\|z(\theta_t\omega)\|^2 +\|u\|^4)\\
&\quad +c(|u|^{2} \|u\|^4+\|z(\theta_t\omega)\|_{H^2}^{2}\|u\|^2
 +|u|^2\|z(\theta_t\omega)\|^4\\
&\quad +\|z(\theta_t\omega)\|_{H^2}^{2} \|z(\theta_t\omega)\|^2).
\end{aligned}
\end{equation}
Then we estimate $I_2$ in \eqref{e4.13}, by writing it as
\begin{equation} \label{e4.42}
\begin{aligned}
I_2&=(B_2(u, Z(\theta_t\omega)), A_2U_m)+(B_2(u, U), A_2U_m) \\
&\quad +(B_2(z(\theta_t\omega), Z(\theta_t\omega)),
A_2U_m)+(B_2(z(\theta_t\omega), U), A_2U_m),
\end{aligned}
\end{equation}
where
\begin{gather} \label{e4.43}
\begin{aligned}
&|(B_2(u, U), A_2U_m)|\\
&\leq |(B_2({P_mu}, U), A_2U_m)|+ |(B_2({u_m}, U), A_2U_m)| \\
&\leq \|P_mu\|_{L^\infty}\|U\|\ |A_2U_m|+
\|u_m\|_{L^\infty}\|U\|\ |A_2U_m| \\
&\leq c\|u\|(1+\log\frac{\lambda_{m+1}}{\lambda_1})^{1/2}\|U\|\ |A_2U_m|
 +c|u|^{1/2}|A_1u_m|^{1/2}\|U\|\ |A_2U_m| \\
&\leq \frac{1}{32} |A_2U_m|^2+c(1+\log\frac{\lambda_{m+1}}{\lambda_1})\|u\|^2\|U\|^2
+c|u|\ |A_1u_m|\ \|U\|^2 \\
&\leq \frac{1}{32} |A_2U_m|^2+c(1+\log\frac{\lambda_{m+1}}{\lambda_1})
\|u\|^2\|U\|^2+\frac{1}{32}
|A_1u_m|^2+c|u|^2\|U\|^4,
\end{aligned} \\
 \label{e4.44}
\begin{aligned}
|(B_2(u, Z(\theta_t\omega)), A_2U_m)|
&\leq \frac{1}{32}
|A_2U_m|^2+c(1+\log\frac{\lambda_{m+1}}{\lambda_1})\|u\|^2\|Z(\theta_t\omega)\|^2 \\
&\quad +\frac{1}{32} |A_1u_m|^2+c|u|^2\|Z(\theta_t\omega)\|^4,
\end{aligned}\\
\label{e4.45}
|(B_2( z(\theta_t\omega), Z(\theta_t\omega)), A_2U_m)|\leq
\frac{1}{32}|A_2U_m|^2+c\|
z(\theta_t\omega)\|_{H^2}^{2}\|Z(\theta_t\omega)\|^2,\\
 \label{e4.46}
|(B_2( z(\theta_t\omega), U), A_2U_m)|\leq
\frac{1}{32}|A_2U_m|^2+c\|z(\theta_t\omega)\|_{H^2}^{2}\| U\|^{2}.
\end{gather}
Then it follows from \eqref{e4.42}--\eqref{e4.46} that
\begin{equation} \label{e4.47}
\begin{aligned}
I_2&\leq\frac{1}{8}|A_2U_m|^2+\frac{1}{8}|A_1u_m|^2
+c(1+\log\frac{\lambda_{m+1}}{\lambda_1})\big(\|u\|^2\|U\|^2\\
&\quad +\|u\|^2\|Z(\theta_t\omega)\|^2\big)
+c(|u|^{2}\|U\|^{4}+|u|^{2}\| Z(\theta_t\omega)\|^{4}+
c\|z(\theta_t\omega)\|_{H^2}^2\|U\|^{2}\\
&\quad +\|z(\theta_t\omega)\|_{H^2}^{2} \|Z(\theta_t\omega)\|^2).
\end{aligned}
\end{equation}
Then we incorporate \eqref{e4.34}, \eqref{e4.35}, \eqref{e4.41} and
\eqref{e4.47} into \eqref{e4.33} to give
\begin{equation} \label{e4.48}
\begin{aligned}
\frac{d}{dt}((\nu+\kappa)\|u_m\|^2+\gamma\|U_m\|^2)
&\leq-\lambda_{m+1}((\nu+\kappa)\|u_m\|^2+\gamma\|U_m\|^2) \\
&\quad+(1+\log\frac{\lambda_{m+1}}{\lambda_1})P(t,\omega) +Q(t,\omega),
\end{aligned}
\end{equation}
where
\begin{gather*}
P(t,\omega)=c(\|u\|^4+\|u\|^2\|z(\theta_t\omega)\|^2+\|u\|^2\|U\|^2
+\|u\|^2\|Z(\theta_t\omega)\|^2), \\
\begin{aligned}
 Q(t,\omega)
&=c(\|\mathbf{U}\|^2+|u|^2\|u\|^4+|u|^2\|z(\theta_t\omega)\|^4
+|u|^2\|U\|^4+\|u\|^2\|Z(\theta_t\omega)\|^4 \\
&\quad +\|z(\theta_t\omega)\|_{H^2}^{2}\|
U\|^{2}+\| z(\theta_t\omega)\|_{H^2}^{2}
\|Z(\theta_t\omega)\|^2+\|z(\theta_t\omega)\|_{H^2}^{2}
\|z(\theta_t\omega)\|^2 \\
&\quad +\|z(\theta_t\omega)\|_{H^2}^{2}
\|u\|^2+|Z(\theta_t\omega)|^2+\|z(\theta_t\omega)\|^2
+\|Z(\theta_t\omega)\|^2+|F|^2).
\end{aligned}
\end{gather*}
 Multiplying \eqref{e4.48} by $e^{\lambda_{m+1}t}$ and then integrating
 over the interval $[t,t+1]$,  we infer that
\begin{equation} \label{e4.49}
\begin{aligned}
&\|\mathbf{U}_m(t+1, \omega,
\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))\|^2\\
&\leq c(1+\log\frac{\lambda_{m+1}}{\lambda_1})e^{-\lambda_{m+1}(t+1)}\int
_t^{t+1}e^{\lambda_{m+1} s}P(s,\omega)ds\\
&\quad +e^{-\lambda_{m+1}(t+1)}\int_t^{t+1}e^{\lambda_{m+1}
s}Q(s,\omega)ds \\
&\quad +e^{-\lambda_{m+1}(t+1)}\int_t^{t+1}e^{\lambda_{m+1}s}\|\mathbf{U}_m(s,
\omega,\mathbf{V}_0(\omega)-\mathbf{Z}(\omega))\|^2ds.
\end{aligned}
\end{equation}
According to Lemma \ref{lem4.3}, there exists $T=T(D,\omega)>0$ such that for
all $t\geq T$,
\begin{equation} \label{e4.50}
\begin{aligned}
&\int_t^{t+1}e^{\lambda_{m+1} s}P(s,\theta_{-t-1}\omega)ds \\
&\leq c\int_t^{t+1}e^{\lambda_{m+1} s}\Big(R_2(\omega)^4
 +R_2(\omega)^2\|z(\theta_{s-t-1}\omega)\|^2\\
&\quad +R_2(\omega)^4 +R_2(\omega)^2\|Z(\theta_{s-t-1}\omega)\|^2\Big)ds \\
&\leq
\frac{c}{\lambda_{m+1}}e^{\lambda_{m+1}(t+1)}(2R_2(\omega)^4
 +R_2(\omega)^2\sup_{-1\leq s\leq0}\|z(\theta_{s}\omega)\|^2 \\
&\quad  +R_2(\omega)^2
 \sup_{-1\leq s\leq0}\|Z(\theta_{s}\omega)\|^2) \\
&:=\frac{\hat{R}(\omega)}{\lambda_{m+1}}e^{\lambda_{m+1}(t+1)} ,
\end{aligned}
\end{equation}
where
\[
\hat{R}(\omega)
=c(2R_2(\omega)^4+R_2(\omega)^2\sup_{-1\leq s\leq0}\|z(\theta_{s}\omega)\|^2
+R_2(\omega)^2\sup_{-1\leq s\leq0}\|Z(\theta_{s}\omega)\|^2)
\]
 is independent of $\lambda_{m+1}$.
By a similar calculation as \eqref{e4.50}, we find that there exists a
random variable $\hat{\hat{R}}(\omega)$ such that for all $t\geq T$,
\begin{equation} \label{e4.51}
\int_t^{t+1}e^{\lambda_{m+1} s}Q(t,\theta_{-t-1}\omega)ds\leq
\frac{\hat{\hat{R}}(\omega)}{\lambda_{m+1}}e^{\lambda_{m+1}(t+1)},
\end{equation}
and
\begin{equation} \label{e4.52}
\int_t^{t+1}e^{\lambda_{m+1} s}\|\mathbf{U}_m(s,
\theta_{-t-1}\omega)\|^2ds\leq
\frac{R_2(\omega)^2}{\lambda_{m+1}}e^{\lambda_{m+1}(t+1)},
\end{equation}
where $R_2(\omega)$ is in Lemma \ref{lem4.3}.
 Then \eqref{e4.49}, together with \eqref{e4.50}--\eqref{e4.52},
implies that for all $t\geq T$,
\begin{align*}
&\|\mathbf{U}_m(t+1,\theta_{-t-1}\omega,
 \mathbf{V}_0(\theta_{-t-1}\omega)-\mathbf{Z}(\theta_{-t-1}\omega))\|^2 \\
&\leq \frac{1}{\lambda_{m+1}}(1+\log\frac{\lambda_{m+1}}{\lambda_1})\hat{R}(\omega)
+\frac{1}{\lambda_{m+1}}(\hat{\hat{R}}(\omega)+R_2(\omega)^2)\to0,
\end{align*}
as $m\to +\infty$. As a consequence,  for every
$\varepsilon>0$ and $D\in \mathcal{D}$, there exists an integer $N$
and positive constant $K=K(\omega)$ such that for all $t\geq T$ and
$m\geq N$,
\begin{gather*}
\|(I-P_m)\mathbf{U}(t,\theta_{-t}\omega,\mathbf{V}_0(\theta_{-t}\omega)
-\mathbf{Z}(\theta_{-t}\omega))\|\leq \varepsilon, \\
\|P_m\mathbf{U}(t,
\theta_{-t}\omega,\mathbf{V}_0(\theta_{-t}\omega)
-\mathbf{Z}(\theta_{-t}\omega))\|\leq K.
\end{gather*}
This concludes the proof.
\end{proof}

\begin{lemma} \label{lem4.5}
Assume that the expectation $\mathbb{E}M<0$, where $M$ is in \eqref{e3.4}.
Then the RDS $\varphi$ generated by the solution of stochastic michropolar fluid
flows \eqref{e1.1} is omega-limit compact in $\mathcal{V}$; i.e., for every
$\varepsilon>0$ and an arbitrary $D\in \mathcal{D}$, there is an
$T=T(\varepsilon,D,\omega)>0$ such that for
$\mathbb{P}$-a.e. $\omega\in \Omega$,
\[
k\Big(\cup_{t\geq T}\varphi(t,\theta_{-t}\omega)D(\theta_{-t}\omega)\Big)
\leq\varepsilon,
\]
where $\mathcal{D}$ is the collection of tempered random subsets of
$\mathcal{H}$.
\end{lemma}



\begin{proof}
 By Lemma \ref{lem4.4}, there exist constants  $\hat{K}(\omega)$ and
 $T=T(\varepsilon,D,\omega)$ and $N_1\in \mathbb{N}$ such that for all
$t\geq T$ and $m\geq  N_1$, there hold
$\|P_m\mathbf{U}(t, \theta_{-t}\omega,
\mathbf{V}_0(\theta_{-t}\omega)-\mathbf{Z}(\theta_{-t}\omega))\|\leq
\hat{K}(\omega)$ and
\begin{equation} \label{e4.53}
 \|(I-P_m)\mathbf{U} (t,\theta_{-t}\omega,\mathbf{V}_0(\theta_{-t}\omega)
-\mathbf{Z}(\theta_{-t}\omega))\|\leq \frac{\varepsilon}{2}.
\end{equation}
Note that $\|P_m\mathbf{Z}(\omega)\|\leq \|\mathbf{Z}(\omega)\|$,
and
$$
\|(I-P_m)\mathbf{Z}(\omega)\|\leq
\frac{1}{\lambda_{m+1}}\|\mathbf{Z}(\omega)\|_{H^2}\to0
$$
as $m\to\infty$; and then there exists
$N_2\in \mathbb{N}$ such that for every $m\geq N_2$,
\begin{equation} \label{e4.54}
\|P_m\mathbf{Z}(\omega)\|\leq \|\mathbf{Z}(\omega)\|,\quad
\|(I-P_m)\mathbf{Z}(\omega)\|\leq \frac{\varepsilon}{2}.
\end{equation}
Put $N=\max\{N_1,N_2\}$, by \eqref{e4.53} and \eqref{e4.54} we find that there
exist $K(\omega)=\hat{K}(\omega)+\|\mathbf{Z}(\omega)\|^2$ and
$T=T(\varepsilon,D,\omega)>0$ such that for all $t\geq T$,
\begin{gather} \label{e4.55}
\|P_N\varphi(t,\theta_{-t}\omega)D(\theta_{-t}\omega)\|\leq K(\omega), \\
\label{e4.56}
\|(I-P_N)\varphi(t,\theta_{-t}\omega)D(\theta_{-t}\omega)\|\leq \varepsilon.
\end{gather}
That is to say the RDS $\varphi$ satisfies the  flattening
conditions in $\mathcal{V}$; see \cite{Kloe}. By utilizing the
additive property of Kuratowski measure of non-compactness; see,
\cite[Lemma 2.5 (ii)]{Yan1}, it follows from \eqref{e4.55} and \eqref{e4.56}
that for $\mathbb{P}$-a.e. $\omega\in \Omega$,
\begin{align*}
k\Big(\cup_{t\geq T}\varphi(t,\theta_{-t}\omega)D(\theta_{-t}\omega)\Big)
&\leq k\Big(P_N\Big(\cup_{t\geq T}\varphi(t,\theta_{-t}\omega)
D(\theta_{-t}\omega)\Big)\Big) \\
&\quad +k\Big((I-P_N)\Big(\cup_{t\geq T}\varphi(t,\theta_{-t}\omega)
 D(\theta_{-t}\omega)\Big)\Big) \\
&\leq 0+k(B_{\mathcal{V}}(0,\varepsilon))=2\varepsilon,
 \end{align*}
where $B_{\mathcal{V}}(0,\varepsilon)$ is the $\varepsilon$-
neighborhood at centre $0$ in $\mathcal{V}$. This completes the
proof.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm3.3}]
According to Lemma \ref{lem4.5}, by the
embedding relation $\mathcal{V}\hookrightarrow \mathcal{H}$, we
can show that $\varphi$ is omega-limit compact in $\mathcal{H}$. But
by Lemma \ref{lem4.2}, $\varphi$ possesses a tempered random absorbing set
$\hat{D}\in \mathcal{D}$ in $\mathcal{H}$. Thus by the first
conclusion in Theorem \ref{thm2.3}, we know that $\varphi$ admits an
$(\mathcal{H},\mathcal{H})$-random attractor $\mathcal{A}_\mathcal{H}$.
From Lemma \ref{lem4.5}, $\varphi$ is $(\mathcal{H},\mathcal{V})$-omega-limit compact.
Then the second conclusion in Theorem \ref{thm2.3}
implies that the existence of $(\mathcal{H},\mathcal{V})$-random
attractor $\mathcal{A}_\mathcal{V}$. Furthermore,
$\mathcal{A}_\mathcal{V}=\mathcal{A}_\mathcal{H}$.
\end{proof}

\subsection*{Acknowledgments}
This research  was supported by the China NSF Grant (No. 11271388),
the Science and Technology Funds of Chongqing Educational Commission (No. KJ130703),
 the Chongqing Basis and Frontier Research Project of China (No. cstc2014jcyjA00035).


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\end{document}