\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 59, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/59\hfil Periodic random  attractors]
{Periodic random  attractors for stochastic  Navier-Stokes equations
on unbounded domains}

\author[B. Wang \hfil EJDE-2012/59\hfilneg]
{Bixiang Wang}

\address{Bixiang Wang \newline
Department of Mathematics\\
New Mexico Institute of Mining and Technology  \\
Socorro,  NM 87801, USA}
\email{bwang@nmt.edu}

\thanks{Submitted February 13, 2012. Published April 12, 2012.}
\subjclass[2000]{35B40, 35B41, 37L30}
\keywords{Random attractor; stochastic  Navier-Stokes equation;
      \hfill\break\indent  unbounded domain; complete solution}

\begin{abstract}
 This article concerns the asymptotic behavior of solutions
 to the two-dimensional Navier-Stokes equations with both
 non-autonomous deterministic and stochastic terms defined on
 unbounded domains. First we  introduce a continuous cocycle
 for the equations and then prove  the existence and uniqueness
 of tempered  random  attractors.
 We also characterize the structures of the random  attractors
 by complete solutions.  When deterministic forcing terms are periodic,
 we show that the tempered  random attractors  are also periodic.
 Since the Sobolev embeddings on unbounded domains  are not compact,
 we establish the pullback asymptotic compactness of solutions  by
 Ball's idea of  energy equations.
\end{abstract}

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


\section{Introduction}

 In this article,  we investigate the pullback attractors for the
 two-dimensional Navier-Stokes equations on unbounded domains
 with  non-autonomous deterministic and stochastic terms.
 Let $Q$ be an unbounded open set in $\mathbb{R}^2$ with boundary
 $\partial Q$.  Given   $\tau \in\mathbb{R}$,  consider   the stochastic
 Navier-Stokes equations with multiplicative noise:
 \begin{gather} \label{intro1}
 \frac{\partial u}{\partial t}
 -\nu \Delta u  + (u \cdot \nabla) u
 = f(x,t) -\nabla p
 +  \alpha u \circ \frac{d w}{dt}, \quad x \in Q  \text{ and }  t >\tau, \\
\label{intro2}
 \operatorname{div} u =0,
  \quad x \in Q  \text{ and }  t >\tau,
 \end{gather}
 together with  homogeneous  Dirichlet boundary condition,
 where $\nu,  \alpha \in \mathbb{R}$  with  $\nu>0$,  $f$ is a
 given function defined on $Q \times \mathbb{R}$,  and $w$
 is  a  two-sided real valued Wiener process
 defined in a probability space.
 The  stochastic  equation \eqref{intro1}  is
 understood in the sense of Stratonovich  integration.

The attractors of the Navier-Stokes equations have been extensively
studied  in the literature; see, e.g.
\cite{bab1, bal1, car3, car4, hal1, ros1, sel1, tem1}
for deterministic   equations
and  \cite{ cra2, fla1, schm1} for stochastic  equations.
Particularly, in the deterministic case (i.e.,  $\alpha =0$),
the autonomous global  attractors and
the non-autonomous pullback attractors
of     \eqref{intro1}-\eqref{intro2} on \emph{unbounded}  domains
have been studied in \cite{ros1} and \cite{car3, car4}, respectively.
For the  stochastic equations with additive noise
and  time-independent  $f$,  the asymptotic compactness
of solutions  on \emph{unbounded}
  domains has been investigated in \cite{brz1}.
As far as the author is aware,  there is no  result available in the literature
on the existence of random attractors for the stochastic equations
\eqref{intro1}-\eqref{intro2}   with   \emph{time-dependent}  $f$
even on bounded domains.
The purpose of the present  article is  to  investigate  this problem
 and examine the periodicity of random
attractors  when $f$ is periodic in time.

It is  worth mentioning  that  the concept of pullback attractors
for random  systems  with time-independent $f$
was introduced  in \cite{cra2, fla1, schm1}  and
the existence of such attractors for compact systems was proved  in
\cite{arn1,  car2, chu2, cra1, cra2, fla1, huang1, kloe1, schm1}
 and the references therein.
For non-compact systems,  the existence of pullback attractors
was established in \cite{bat1, bat2,  wan2, wan3}.
In the present paper,    we study pullback   attractors
for the stochastic equations \eqref{intro1}-\eqref{intro2}
on unbounded domains with time-dependent $f$.
In this case,  the random dynamical  systems
associated with the equations are non-compact.


To deal with  the  stochastic equations with
 non-autonomous $f$,   we need
to combine  the  ideas of non-autonomous deterministic
dynamical systems  and  that of random dynamical systems.
Particularly,   the concept of dynamical systems defined  over
two parametric spaces, say $\Omega_1$ and $\Omega_2$,   is needed,
where $\Omega_1$ is a nonempty set
used to deal with the non-autonomous deterministic terms,
and $\Omega_2$ is  a probability space  responsible for the stochastic terms.
The existence and uniqueness of random attractors
for dynamical systems over two parametric spaces have
been recently established in \cite{wan4}.
For the stochastic Navier-Stokes equations
\eqref{intro1}-\eqref{intro2}, we may take
$\Omega_1$   as the set of all    translations of $f$.
We   can also take  $\Omega_1$ as  the collection
of all initial times; i.e., $\Omega_1 =\mathbb{R}$.
In this paper, we will   choose  $\Omega_1 =\mathbb{R}$.
We  first  define  a  continuous
cocycle   for \eqref{intro1}-\eqref{intro2}
over $\Omega_1$   and $\Omega_2$, and then
prove the existence of tempered random absorbing sets.
Since the Sobolev embeddings on unbounded domains are no
longer compact, we have to appeal   to the idea of energy equations
to establish   the pullback asymptotic compactness of solutions.
This  method    was introduced by
Ball in \cite{bal1} for deterministic equations, and used by the authors
in \cite{car3, car4, ros1}    for  the deterministic Navier-Stokes equations
on unbounded domains and in \cite{brz1}   for  the
stochastic equations with time-independent $f$.
We will adapt this approach to the stochastic equations
\eqref{intro1}-\eqref{intro2}   with time-dependent $f$,
and prove the existence of tempered random attractors
 for the equations.
We also consider
the random attractors  in   the case where $f$ is a periodic function in time.
If $f$ is periodic, we will show   that the  tempered random attractors
are also periodic in  some sense.
Following \cite{wan4}, the structures of the
 tempered random attractors  will  be characterized by
  the tempered complete solutions.


 In the next   section,  we will recall some results
 on pullback attractors  for random dynamical
 systems over two parametric spaces.  A continuous
 cocycle for the stochastic Navier-Stokes equations
 \eqref{intro1}-\eqref{intro2} with non-autonomous $f$
 is defined in Section  3.  We   then
  derive uniform estimates
 of the  solutions in Section 4 and prove the existence
 and uniqueness  of pullback
 attractors in Section 5.

In    the  sequel,    we  will use
$\| \cdot \|$ and $(\cdot, \cdot)$ to denote  the norm and the inner product
of $L^2(Q)$,  respectively.   The
norm of a   Banach space $X$  is generally  written as    $\|\cdot\|_X$.
   The letters $c$ and $c_i$ ($i=1, 2, \dots$)
are  used  to denote   positive constants
whose  values are not significant in the context.

\section{Theory of pullback attractors}

In this section,  we recall some   results
on pullback   attractors    for random dynamical
systems  with two parametric spaces   as presented
in \cite{wan4}. This sort of dynamical systems can be generated
by differential  equations  with both deterministic
and stochastic  non-autonomous
external terms.
All results given in this section are not original and they
are presented here   just   for the reader's   convenience.
We also  refer   the reader to
  \cite{bat1, cra1, cra2, fla1, schm1} for the theory of pullback
attractors  for random dynamical  systems with one parametric
space.

Let $\Omega_1$ be a nonempty  set
and $\{\theta_{1,t}\}_{t \in \mathbb{R}}$  be a family of
mappings   from $\Omega_1$ into itself   such that
 $\theta_{1, 0}  $ is the identity on $\Omega_1$
and $\theta_{1,  s+t}  = \theta_{1,t,}  \circ \theta_{1,s}  $ for all
$t, s \in \mathbb{R}$.
Let $(\Omega_2, \mathcal{F}_2, P)$
be a probability space and
 $\theta_2 : \mathbb{R}\times \Omega_2 \to \Omega_2$
 be  a    $(\mathcal{B} (\mathbb{R}) \times \mathcal{F}_2, \mathcal{F}_2)$-measurable mapping
 such that $\theta_2(0,\cdot) $ is the
identity on $\Omega_2$,
$\theta_2 (s+t,\cdot) = \theta_2 (t,\cdot) \circ \theta_2 (s,\cdot)$ for all
$t, s \in \mathbb{R}$ and $P \theta_2 (t,\cdot)  =P$
for all $t \in \mathbb{R}$.
We usually  write $\theta_2 (t, \cdot)$ as $\theta_{2,t}$
and   call  both
$(\Omega_1, \{\theta_{1,t}\}_{t \in \mathbb{R}})$ and
$(\Omega_2, \mathcal{F}_2, P,  \{\theta_{2,t}\}_{t \in \mathbb{R}})$
a parametric   dynamical system.

Let  $(X, d)$   be  a complete
separable  metric space with  Borel $\sigma$-algebra $\mathcal{B} (X)$.
Given $r>0$     and $D \subseteq  X$,
the neighborhood of $D$  with radius   $r$ is written  as
$\mathcal{N}_r(D)$.
 Denote by $2^X$   the collection of all subsets of $X$. A set-valued mapping
$K: \Omega_1 \times \Omega_2 \to  2^X$   is called measurable
with respect to $\mathcal{F}_2$
in $\Omega_2$
if  the value  $K(\omega_1, \omega_2)$
is a closed  nonempty subset  of $X$
for all $\omega_1 \in \Omega_1$ and $\omega_2 \in \Omega_2$,
 and  the mapping
$ \omega_2 \in  \Omega_2
 \to d(x, K(\omega_1, \omega_2) )$
is $(  \mathcal{F}_2, \ \mathcal{B}(\mathbb{R}) )$-measurable
for every  fixed $x \in X$ and $\omega_1 \in \Omega_1$.
If $K$ is  measurable  with respect to $\mathcal{F}_2$
in $\Omega_2$,   then we   say    that      the family
$\{K(\omega_1, \omega_2): \omega_1 \in \Omega_1, \omega_2 \in \Omega_2 \}$
  is measurable
with respect to $\mathcal{F}_2$
 in $\Omega_2$.
 We now   define a cocycle on $X$ over two parametric spaces.

\begin{definition} \label{ds1} \rm
 Let
$(\Omega_1,  \{\theta_{1,t}\}_{t \in \mathbb{R}})$
and
$(\Omega_2, \mathcal{F}_2, P,  \{\theta_{2,t}\}_{t \in \mathbb{R}})$
be parametric  dynamical systems.
A mapping $\Phi$: $ \mathbb{R}^+ \times \Omega_1 \times \Omega_2 \times X
\to X$ is called a continuous  cocycle on $X$
over $(\Omega_1,  \{\theta_{1,t}\}_{t \in \mathbb{R}})$
and
$(\Omega_2, \mathcal{F}_2, P,  \{\theta_{2,t}\}_{t \in \mathbb{R}})$
if   for all
  $\omega_1\in \Omega_1$,
  $\omega_2 \in   \Omega_2 $
  and    $t, \tau \in \mathbb{R}^+$,  the following conditions (i)-(iv)  are satisfied:
\begin{itemize}
\item [(i)]   $\Phi (\cdot, \omega_1, \cdot, \cdot): \mathbb{R}^+ \times \Omega_2 \times X
\to X$ is
 $(\mathcal{B} (\mathbb{R}^+)   \times \mathcal{F}_2 \times \mathcal{B} (X), \
\mathcal{B}(X))$-measurable;

\item[(ii)]    $\Phi(0, \omega_1, \omega_2, \cdot) $ is the identity on $X$;

\item[(iii)]    $\Phi(t+\tau, \omega_1, \omega_2, \cdot) = \Phi(t, \theta_{1,\tau} \omega_1,  \theta_{2,\tau} \omega_2, \cdot) \circ \Phi(\tau, \omega_1, \omega_2, \cdot)$;

\item[(iv)]    $\Phi(t, \omega_1, \omega_2,  \cdot): X \to  X$ is continuous.
    \end{itemize}
If,  in addition,  there exists  a
    positive number   $T $ such that
    for every $t\ge 0$, $\omega_1 \in \Omega_1$  and $\omega_2 \in \Omega_2$,
$$
\Phi(t, \theta_{1, T} \omega_1, \omega_2, \cdot)
= \Phi(t, \omega_1,  \omega_2, \cdot ),
$$
then $\Phi$ is called
a  continuous periodic  cocycle  on $X$ with period $T$.
\end{definition}


In the sequel, we use  $\mathcal{D}$ to denote
 a  collection  of  some families of  nonempty subsets of $X$:
\begin{equation}
\label{defcald}
{\mathcal{D}} = \{ D =\{ D(\omega_1, \omega_2 ) \subseteq X: \
D(\omega_1, \omega_2 ) \neq \emptyset,  \
  \omega_1 \in \Omega_1, \
  \omega_2 \in \Omega_2\} \}.
\end{equation}
Two elements  $D_1$ and $D_2$  of  $\mathcal{D}$
are said  to be equal if
$D_1(\omega_1, \omega_2) =  D_2(\omega_1, \omega_2)$
for any $\omega_1 \in \Omega_1$   and
$\omega_2 \in \Omega_2$.
Sometimes, we require  that $\mathcal{D}$
is  neighborhood closed
which is defined as follows.



\begin{definition} \label{defepsneigh1} \rm
A collection $\mathcal{D}$ of some families
of nonempty subsets of $X$
is said  to be   neighborhood closed if   for each
$D=\{D(\omega_1, \omega_2):
\omega_1 \in \Omega_1, \omega_2 \in \Omega_2 \}
\in \mathcal{D}$,   there exists a positive number
$\varepsilon$ depending on $D$ such that  the family
\begin{equation}\label{defepsneigh2}
\begin{split}
 &\big\{ {B}(\omega_1, \omega_2) :
 {B}(\omega_1, \omega_2) \text{ is a  nonempty subset of }
 \mathcal{N}_\varepsilon ( D (\omega_1, \omega_2) ), \\
&\quad \forall \
 \omega_1 \in \Omega_1,  \forall\  \omega_2
\in  \Omega_2\big\}
\end{split}
\end{equation}
also belongs to $\mathcal{D}$.
\end{definition}


\begin{definition} \label{temset} \rm
Let
$D=\{D(\omega_1, \omega_2): \omega_1 \in \Omega_1, \omega_2 \in \Omega_2 \}$
be a family of    nonempty subsets of $X$.
We say $D$ is tempered in $X$
with respect to $(\Omega_1,  \{\theta_{1,t}\}_{t \in \mathbb{R}})$
and
$(\Omega_2, \mathcal{F}_2, P,  \{\theta_{2,t}\}_{t \in \mathbb{R}})$
if  there exists $x_0 \in X$ such that for every  $c>0$,
$\omega_1 \in \Omega_1$ and $\omega_2 \in \Omega_2$,
$$
\lim_{t \to -\infty}
e^{c t} d (x_0, D(\theta_{1,t} \omega_1, \theta_{2,t} \omega_2))
=0.
$$
\end{definition}


\begin{definition} \label{defTlation}
Suppose   $T  \in \mathbb{R}$     and
  $\mathcal{D}$  is  a collection  of some families of nonempty subsets of $X$
  as given by \eqref{defcald}.
  For  every  $D=\{D(\omega_1, \omega_2):
\omega_1 \in \Omega_1, \omega_2 \in \Omega_2 \} \in \mathcal{D}$,  we write
$$
D_T = \{ D_T(\omega_1, \omega_2): \  \
 D_T(\omega_1, \omega_2) = D(\theta_{1, T} \omega_1,  \omega_2), \
 \omega_1 \in \Omega_1, \
\omega_2 \in \Omega_2
\}.
$$
The family  $D_T$  is called    the  $T$-translation of
$D$.
  Let $\mathcal{D}_T$ be the collection of
    $T$-translations of  all elements of $\mathcal{D}$, that is,
  $$
  \mathcal{D}_T = \{ D_T: D_T \text{ is the $T$-translation of } D,
  \;  D \in \mathcal{D} \}.
  $$
  Then $\mathcal{D}_T$ is called the $T$-translation of
  the collection $\mathcal{D}$.
  If $\mathcal{D}_T \subseteq \mathcal{D}$,   we say $\mathcal{D}$ is
      $T$-translation   closed.
      If  $\mathcal{D}_T =  \mathcal{D}$,   we say $\mathcal{D}$ is
      $T$-translation  invariant.
\end{definition}

One can check    that
  $\mathcal{D}$ is $T$-translation  invariant
  if    and only if $\mathcal{D}$ is both $-T$-translation closed   and $T$-translation
  closed.
 For later purpose, we need the concept of a complete orbit
 of $\Phi$ which is given below.


\begin{definition}\label{comporbit} \rm
 Let $\mathcal{D}$ be a collection of some families of
 nonempty  subsets of $X$. A mapping 
 $\psi: \mathbb{R}\times \Omega_1 \times \Omega_2$
 $\to X$ is called a complete orbit of $\Phi$ if for every $\tau \in \mathbb{R}$, $t \ge 0$,
 $\omega_1 \in \Omega_1$ and $\omega_2 \in \Omega_2$,  the following holds:
\begin{equation}
\label{comporbit1}
 \Phi (t, \theta_{1, \tau} \omega_1, \theta_{2, \tau} \omega_2,
  \psi (\tau, \omega_1, \omega_2) )
  = \psi (t + \tau, \omega_1, \omega_2 ).
\end{equation}
 If, in  addition,    there exists 
 $D=\{D(\omega_1, \omega_2): \omega_1 \in \Omega,
 \omega_2 \in \Omega_2 \}\in \mathcal{D}$ such that
 $\psi(t, \omega_1, \omega_2)$ belongs to
 $D(\theta_{1,t} \omega_1, \theta_{2, t} \omega_2 )$
 for every  $t \in \mathbb{R}$, $\omega_1 \in \Omega_1$
 and $\omega_2 \in \Omega_2$, then $\psi$ is called a
 $\mathcal{D}$-complete orbit of $\Phi$.
 \end{definition}


\begin{definition} \label{defomlit} \rm
Let $B=\{B(\omega_1, \omega_2): \omega_1 \in \Omega_1, \ \omega_2  \in \Omega_2\}$
be a family of nonempty subsets of $X$.
For every $\omega_1 \in \Omega_1$ and
$\omega_2 \in \Omega_2$,  let
\begin{equation}\label{omegalimit}
\Omega (B, \omega_1, \omega_2)
= \cap_{\tau \ge 0}
 \overline{ \cup_{t\ge \tau} \Phi(t, \theta_{1,-t} \omega_1, \theta_{2, -t} 
 \omega_2, B(\theta_{1,-t} \omega_1, \theta_{2,-t}\omega_2  ))}.
\end{equation}
Then
the  family
 $\{\Omega (B, \omega_1, \omega_2): \omega_1 \in \Omega_1, \omega_2 \in \Omega_2 \}$
 is called the $\Omega$-limit set of $B$
 and is denoted by $\Omega(B)$.
 \end{definition}


\begin{definition} \rm
Let $\mathcal{D}$ be a collection of some families of nonempty subsets of 
$X$ and
$K=\{K(\omega_1, \omega_2): \omega_1 \in \Omega_1, \ \omega_2  
\in \Omega_2\} \in \mathcal{D}$. Then
$K$  is called a  $\mathcal{D}$-pullback
 absorbing
set for   $\Phi$   if
for all $\omega_1 \in \Omega_1$,
$\omega_2 \in \Omega_2 $
and  for every $B \in \mathcal{D}$,
 there exists $T= T(B, \omega_1, \omega_2)>0$ such
that
\begin{equation}
\label{abs1}
\Phi(t, \theta_{1,-t} \omega_1, \theta_{2, -t} \omega_2, B(\theta_{1,-t}
 \omega_1, \theta_{2,-t} \omega_2  ))  \subseteq  K(\omega_1, \omega_2)
\quad \text{for all} \ t \ge T.
\end{equation}
If, in addition, for all $\omega_1 \in \Omega_1$ and $\omega_2 \in \Omega_2$,
   $K(\omega_1, \omega_2)$ is a closed nonempty subset of $X$
   and $K$ is measurable with respect to the $P$-completion of $\mathcal{F}_2$
   in $\Omega_2$,
 then we say $K$ is a  closed measurable
  $\mathcal{D}$-pullback absorbing  set for $\Phi$.
\end{definition}


\begin{definition} \label{asycomp} \rm
 Let $\mathcal{D}$ be a collection of  some families of  nonempty
 subsets of $X$.
 Then
$\Phi$ is said to be  $\mathcal{D}$-pullback asymptotically
compact in $X$ if
for all $\omega_1 \in \Omega_1$ and
$\omega_2 \in \Omega_2$,    the sequence
\begin{equation}
\label{asycomp1}
\{\Phi(t_n, \theta_{1, -t_n} \omega_1, \theta_{2, -t_n} \omega_2,
x_n)\}_{n=1}^\infty \text{  has a convergent  subsequence  in }   X
\end{equation}
 whenever
  $t_n \to \infty$, and $ x_n\in   B(\theta_{1, -t_n}\omega_1,
  \theta_{2, -t_n} \omega_2 )$   with
$\{B(\omega_1, \omega_2): \omega_1 \in \Omega_1, \ \omega_2 \in \Omega_2
\}   \in \mathcal{D}$.
\end{definition}


\begin{definition} \label{defatt} \rm
 Let $\mathcal{D}$ be a collection of some families of
 nonempty  subsets of $X$
 and
 $\mathcal{A} = \{\mathcal{A} (\omega_1, \omega_2): \omega_1 \in \Omega_1,
  \omega_2 \in \Omega_2 \} \in \mathcal{D} $.
Then     $\mathcal{A}$
is called a    $\mathcal{D}$-pullback    attractor  for
  $\Phi$
if the following  conditions (i)-(iii) are  fulfilled:
\begin{itemize}
\item [(i)]   $\mathcal{A}$ is measurable
with respect to the $P$-completion of $\mathcal{F}_2$ in $\Omega_2$ and
 $\mathcal{A}(\omega_1, \omega_2)$ is compact for all $\omega_1 \in \Omega_1$
and    $\omega_2 \in \Omega_2$.

\item[(ii)]   $\mathcal{A}$  is invariant, that is,
for every $\omega_1 \in \Omega_1$ and
 $\omega_2 \in \Omega_2$,
$$ \Phi(t, \omega_1, \omega_2, \mathcal{A}(\omega_1, \omega_2)   )
= \mathcal{A} (\theta_{1,t} \omega_1, \theta_{2,t} \omega_2
), \ \  \forall \   t \ge 0.
$$

\item[(iii)]   $\mathcal{A}  $
attracts  every  member   of   $\mathcal{D}$,  that is, for every
 $B = \{B(\omega_1, \omega_2): \omega_1 \in \Omega_1, \omega_2 \in \Omega_2\}
 \in \mathcal{D}$ and for every $\omega_1 \in \Omega_1$ and
 $\omega_2 \in \Omega_2$,
$$ \lim_{t \to  \infty} d (\Phi(t, \theta_{1,-t}\omega_1, 
\theta_{2,-t}\omega_2, B(\theta_{1,-t}\omega_1, \theta_{2,-t}\omega_2) ) , 
\mathcal{A} (\omega_1, \omega_2 ))=0.
$$
 \end{itemize}
 If, in addition, there exists $T>0$ such that
 $$
 \mathcal{A}(\theta_{1, T} \omega_1, \omega_2) = \mathcal{A}(\omega_1, \omega_2 ),
 \quad \forall   \omega_1 \in \Omega_1, \forall \
  \omega_2 \in \Omega_2,
 $$
 then we say $\mathcal{A}$ is periodic with period $T$.
\end{definition}


The following result on the  existence and uniqueness of
 $\mathcal{D}$-pullback attractors
for $\Phi$ can be found in \cite{wan4}.
The reader is  referred to
\cite{bat1,  cra2, fla1, schm1} for similar results
for random dynamical   systems.


 \begin{proposition} \label{att}
 Let $\mathcal{D}$ be a   neighborhood closed
 collection of some  families of   nonempty subsets of
$X$,  and $\Phi$  be a continuous   cocycle on $X$
over $(\Omega_1,  \{\theta_{1,t}\}_{t \in \mathbb{R}})$
and
$(\Omega_2, \mathcal{F}_2, P,  \{\theta_{2,t}\}_{t \in \mathbb{R}})$.
Then
$\Phi$ has a  $\mathcal{D}$-pullback
attractor $\mathcal{A}$  in $\mathcal{D}$
if and only if
$\Phi$ is $\mathcal{D}$-pullback asymptotically
compact in $X$ and $\Phi$ has a  closed
   measurable (w.r.t. the $P$-completion of $\mathcal{F}_2$)
     $\mathcal{D}$-pullback absorbing set
  $K$ in $\mathcal{D}$.
  The $\mathcal{D}$-pullback
attractor $\mathcal{A}$   is unique   and is given  by,
for each $\omega_1  \in \Omega_1$   and
$\omega_2 \in \Omega_2$,
\begin{align}\label{attform1}
\mathcal{A} (\omega_1, \omega_2)
&=\Omega(K, \omega_1, \omega_2)
=\cup_{B \in \mathcal{D}} \Omega(B, \omega_1, \omega_2) \\
\label{attform2}
& =\{\psi(0, \omega_1, \omega_2): \psi \text{ is a  $\mathcal{D}$-complete orbit of } \Phi\} .
 \end{align}
  \end{proposition}


The periodicity of   $\mathcal{D}$-pullback attractors  is
proved in \cite{wan4} as given  below.



\begin{proposition} \label{periodatt}
Let   $T$   be   a positive number.
Suppose  $\Phi$   is  a continuous  periodic   cocycle
with period $T$  on $X$
over $(\Omega_1,  \{\theta_{1,t}\}_{t \in \mathbb{R}})$
and
$(\Omega_2, \mathcal{F}_2, P,  \{\theta_{2,t}\}_{t \in \mathbb{R}})$.
 Let   $\mathcal{D}$   be  a  neighborhood closed
   and $T$-translation invariant collection of
    some  families of   nonempty subsets of
$X$.
 If
$\Phi$ is $\mathcal{D}$-pullback asymptotically
compact in $X$ and $\Phi$ has a  closed
   measurable (w.r.t. the $P$-completion of $\mathcal{F}_2$)
     $\mathcal{D}$-pullback absorbing set
  $K$ in $\mathcal{D}$, then $\Phi$
  has a unique periodic
   $\mathcal{D}$-pullback
attractor $\mathcal{A} \in \mathcal{D}$    with period $T$; i.e.,
$\mathcal{A} (\theta_{1, T} \omega_1,  \omega_2)
=\mathcal{A}(\omega_1, \omega_2)$.
\end{proposition}

\section{Cocycles  for Navier-Stokes equations on unbounded domains}

This section is devoted  to the existence of
a continuous cocycle for the stochastic Navier-Stokes
equations with  non-autonomous deterministic terms.
Suppose   $Q$  is  an unbounded open set in $\mathbb{R}^2$ with boundary
$\partial Q$.
 Then consider   the following   stochastic
   equations with multiplicative noise
    defined on $Q \times (\tau, \infty)$
   with  $\tau \in \mathbb{R}$:
 \begin{gather} \label{nse1}
 {\frac{\partial u}{\partial t}}
 -\nu \Delta u  + (u \cdot \nabla) u
 = f(x,t) -\nabla p
 +  \alpha u \circ {\frac{d w}{dt}}, \quad x \in Q  \text{ and }  t >\tau, \\
\label{nse2}
 \operatorname{div}u =0,
  \quad x \in Q \text{ and }t >\tau,
\end{gather}
 with boundary condition
 \begin{equation}\label{nse3}
 u = 0,   \quad x \in  \partial Q \text{ and }t >\tau,
 \end{equation}
 and initial condition
 \begin{equation}  \label{nse4}
 u(x, \tau) =u_\tau (x),  \quad x \in Q ,
 \end{equation}
 where $\nu$ and $\alpha$
 are constants, $\nu>0$,  $f$ is a
 given function defined on $Q \times \mathbb{R}$,
 and $w$  is  a  two-sided real valued Wiener process
 defined in a  probability space.
 Note that equation \eqref{nse1} must be
 understood in the sense of Stratonovich  integration.

 To reformulate problem \eqref{nse1}-\eqref{nse4},  we
 recall  the  standard function space:
   $$
   {\mathcal{V}} = \{ u \in C_0^\infty (Q) \times C_0^\infty (Q):
   \operatorname{div}u =0 \}.
   $$
   Let
    $H $
    and  $V$   be  the closures of ${\mathcal{V}}$ in
    $L^2(Q) \times L^2(Q)$ and $H^1_0(Q) \times H^1_0(Q)$,
    respectively.
    The dual space  of $V$ is denoted by $V^*$ with
    norm $\| \cdot \|_{V^*}$.
    The duality pair between $V$   and
    $V^*$ is denoted by
    $\langle\cdot, \cdot \rangle$.
      Given $u, v \in V$, we    set
    $$
    (Du, Dv) = \sum_{i, j=1}^2 \int_Q
    {\frac{\partial u_i}{\partial x_j}}
 {\frac{\partial v_i}{\partial x_j}}  dx
 \quad {\rm and } \quad
 \| Du \| = (Du, Du)^{1/2}.
 $$
 For convenience, we write, for each
 $u, v, w \in V$,
 $$
 b(u,v,w) =  \sum_{i, j=1}^2 \int_Q
 u_i {\frac{\partial u_j}{\partial x_i}} w_j dx .
 $$


Let $\{\theta_{1,t}\}_{t \in \mathbb{R}}$  be a family
of shift operators on $\mathbb{R}$   which is given by,
for each $t \in \mathbb{R}$,
 \begin{equation}
 \label{shiftr}
 \theta_{1,t}  (\tau) =   \tau + t, \quad \text{for all }  \tau \in  \mathbb{R}.
 \end{equation}
 For  the  probability space we will use later, we write
 $$
\Omega = \{ \omega   \in C(\mathbb{R}, \mathbb{R}):  \omega(0) =  0 \}.
$$
Let ${\mathcal{F}}_1$  be
 the Borel $\sigma$-algebra induced by the
compact-open topology of $\Omega$, and $P$
be  the corresponding Wiener
measure on $(\Omega, {\mathcal{F}}_1)$.
As usual, for each $t \in \mathbb{R}$  and
$\omega \in \Omega$, we may write
$w_t (\omega) = \omega (t)$.
Denote by    $\{\theta_{2,t} \}_{t \in \mathbb{R}}$
the standard group  on
 $(\Omega, {\mathcal{F}}_1, P)$:
\begin{equation}\label{shiftome}
 \theta_{2,t} \omega (\cdot) = \omega (\cdot +t) - \omega (t), \quad  \omega \in \Omega,
 \; t \in \mathbb{R}.
\end{equation}
Then $(\Omega, \mathcal{F}, P, \{\theta_{2,t}\}_{t\in \mathbb{R}})$ is a  parametric
dynamical  system. In addition,
  there exists a $\theta_{2,t}$-invariant set
  $\tilde{\Omega}\subseteq \Omega$
of full $P$ measure  such that
for each $\omega \in \tilde{\Omega}$,
\begin{equation}\label{aspomega}
{\frac{\omega (t)}{t}} \to 0 \quad \text{as } \ t \to \pm \infty.
\end{equation}
From now on, we only consider the space
$\tilde{\Omega}$ instead of $\Omega$, and hence
we  will write  $\tilde{\Omega}$ as
$\Omega$  for   convenience.

Next, we  define a continuous   cocycle   for \eqref{nse1}-\eqref{nse4}
     in $H$  over
  $(\mathbb{R}, \{\theta_{1,t}\}_{t \in \mathbb{R}})$
  and $(\Omega, {\mathcal{F}}_1, P, \{\theta_{2,t}\}_{t \in \mathbb{R}})$.
  To  this  end,   we need to
  transfer the stochastic equation into a deterministic one with
  random parameters.
  Given $t \in \mathbb{R}$  and $\omega \in \Omega$,   let
  $ z(t, \omega) = e^{-\alpha \omega  (t)}$.
  Then   we find that  $z$ is a solution of  the equation
  \begin{equation}\label{zequ1}
  d z = - \alpha z \circ d w.
 \end{equation}
 Let $v$ be a new variable given by
 \begin{equation}\label{vu}
 v(t, \tau, \omega, v_\tau) = z(t, \omega)
 u(t, \tau, \omega, u_\tau)
 \quad \text{with } \ v_\tau = z(\tau, \omega) u_\tau.
\end{equation}
 Formally,  from \eqref{nse1}-\eqref{nse4} and \eqref{zequ1}
 we get that
  \begin{gather} \label{v1}
 {\frac{\partial v}{\partial t}}
 -\nu \Delta v  + {\frac 1{z(t,\omega)}}  (v \cdot \nabla) v
 = z(t, \omega) \left (f(x,t) -\nabla p \right ),
 \quad x \in Q \text{ and }t >\tau, \\
 \label{v2}
 \operatorname{div}v =0,
  \quad x \in Q \text{ and }t >\tau,
 \end{gather}
 with boundary condition
 \begin{equation}\label{v3}
 v = 0,   \quad x \in  \partial Q \text{ and }t >\tau,
 \end{equation}
 and initial condition
 \begin{equation}
 \label{v4}
 v(x, \tau) =v_\tau (x),  \quad x \in Q .
 \end{equation}


 Let  $\tau \in \mathbb{R}$, $\omega \in \Omega$,
  and
  $v_\tau \in H$.
   A mapping $v (\cdot, \tau, \omega, v_\tau)$: $[\tau, \infty)
     \to H$  is called   a solution of problem
     \eqref{v1}-\eqref{v4} if  for every
   $T>0$,
   $$
v(\cdot, \tau, \omega, u_\tau)
     \in C([\tau, \infty), H)  \cap    L^2 ((0,T), V)
 $$
and $v$  satisfies
\begin{equation}\label{solv1}
\begin{split}
&(v(t), \zeta)
+ \nu \int_\tau^t (D v,  D \zeta) ds
+ \int_\tau^t {\frac 1{z(s, \omega)}}
b(v , v, \zeta) ds\\
&=  (v_\tau, \zeta) + \int_\tau^t
   z(s, \omega)  \langle    f(\cdot, s),
    \zeta \rangle  ds,
\end{split}
\end{equation}
for every $t \ge \tau$   and
 $\zeta \in V$.
 If,   in addition,
 $v$ is
 $({\mathcal{F}}_1, \mathcal{B}(H))$-measurable
 with respect to $\omega \in \Omega$,
  we say   $v$ is a measurable solution of
  problem
     \eqref{v1}-\eqref{v4}.
     Since    \eqref{v1} is  a deterministic equation, it follows from
     \cite{tem1}  that  for every $\tau \in \mathbb{R}$,
      $v_\tau \in H$   and $\omega \in \Omega$, problem
      \eqref{v1}-\eqref{v4} has a unique solution
      $v$  in the sense
      of \eqref{solv1}
      which  continuously  depends  on $v_\tau$
      with the respect to the norm of $H$. Moreover,
       the solution  $v$  is $({\mathcal{F}}_1, \mathcal{B} (H))$-measurable
      in $\omega \in \Omega$.
      This enables us to     define a cocycle
       $\Phi: \mathbb{R}^+ \times \mathbb{R}\times \Omega \times H$
$\to H$ for      problem
\eqref{nse1}-\eqref{nse4} by using \eqref{vu}.
Given $t \in \mathbb{R}^+$,  $\tau \in \mathbb{R}$, $\omega \in \Omega$
 and $u_\tau \in H$,
let
 \begin{equation} \label{nsephi}
 \Phi (t, \tau,  \omega, u_\tau) =
  u (t+\tau,  \tau, \theta_{2, -\tau} \omega, u_\tau)
  = {\frac 1{z(t+\tau, \theta_{2, -\tau} \omega)}}
v(t+\tau, \tau,  \theta_{2, -\tau} \omega,  v_\tau),
\end{equation}
where $v_\tau = z(\tau, \theta_{2, -\tau} \omega) u_\tau $.
By \eqref{nsephi} we have,   for every
$t \ge 0$, $\tau \ge 0$, $r \in \mathbb{R}$,
$\omega \in \Omega$ and $u_0 \in H$,
\begin{equation}
\label{pcon1}
\Phi (t + \tau, r, \omega,  u_0)
=  {\frac 1{z(t+\tau +r, \theta_{2, -r} \omega)}}
v(t+\tau +r,  r,   \theta_{2, -r} \omega,  v_0),
\end{equation}
where $v_0 = z(r, \theta_{2,-r} \omega ) u_0$.
Similarly,  we have
\begin{equation}  \label{pcon2}
\begin{split}
&\Phi \left (t , \tau + r,  \theta_{2,\tau}\omega,
 \Phi (\tau, r, \omega, u_0)
\right ) \\
&=  {\frac 1{z(t+\tau +r, \theta_{2, -r} \omega)}}
v(t+\tau +r,  \tau +r,   \theta_{2, -r} \omega,
 z(\tau +r, \theta_{2, -r} \omega)  \Phi (\tau, r, \omega, u_0)  ) \\
&=  {\frac 1{z(t+\tau +r, \theta_{2, -r} \omega)}}
v(t+\tau +r,  \tau +r,   \theta_{2, -r} \omega,
  v(\tau +r, r,  \theta_{2, -r} \omega,  v_0)\\
&=  {\frac 1{z(t+\tau +r, \theta_{2, -r} \omega)}}
v(t+\tau +r,  r,   \theta_{2, -r} \omega,  v_0).
\end{split}
\end{equation}
It follows from \eqref{pcon1}-\eqref{pcon2} that
\begin{equation}
\label{pcon3}
\Phi (t + \tau, r, \omega,  u_0)
=\Phi \left (t , \tau + r,  \theta_{2,\tau}\omega,
 \Phi (\tau, r, \omega, u_0)
\right ).
\end{equation}
Since $v$  is  the   measurable solution of problem
\eqref{v1}-\eqref{v4} which is continuous in initial data in $H$,
we find   from \eqref{pcon3}  that  $\Phi$
is a continuous   cocycle on $H$ over $(\mathbb{R},  \{\theta_{1,t}\}_{t \in \mathbb{R}})$
and
$(\Omega, {\mathcal{F}}_1,
P, \{\theta_{2,t} \}_{t\in \mathbb{R}})$.
The rest of this paper is devoted to the
existence of pullback attractors
for $\Phi$   in $H$.  To  this end, we assume that   the open set
$Q$ is a Poincare domain in the sense    that   there
exists a positive number $\lambda$ such that
\begin{equation} \label{poincare}
\int_Q  |\nabla \phi (x) |^2  dx
\ge \lambda \int_Q |\phi  (x) |^2 dx,
\quad\text{for   all } \ \phi \in H^1_0 (Q).
\end{equation}


 Given a  bounded nonempty  subset
 $B$  of $H$,  we write
   $  \| B\| = \sup\limits_{\phi \in B}    \| \phi\|_{H }$.
Suppose
   $D =\{ D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}$
    is   a  tempered family of
  bounded nonempty   subsets of $H $,  that is,
  for every  $c>0$, $\tau \in \mathbb{R}$   and $\omega \in \Omega$,
 \begin{equation}
 \label{attdom1}
 \lim_{r \to   \infty} e^{ -  c  r}
 \| D( \tau  -r, \theta_{2, -r} \omega ) \|  =0.
 \end{equation}
Let $\mathcal{D}$  be      the  collection of all  tempered families of
bounded nonempty  subsets of $H$; i.e.,
 \begin{equation}  \label{dnse}
\mathcal{D} = \{
   D =\{ D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}:
 D   \text{ satisfies }  \eqref{attdom1} \} .
\end{equation}
We see  that $\mathcal{D}$ is  neighborhood closed.
For later purpose,   we assume that   the external term
$f$ satisfies    the  following condition:   there exists
a number $\delta \in [0, \nu\lambda)$
such that
 \begin{equation}
 \label{fcond1}
 \int_{-\infty}^\tau e^{\delta  r } \| f(\cdot, r)\|^2_ {V^*}d r
<  \infty, \quad \forall \ \tau \in \mathbb{R}.
 \end{equation}
 When proving the existence of tempered pullback absorbing
 sets for the Navier-Stokes equations, we also assume  that
 there   exists
  $\delta \in [0, \nu\lambda)$
such that  for every positive number $c$,
 \begin{equation}
 \label{fcond2}
 \lim_{r \to -\infty} e^{c r}
 \int_{-\infty}^0   e^{\delta  s}
  \|f(\cdot,  s+r) \|^2_{V^*}    d s  =0.
  \end{equation}
  Note that \eqref{fcond2} implies \eqref{fcond1} if
  $f \in L^2_{loc} (\mathbb{R}, V^*)$.
  It is worth pointing out that
 both  conditions  \eqref{fcond1}
 and \eqref{fcond2}
do   not require that    $f$
is  bounded in $V^*$
at $ \pm \infty$.  For instance,  for any $\beta \ge 0$  and
$f_1 \in V^*$,  the function $ f(\cdot, t) = t^\beta f_1  $
satisfies both     \eqref{fcond1}
 and \eqref{fcond2}.

\section{Uniform estimates of solutions}

In this section,  we  derive uniform estimates  on the
  solutions  of  problem \eqref{v1}-\eqref{v4}
  and then prove the $\mathcal{D}$-pullback asymptotic compactness
  of the solutions    by the    idea  of energy equations
as introduced by Ball in \cite{bal1} for deterministic systems.

\begin{lemma} \label{lem1}
 Suppose  \eqref{poincare} and \eqref{fcond1} hold.
Then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$   and $D=\{D(\tau, \omega)
: \tau \in \mathbb{R},  \omega \in \Omega\}  \in \mathcal{D}$,
 there exists  $T=T(\tau, \omega,  D)>0$ such that for all $t \ge T$ and
 $s  \ge  \tau -t $, the solution
 $v$ of  problem  \eqref{v1}-\eqref{v4}  with $\omega$ replaced by
 $\theta_{2, -\tau} \omega$  satisfies
$$
\| v(s , \tau -t,  \theta_{2, -\tau} \omega, v_{\tau -t}  ) \|^2
 \le
 e^{\nu \lambda (\tau -s)}
 + {\frac 2\nu} e^{-\nu \lambda s}
 \int_{-\infty}^s
 e^{\nu\lambda r}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr,
$$
and
\begin{align*}
&\int_{\tau -t}^s e^{ \nu \lambda r}
  \| Dv(r, \tau -t,\theta_{2, -\tau} \omega, v_{\tau -t} ) \|^2 dr\\
&\le {\frac 2\nu}
e^{\nu \lambda \tau  }
 +  {\frac 4{\nu^2}}
 \int_{-\infty}^s
 e^{\nu\lambda r}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr,
\end{align*}
 where $v_{\tau -t}\in D(\tau -t, \theta_{2, -t} \omega)$.
\end{lemma}

\begin{proof}
 Formally, it follows   from \eqref{v1}-\eqref{v3}  that for
each $\tau \in \mathbb{R}$, $t \ge 0$  and $\omega \in \Omega$,
\begin{equation}\label{plem1_1}
{\frac 12} {\frac d{dt}} \| v\|^2  + \nu \| Dv \|^2
= z(t, \omega) \langle f(\cdot, t), v \rangle .
\end{equation}
The right-hand side of \eqref{plem1_1} is bounded by
$$|z(t, \omega) \langle f(\cdot, t), v \rangle |
\le {\frac 14}\nu \| D v\|^2
+ {\frac 1\nu} z^2(t,  \omega ) \| f(\cdot, t)\|^2_{V^*}.
$$
Therefore,  from \eqref{plem1_1}   we get
\begin{equation}\label{plem1_2}
 {\frac d{dt}} \| v\|^2  + {\frac 32} \nu \| Dv \|^2
\le
{\frac 2\nu} z^2(t,  \omega ) \| f(\cdot, t)\|^2_{V^*}.
\end{equation}
By \eqref{poincare} and \eqref{plem1_2}   we have
\begin{equation}\label{plem1_3}
 {\frac d{dt}} \| v\|^2
 +\nu\lambda \| v \|^2 +
{\frac 12} \nu \| Dv \|^2
\le
{\frac 2\nu} z^2(t,  \omega ) \| f(\cdot, t)\|^2_{V^*}.
\end{equation}
Multiplying \eqref{plem1_3}   by $e^{\nu\lambda t}$ and then
integrating the inequality on $[\tau -t, s]$,
we obtain
\begin{align*}
&\| v(s, \tau -t, \omega, v_{\tau -t} )\|^2
+{\frac 12} \nu
\int_{\tau -t}^s
e^{\nu \lambda (r-s)}
\|D v(r, \tau -t, \omega, v_{\tau-t}) \|^2 dr \\
&\le
e^{\nu\lambda (\tau -s)} e^{-\nu \lambda t} \| v_{\tau -t} \|^2
+{\frac 2\nu} \int_{\tau -t} ^s
e^{\nu\lambda (r-s)} z^2(r, \omega) \| f(\cdot, r)\|^2_{V^*} dr.
\end{align*}
Replacing $\omega$   by
$\theta_{2, -\tau} \omega$ in the above,   we get that
\begin{equation}\label{plem1_5}
\begin{split}
&\| v(s, \tau -t, \theta_{2, -\tau} \omega, v_{\tau -t} )\|^2
+{\frac 12} \nu
\int_{\tau -t}^s
e^{\nu \lambda (r-s)}
\|D v(r, \tau -t,  \theta_{2, -\tau} \omega, v_{\tau-t}) \|^2 dr\\
&\le
e^{\nu\lambda (\tau -s)} e^{-\nu \lambda t} \| v_{\tau -t} \|^2
+{\frac 2\nu} e^{- \nu\lambda  s }
\int_{\tau -t} ^s
e^{\nu\lambda r} z^2(r,  \theta_{2, -\tau} \omega) \| f(\cdot, r)\|^2_{V^*} dr.
\end{split}
\end{equation}
We now  estimate   the last   term
on the right-hand side  of \eqref{plem1_5}.
Let $\tilde{\omega} = \theta_{2, -\tau}  \omega$. Then
by \eqref{aspomega} we find that there  exists $R<0$ such that
for all $r \le R$,
$$
-2\alpha \tilde{\omega} (r)
\le -(\nu \lambda -\delta) r,
$$
where $\delta$ is the positive constant
in \eqref{fcond1}. Therefore, for   all $r \le R$,
\begin{equation}
\label{plem1_6}
z^2(r, \tilde{\omega})
=e^{-2\alpha \tilde{\omega} (r)}
 \le e^{-(\nu\lambda -\delta)r}.
\end{equation}
By \eqref{plem1_6}   we have for all $r \le R$,
$$
e^{\nu\lambda r} z^2(r,  \theta_{2, -\tau} \omega) \| f(\cdot, r)\|^2_{V^*}
=
e^{(\nu\lambda-\delta) r}
z^2(r,  \tilde{\omega})  e^{\delta r} \| f(\cdot, r)\|^2_{V^*}
\le e^{\delta r} \| f(\cdot, r)\|^2_{V^*} ,
$$
which along with \eqref{fcond1}  shows    that
for every $s \in \mathbb{R}$, $\tau \in \mathbb{R}$    and
$\omega \in \Omega$,
\begin{equation}\label{plem1_7}
\int_{-\infty}^s
e^{\nu\lambda r} z^2(r,  \theta_{2, -\tau} \omega) \| f(\cdot, r)\|^2_{V^*}  dr
< \infty.
\end{equation}
On the other hand,
since $v_{\tau -t} \in D(\tau -t, \theta_{2, -t} \omega)$,
  for the first term on the
right-hand side of \eqref{plem1_5}, we have
$$
e^{-\nu \lambda t} \| v_{\tau -t} \|^2
\le
e^{-\nu \lambda t} \| D(\tau -t, \theta_{2, -t} \omega ) \|^2
\to 0,\quad \text{as  }    t \to  \infty.$$
This shows  that there exists
$T= T(\tau, \omega, D)>0$ such that
$e^{-\nu \lambda t} \| v_{\tau -t} \|^2 \le 1$
for   all $t \ge T$.
Thus, the   first term on the
right-hand side of \eqref{plem1_5} satisfies
\begin{equation}\label{plem1_8}
e^{\nu\lambda (\tau -s)} e^{-\nu \lambda t} \| v_{\tau -t} \|^2
\le
e^{\nu\lambda (\tau -s)},
\quad\text{for all }  t\ge T.
\end{equation}
From  \eqref{plem1_5}, \eqref{plem1_7} and \eqref{plem1_8},
the lemma    follows.
 \end{proof}


As an immediate consequence of Lemma \ref{lem1}, we have the following estimates
on the solutions of problem \eqref{v1}-\eqref{v4}.

\begin{lemma} \label{lem2}
 Suppose  \eqref{poincare} and \eqref{fcond1} hold.
Then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$   and $D=\{D(\tau, \omega)
: \tau \in \mathbb{R},  \omega \in \Omega\}  \in \mathcal{D}$,
 there exists  $T=T(\tau, \omega,  D)>0$ such that for
 every $k \ge 0$   and    for all $t \ge T +k$,
 the solution
 $v$ of  problem  \eqref{v1}-\eqref{v4}  with $\omega$ replaced by
 $\theta_{2, -\tau} \omega$  satisfies
\begin{align*}
&\| v(\tau -k , \tau -t,  \theta_{2, -\tau} \omega, v_{\tau -t}  ) \|^2\\
& \le
 e^{\nu \lambda k}
 + {\frac 2\nu} e^{\nu \lambda (k -\tau)}
 \int_{-\infty}^{\tau -k}
 e^{\nu\lambda r}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr,
 \end{align*}
 where $v_{\tau -t}\in D(\tau -t, \theta_{2, -t} \omega)$.
\end{lemma}

\begin{proof}
Given $\tau \in \mathbb{R}$    and $k \ge 0$, let
$s= \tau -k$.   Let $T= T(\tau, \omega, D)$ be   the positive constant
claimed in Lemma \ref{lem1}.  If
  $t \ge T + k$,  then   we have
  $ t\ge T$   and $ s \ge \tau -t$.
Thus,
the desired result follows   from Lemma \ref{lem1}.
\end{proof}

 Next,  we prove  the $\mathcal{D}$-pullback asymptotic compactness of the
 solutions  of problem \eqref{v1}-\eqref{v4}.  For this purpose, we need
 the following weak continuity of solutions in initial data, which can
 be established
 by  the standard  methods  as in \cite{ros1}.

 \begin{lemma}  \label{lem3}
 Suppose  \eqref{poincare} holds
 and $f \in L^2_{loc} (\mathbb{R}, V^*)$. Let
 $\tau \in \mathbb{R}$, $\omega \in \Omega$,
 $v_\tau $, $v_{\tau, n} \in  H$ for
 all $n \in \mathbb{N}$.
 If $v_{\tau, n}   \rightharpoonup v_\tau$ in $H$,
  then  the solution $v$   of problem \eqref{v1}-\eqref{v4}
  has  the  properties:
  $$
  v(r, \tau, \omega, v_{\tau, n}) \rightharpoonup
  v(r, \tau, \omega, v_\tau) \quad\text{in } \  H
  \ \text{ for   all } \  r \ge \tau,
  $$
  and
  $$
   v(\cdot,  \tau, \omega, v_{\tau, n}) \rightharpoonup
  v(\cdot, \tau, \omega, v_\tau) \quad\text{in } \  L^2 ((\tau, \tau +T), V)
  \ \text{ for   every } \   T>0.
  $$
 \end{lemma}

 The next lemma   is concerned with the
  pullback asymptotic compactness of
 problem \eqref{v1}-\eqref{v4}.

 \begin{lemma}  \label{lem4}
Suppose  \eqref{poincare} and \eqref{fcond1} hold.
Then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$,  $D=\{D(\tau, \omega)
: \tau \in \mathbb{R},  \omega \in \Omega\}  \in \mathcal{D}$ and
$t_n \to \infty$,
 $v_{0,n}  \in D(\tau -t_n, \theta_{2, -t_n} \omega )$,  the sequence
 $v(\tau, \tau -t_n,  \theta_{2, -\tau} \omega,   v_{0,n}  ) $
 of solutions of problem \eqref{v1}-\eqref{v4}
   has a    convergent subsequence in   $H$.
\end{lemma}

\begin{proof}
It follows   from Lemma \ref{lem2} with $k =0$ that, there  exists
$T=T(\tau, \omega, D)>0$    such that   for   all $t \ge T$,
\begin{equation}\label{plem4_1a1}
\| v(\tau, \tau -t, \theta_{2, -\tau} \omega, v_{\tau -t} ) \|^2
\le 1
 + {\frac 2\nu} e^{- \nu \lambda \tau }
 \int_{-\infty}^{\tau }
 e^{\nu\lambda r}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr,
 \end{equation}
 with $v_{\tau -t} \in D(\tau -t, \theta_{2, -t} \omega )$.
 Since
 $t_n \to \infty$,  there  exists  $N_0 \in \mathbb{N}$ such that
   $t _n \ge T$   for    all   $ n \ge N_0$.
   Due to   $v_{0,n}  \in D(\tau -t_n, \theta_{2, -t_n} \omega )$,
   we get  from    \eqref{plem4_1a1}
   that   for all $n \ge   \mathbb{N}_0$,
 \begin{equation}\label{plem4_1}
 \| v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} ) \|^2
\le 1
 + {\frac 2\nu} e^{- \nu \lambda \tau }
 \int_{-\infty}^{\tau }
 e^{\nu\lambda r}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr.
 \end{equation}
  By \eqref{plem4_1} there exists $\tilde{v} \in H$
  and a subsequence (which  is not relabeled) such that
  \begin{equation}
  \label{plem4_2}
  v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )
   \rightharpoonup \tilde{v} \quad\text{in }  H.
   \end{equation}
   We now prove that  the weak convergence  of \eqref{plem4_2}
   is actually   a  strong convergence,  which will complete
   the proof.
   Note    that \eqref{plem4_2}   implies
   \begin{equation}
   \label{plem4_3}
   \liminf_{n \to \infty}
    \|v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} ) \|
    \ge \| \tilde{v} \|.
    \end{equation}
    So we  only need   to show
     \begin{equation}
   \label{plem4_4}
   \limsup_{n \to \infty}
    \|v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} ) \|
     \le  \| \tilde{v} \|.
    \end{equation}
    We  will establish   \eqref{plem4_4} by the method of  energy equations
    due to Ball \cite{bal1}.
    Given $k \in \mathbb{N}$   we have
    \begin{equation}
    \label{plem4_5}
    v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )
    = v(\tau, \tau - k, \theta_{2, -\tau} \omega, \
        v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )    ).
        \end{equation}
        For each $k$, let $N_k$   be large enough such that
        $t_n \ge T+ k$   for all $ n \ge N_k$.
       Then it   follows   from Lemma  \ref{lem2}    that
        for  $n \ge N_k $,
\begin{align*}
&\| v(\tau -k , \tau -t_n,  \theta_{2, -\tau} \omega, v_{0,n}  ) \|^2\\
& \le
 e^{\nu \lambda k}
 + {\frac 2\nu} e^{\nu \lambda (k -\tau)}
 \int_{-\infty}^{\tau -k}
 e^{\nu\lambda r}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr,
\end{align*}
 which     shows   that, for each fixed $k \in \mathbb{N}$,   the sequence
 $ v(\tau -k , \tau -t_n,  \theta_{2, -\tau} \omega, v_{0,n}  ) $
 is bounded in $H$. By a diagonal process,   one can  find
 a  subsequence (which we do not relabel)
and a point
 $\tilde{v}_k \in H$  for   each $k \in \mathbb{N}$      such that
 \begin{equation}
 \label{plem4_6}
  v(\tau -k , \tau -t_n,  \theta_{2, -\tau} \omega, v_{0,n}  )
  \rightharpoonup \tilde{v}_k
  \quad\text{in } \ H.
  \end{equation}
  By \eqref{plem4_5}-\eqref{plem4_6}   and Lemma \ref{lem3}
  we get that
  for  each $k \in \mathbb{N}$,
  \begin{equation}
  \label{plem4_10}
    v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )
    \rightharpoonup
    v(\tau, \tau - k, \theta_{2, -\tau} \omega,  \tilde{v}_k )
    \quad\text{in } \ H,
    \end{equation}
    and
    \begin{equation}
    \label{plem4_11}
    v(\cdot, \tau - k, \theta_{2, -\tau} \omega, \
        v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )    )
        \rightharpoonup
        v(\cdot, \tau - k, \theta_{2, -\tau} \omega,   \tilde{v}_k )
\end{equation}
 in $ L^2((\tau -k,  \tau), V) $.
By \eqref{plem4_2}  and \eqref{plem4_10}   we have
\begin{equation}
\label{plem4_12}
v(\tau, \tau - k, \theta_{2, -\tau} \omega,  \tilde{v}_k ) =\tilde{v}.
\end{equation}
Note that \eqref{plem1_1}  implies  that
  \begin{equation}\label{plem4_20}
 {\frac d{dt}} \| v\|^2  +
\nu \lambda  \| v \|^2 + \psi (v)
= 2 z(t, \omega) \langle f(\cdot, t), v \rangle,
\end{equation}
where $\psi$   is a   functional on $V$  given   by
$$ \psi (v)  = 2\nu \| Dv \|^2 - \nu \lambda \| v \|^2,
\quad\text{for  all }  v \in V.
$$
By \eqref{poincare}   we see   that
$$ \nu \| D v \|^2  \le \psi (v) \le 2 \nu \| Dv \|^2,
\quad\text{for  all }  v \in V.
$$
This indicates    that $\psi (\cdot) $ is an equivalent norm
of $V$.  It   follows   from \eqref{plem4_20} that
for each $\omega \in \Omega$, $s \in \mathbb{R}$  and $\tau \ge s$,
\begin{equation}\label{plem4_21}
\begin{split}
\| v(\tau, s,  \omega, v_s) \|^2
&=e^{\nu\lambda (s -\tau)} \| v_s \|^2
-\int_s^\tau e^{\nu \lambda (r-\tau)} \psi (v(r, s, \omega, v_s))  dr\\
&\quad
+2 \int_s^\tau  e^{\nu \lambda (r-\tau)}
z(r, \omega) \langle f(\cdot, r), v(r,s,\omega, v_s )\rangle dr.
\end{split}
\end{equation}
By  \eqref{plem4_12} and \eqref{plem4_21} we find that
\begin{equation}\label{plem4_22}
\begin{split}
\| \tilde{v}\|^2
& =  \| v(\tau, \tau -k,  \theta_{2, -\tau} \omega,  \tilde{v}_k) \|^2\\
&=e^{ - \nu\lambda  k } \|  \tilde{v}_k \|^2
- \int_{\tau -k}^\tau e^{\nu \lambda (r-\tau)}
 \psi (v(r, \tau -k, \theta_{2, -\tau} \omega,  \tilde{v}_k ))  dr \\
&\quad
+2 \int_{\tau -k}^\tau  e^{\nu \lambda (r-\tau)}
z(r,  \theta_{2, -\tau} \omega)
 \langle f(\cdot, r), v(r, \tau -k,  \theta_{2, -\tau}\omega,  \tilde{v}_k )\rangle dr.
\end{split}
\end{equation}
    Similarly,  by  \eqref{plem4_5}
    and \eqref{plem4_21} we obtain  that
\begin{equation}\label{plem4_23}
\begin{split}
 &\| v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )  \|^2\\
&= \|v(\tau, \tau - k, \theta_{2, -\tau} \omega, \
        v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )    ) \|^2\\
&  =e^{ - \nu\lambda  k }
   \|  v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )    \|^2\\
&\quad
- \int_{\tau -k}^\tau e^{\nu \lambda (r-\tau)}
 \psi (v(r, \tau -k, \theta_{2, -\tau} \omega,
  v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} ) ) )  dr\\
&\quad +2 \int_{\tau -k}^\tau  e^{\nu \lambda (r-\tau)}
z(r,  \theta_{2, -\tau} \omega)\\
&\quad\times  \langle f(\cdot, r),
 v(r, \tau -k,  \theta_{2, -\tau}\omega,
  v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )  )\rangle dr.
\end{split}
\end{equation}
We now  consider the limit of each  term on the right-hand
side of \eqref{plem4_23}   as $n \to \infty$.
For the first  term,
by \eqref{plem1_5} with $s = \tau -k$ and $t =t_n$
 we get  that
 \begin{equation}  \label{plem4_24}
\begin{split}
&e^{- \nu\lambda k}
\| v(\tau -k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0,n} )\|^2 \\
&\le
e^{- \nu\lambda t_n }   \| v_{0,n} \|^2
+{\frac 2\nu} e^{- \nu\lambda  \tau }
\int_{-\infty} ^{\tau -k}
e^{\nu\lambda r} z^2(r,  \theta_{2, -\tau} \omega) \| f(\cdot, r)\|^2_{V^*} dr.
\end{split}
\end{equation}
Since $v_{0,n} \in D(\tau -t_n, \theta_{2, -t_n} \omega)$ we have
$$
e^{- \nu\lambda t_n }   \| v_{0,n} \|^2
\le  e^{- \nu\lambda t_n }   \| D(\tau -t_n, \theta_{2, -t_n} \omega)  \|^2
\to 0 \quad \text{as }  n \to \infty,
$$
which   along with \eqref{plem4_24} shows   that
\begin{equation}  \label{plem4_30}
\begin{split}
&\limsup_{n \to \infty} e^{- \nu\lambda k}
\| v(\tau -k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0,n} )\|^2 \\
&\le
{\frac 2\nu} e^{- \nu\lambda  \tau }
\int_{-\infty} ^{\tau -k}
e^{\nu\lambda r} z^2(r,  \theta_{2, -\tau} \omega) \| f(\cdot, r)\|^2_{V^*} dr.
\end{split}
\end{equation}
By \eqref{plem4_11} we find   that
  \begin{equation} \label{plem4_40}
\begin{split}
&\lim_{n \to \infty}
\int_{\tau -k}^\tau  e^{\nu \lambda (r-\tau)}
z(r,  \theta_{2, -\tau} \omega)\\
&\quad\times \langle f(\cdot, r),
 v(r, \tau -k,  \theta_{2, -\tau}\omega,
  v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )  )\rangle dr\\
&=  \int_{\tau -k}^\tau  e^{\nu \lambda (r-\tau)}
z(r,  \theta_{2, -\tau} \omega)
 \langle f(\cdot, r),
 v(r, \tau -k,  \theta_{2, -\tau}\omega,  \tilde{v}_k ) \rangle dr,
\end{split}
 \end{equation}
 and
  \begin{equation}\label{plem4_41}
\begin{split}
&\liminf_{n \to \infty}
 \int_{\tau -k}^\tau e^{\nu \lambda (r-\tau)}
 \psi (v(r, \tau -k, \theta_{2, -\tau} \omega,
  v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} ) ) )  dr\\
& \ge   \int_{\tau -k}^\tau e^{\nu \lambda (r-\tau)}
 \psi (v(r, \tau -k, \theta_{2, -\tau} \omega,
   \tilde{v}_k  ))   dr.
\end{split}
\end{equation}
   Note  that \eqref{plem4_41} implies   that
  \begin{equation}\label{plem4_42}
\begin{split}
&\limsup _{n \to \infty} -
 \int_{\tau -k}^\tau e^{\nu \lambda (r-\tau)}
 \psi (v(r, \tau -k, \theta_{2, -\tau} \omega,
  v(\tau-k, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} ) ) )  dr\\
&  \le -   \int_{\tau -k}^\tau e^{\nu \lambda (r-\tau)}
 \psi (v(r, \tau -k, \theta_{2, -\tau} \omega,
   \tilde{v}_k  ))   dr.
\end{split}
\end{equation}
   Taking   the limit of \eqref{plem4_23} as $n \to \infty$, by
   \eqref{plem4_30}, \eqref{plem4_40} and \eqref{plem4_42} we
   obtain  that
\begin{equation}\label{plem4_50}
\begin{split}
&\limsup _{n \to \infty}
      \| v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )  \|^2\\
&\le {\frac 2\nu} e^{- \nu\lambda  \tau } \int_{-\infty} ^{\tau -k}
e^{\nu\lambda r} z^2(r,  \theta_{2, -\tau} \omega) \| f(\cdot, r)\|^2_{V^*} dr\\
&\quad   - \int_{\tau -k}^\tau e^{\nu \lambda (r-\tau)}
 \psi (v(r, \tau -k, \theta_{2, -\tau} \omega,  \tilde{v}_k ))  dr \\
&\quad +2 \int_{\tau -k}^\tau  e^{\nu \lambda (r-\tau)}
z(r,  \theta_{2, -\tau} \omega)
 \langle f(\cdot, r), v(r, \tau -k,  \theta_{2, -\tau}\omega,  \tilde{v}_k )\rangle dr.
\end{split}
\end{equation}
It follows from \eqref{plem4_22}  and \eqref{plem4_50}
that
 \begin{equation}  \label{plem4_60}
\begin{split}
&\limsup _{n \to \infty}
      \| v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )  \|^2\\
&\le \| \tilde{v} \|^2    +
{\frac 2\nu} e^{- \nu\lambda  \tau }
\int_{-\infty} ^{\tau -k}
e^{\nu\lambda r} z^2(r,  \theta_{2, -\tau} \omega) \| f(\cdot, r)\|^2_{V^*} dr.
\end{split}
\end{equation}
Let $ k \to \infty$ in  \eqref{plem4_60} to yield
 \begin{equation}
 \label{plem4_61}
   \limsup _{n \to \infty}
      \| v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )  \|^2
      \le \| \tilde{v} \|^2  .
\end{equation}
By \eqref{plem4_2}-\eqref{plem4_3}   and \eqref{plem4_61}
      we find   that
      $$
\lim _{n \to \infty} v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0, n} )
=    \tilde{v}  \quad \text{in } H .
$$
This completes  the proof.
\end{proof}


\section{Existence of  pullback  attractors }

 In   this  section,  we establish   the existence of  $\mathcal{D}$-pullback attractors
 for the Navier-Stokes equations \eqref{nse1}-\eqref{nse2}.
 Based on  the uniform
 estimates  on the   solutions   of problem \eqref{v1}-\eqref{v4},  we first show
 that the cocycle  $\Phi$    associated  with the  stochastic  system
 \eqref{nse1}-\eqref{nse4}  has a  measurable $\mathcal{D}$-pullback  absorbing   set
 in $H$, and then prove  the $\mathcal{D}$-pullback   asymptotic compactness of $\Phi$.

\begin{lemma} \label{lematt1}
 Suppose  \eqref{poincare} and \eqref{fcond1} hold.
Then for every $\tau \in \mathbb{R}$, $\omega \in \Omega$   and 
$D=\{D(\tau, \omega): \tau \in \mathbb{R},  \omega \in \Omega\}  \in \mathcal{D}$,
 there exists  $T=T(\tau, \omega,  D)>0$ such that
for all $t \ge T $,
 the solution
 $u$ of  problem  \eqref{nse1}-\eqref{nse4}  with $\omega$ replaced by
 $\theta_{2, -\tau} \omega$  satisfies
\begin{align*}
&\| u(\tau  , \tau -t,  \theta_{2, -\tau} \omega, u_{\tau -t}  ) \|^2\\
& \le z^{-2} (\tau, \theta_{2, -\tau} \omega )
 + {\frac 2\nu}z^{-2} (\tau, \theta_{2, -\tau} \omega )
 \int_{-\infty}^{\tau }
 e^{\nu\lambda (r-\tau)}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr,
\end{align*}
 where $u_{\tau -t}\in D(\tau -t, \theta_{2, -t} \omega)$.
\end{lemma}

\begin{proof}
Given $D=\{D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega\}
 \in \mathcal{D}$,  for each $\tau \in \mathbb{R}$   and
 $\omega \in \Omega$,  denote by
 \begin{equation} \label{plematt1_1}
{\tilde{D}}(\tau, \omega)
 = \{ v \in H :  \| v \|
 \le |z(\tau, \theta_{2, -\tau} \omega)| \ \| D(\tau, \omega) \| \}.
 \end{equation}
 Let ${\tilde{D}}$  be a  family
 corresponding to $D$ which  consists of
  the sets given by \eqref{plematt1_1}; i.e.,
 \begin{equation}\label{plematt1_2}
 {\tilde{D}} =
\{ {\tilde{D}}(\tau, \omega) :
  {\tilde{D}}(\tau, \omega) \text{ is    defined by   \eqref{plematt1_1}}, 
 \tau \in \mathbb{R},  \omega \in \Omega \}.
\end{equation}
We now  prove
${\tilde{D}}$  is tempered in $H$
 for  $D \in \mathcal{D}$.
Given  $c>0$, by \eqref{aspomega} we find that
for  each $\omega \in \Omega$,
there   exists $R>0$ such    that   for   all
$r \ge R$,
\begin{equation} \label{plematt1_3}
| -\alpha \omega (-r) | \le {\frac 12} c r.
\end{equation}
Since $D\in \mathcal{D}$,   from  \eqref{plematt1_3} it follows that
\begin{align*}
e^{-cr} \| {\tilde{D}} (\tau -r, \theta_{2,-r} \omega ) \|
&= e^{-cr} |z(\tau -r, \theta_{2, -\tau} \omega )| \
\| D  (\tau -r, \theta_{2,-r} \omega ) \| \\
& \le e^{\alpha \omega (-\tau) }
e^{- {\frac 12} cr}
\| D  (\tau -r, \theta_{2,-r} \omega ) \|
\to 0,
\quad\text{as }   r \to \infty,
\end{align*}
which  shows that ${\tilde{D}} \in \mathcal{D}$.
Since $u_{\tau -t}\in D(\tau -t, \theta_{2, -t} \omega)$, by
\eqref{vu} we  know  that
$$ 
\| v_{\tau -t} \|
= \| z(\tau -t, \theta_{2, -\tau} \omega ) \ u_{\tau -t} \|
\le |  z(\tau -t, \theta_{2, -\tau} \omega )|
 \, \| D(\tau -t, \theta_{2, -t} \omega)\|,
$$
which along with \eqref{plematt1_1} implies  that
$v_{\tau -t} \in {\tilde{D}} (\tau -t, \theta_{2, -t} \omega )$.
Since ${\tilde{D}} $  is tempered, it follows   from Lemma \ref{lem2} with
$k=0$   that there  exists  $T= T(\tau, \omega, D)>0$   such that
for all $ t \ge T$,
$$
\| v(\tau  , \tau -t,  \theta_{2, -\tau} \omega, v_{\tau -t}  ) \|^2
 \le  1
 + {\frac 2\nu}
 \int_{-\infty}^{\tau }
 e^{\nu\lambda (r-\tau)}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr,
 $$
 which along with \eqref{vu} completes   the proof.
\end{proof}



 \begin{lemma} \label{lematt2}
 Suppose  \eqref{poincare}  and   \eqref{fcond2}  hold.
 Then the continuous cocycle $\Phi$ associated with
 problem \eqref{nse1}-\eqref{nse4} has a closed measurable
 $\mathcal{D}$-pullback absorbing set
 $K =\{ K(\tau, \omega): \tau \in \mathbb{R},  \omega \in \Omega\}$
 $\in \mathcal{D}$.
\end{lemma}

\begin{proof}
Given $\tau \in \mathbb{R}$    and $\omega \in \Omega$,   denote by
  \begin{equation}  \label{plematt2_1}
 K(\tau, \omega) = \{ u\in H : \|u \|^2
 \le  M(\tau, \omega) \},
 \end{equation}
   where
 $M(\tau, \omega)$ is  given by
  \begin{equation}  \label{plematt2_2}
\begin{split}
 M(\tau, \omega) 
&= z^{-2} (\tau, \theta_{2, -\tau} \omega )\\
& + {\frac 2\nu}z^{-2} (\tau, \theta_{2, -\tau} \omega )
 \int_{-\infty}^{\tau }
 e^{\nu\lambda (r-\tau)}
 z^2(r, \theta_{2,-\tau} \omega) \| f(\cdot, r)\|_{V^*}^2 dr.
\end{split}
\end{equation}
Since  for each $\tau \in \mathbb{R}$,  $M(\tau, \cdot): \Omega \to \mathbb{R}$
is   $({\mathcal{F}}_1, \mathcal{B}(\mathbb{R}))$-measurable,   we know that
$K(\tau, \cdot): \Omega \to 2^H$ is a  measurable set-valued mapping.
It follows  from Lemma \ref{lematt1} that,  for each
$\tau \in \mathbb{R}$, $\omega \in \Omega$   and $D \in \mathcal{D}$,
  there  exists $T=T(\tau, \omega, D) >0$  such that   for   all
  $t \ge T$,
  $$
  \Phi (t, \tau-t, \theta_{2, -t} \omega, D(\tau -t, \theta_{2, -t} \omega ))
  =u (\tau,  \tau-t, \theta_{2, -\tau} \omega, D(\tau -t, \theta_{2, -t} \omega ))
  \subseteq K(\tau, \omega).
  $$
  Therefore,  $K =\{ K(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}$
  will be a closed measurable  $\mathcal{D}$-pullback absorbing set of $\Phi$ in $H$
  if one can show   that $K$ belongs to $\mathcal{D}$.
  For each $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $r>0$,
  by \eqref{plematt2_1}   we have
\begin{equation}\label{plematt2_3}
\begin{split}
&\| K(\tau -  r, \theta_{2, -r} \omega) \|\\
 &\le  {\frac 1{z(\tau -r, \theta_{2,-\tau} \omega)}}
  \Big(
   1+ {\frac 2\nu} \int_{-\infty} ^{\tau -r}
   e^{\nu\lambda (s -\tau + r)} z^2(s, \theta_{2, -\tau} \omega)
   \| f(\cdot, s ) \|^2_{V^*} ds
  \Big)^{1/2}
\\
&  \le e^{-\alpha \omega (-\tau)}
  e^{\alpha \omega (-r)}\\
&\quad\times  \Big(
   1+ {\frac 2\nu} \int_{-\infty} ^{0}
   e^{\nu\lambda s } z^2(s+\tau -r, \theta_{2, -\tau} \omega)
   \| f(\cdot, s+\tau -r ) \|^2_{V^*} ds  \Big)^{1/2}
\\
&\le e^{-\alpha \omega (-\tau)}
  e^{\alpha \omega (-r)}
   \Big( 1\\
&\quad +
  \Big(\frac{2}{\nu} \int_{-\infty} ^{0}
   e^{(\nu\lambda-\delta) s }
    z^2(s+\tau -r , \theta_{2, -\tau} \omega)
   e^{\delta s} \| f(\cdot, s+\tau - r ) \|^2_{V^*} ds   \Big)^{1/2}
  \Big).
\end{split}
  \end{equation}
  Let $c$ be an arbitrary positive number
  and $\varepsilon   =\min \{ \nu\lambda -\delta,  \ {\frac 12} c\}$.
  By \eqref{aspomega} we see   that  there
  exists $N_1>0$  such that
  \begin{equation}\label{plematt2_5}
  | -2 \alpha  \ \omega (p) |
  \le -\varepsilon p \quad\text{for all }   p \le -N_1.
  \end{equation}
Let $s \le 0$   and $r \ge N_1$. Then
  $p = s-r \le  -N_1$ and hence
  it follows  from \eqref{plematt2_5}
  that
   \begin{equation}\label{plematt2_6}
   -2 \alpha  \ \omega (s-r)
  \le -\varepsilon  (s-r) ,
  \quad\text{for  all } 
  s\le 0 \text{ and }  r \ge N_1 .
  \end{equation}
  By \eqref{plematt2_6}   we have,   for  all $s \le 0$   and $r \ge N_1$,
  \begin{equation}
  \label{plematt2_7}
  e^{(\nu\lambda -\delta) s}
  z^2(s+ \tau -r, \theta_{2, -\tau} \omega)
  \le
  e^{(\nu\lambda -\delta) s} e^{2 \alpha  \omega (-\tau)}
  e^{ -2 \alpha \omega (s-r)}
  \le e^{ 2 \alpha \omega (-\tau)} e^{\varepsilon r}.
  \end{equation}
  From \eqref{plematt2_3}, \eqref{plematt2_5}  and \eqref{plematt2_7} we have
  that,  for all $r \ge   N_1$,
 \begin{equation}\label{plematt2_10}
\begin{split} 
&\| K(\tau -  r, \theta_{2, -r} \omega) \|\\
&\le  e^{\varepsilon r -\alpha \omega (-\tau) }
  +\sqrt{{\frac 2\nu}}  e^{ {\frac 32} \varepsilon r}
  \Big(
  \int_{-\infty}^0
  e^{\delta s} \| f(\cdot, s+\tau - r ) \|^2_{V^*} ds
  \Big)^{1/2}
\\
&\le   e^{{\frac 12 } c r -\alpha \omega (-\tau) }
  +\sqrt{{\frac 2\nu}}  e^{ {\frac 34} c r}
  \Big(
  \int_{-\infty}^0
  e^{\delta s} \| f(\cdot, s+\tau - r ) \|^2_{V^*} ds
  \Big)^{1/2},
\end{split}
 \end{equation}
  where we have used the fact $\varepsilon \le c/2$.
  It follows   from \eqref{plematt2_10}   that,   for  all
  $r \ge N_1$,
\begin{align*}
 & e^{-c  r} \| K(\tau -  r, \theta_{2, -r} \omega) \|\\
& \le
  e^{-{\frac 12 } c r -\alpha \omega (-\tau) }
  +\sqrt{{\frac 2\nu}}  e^{- {\frac 14} c r}
  \Big(
  \int_{-\infty}^0
  e^{\delta s} \| f(\cdot, s+\tau - r ) \|^2_{V^*} ds
  \Big)^{1/2} \\
& \le
  e^{-{\frac 12 } c r -\alpha \omega (-\tau) }
  +\sqrt{{\frac 2\nu}}  e^{- {\frac 14} c \tau}
  \left ( e^{{\frac 12} c (\tau -r)}
  \int_{-\infty}^0
  e^{\delta s} \| f(\cdot, s+\tau - r ) \|^2_{V^*} ds
  \right )^{1/2},
\end{align*}
 which along with \eqref{fcond2} shows   that
 for every positive constant $c$,
 $$
 \lim_{r \to \infty}  e^{-c  r} \| K(\tau -  r, \theta_{2, -r} \omega) \|
 =0,
 $$
 and hence
 $K = \{ K(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}$
 is tempered.  This completes   the proof.
  \end{proof}

We now prove   the $\mathcal{D}$-pullback   asymptotic
  compactness of solutions of the stochastic   equations
  \eqref{nse1}-\eqref{nse2}.


\begin{lemma} \label{lematt3}
 Suppose  \eqref{poincare}  and   \eqref{fcond2}  hold.
 Then the continuous cocycle $\Phi$ associated with
 problem \eqref{nse1}-\eqref{nse4}
  is $\mathcal{D}$-pullback
 asymptotically compact  in $H$,
that is, for  every $\tau \in \mathbb{R}$, $\omega \in \Omega$,
 $D=\{D(\tau, \omega): \tau \in \mathbb{R}, \omega \in \Omega \}$
 $  \in \mathcal{D}$,
and $t_n \to \infty$,
 $u_{0,n}  \in D(\tau -t_n, \theta_{2, -t_n} \omega )$,  the sequence
 $\Phi(t_n, \tau -t_n,  \theta_{2, -t_n} \omega,   u_{0,n}  ) $   has a
   convergent
subsequence in $H $.
\end{lemma}

\begin{proof}
Since $D \in \mathcal{D}$  and
$u_{0,n}  \in D(\tau -t_n, \theta_{2, -t_n} \omega )$, by
the proof of Lemma  \ref{lematt1} we find   that
for   each $n \in \mathbb{N}$,
$v_{0,n} = z(\tau - t_n, \theta_{2, -\tau} \omega ) u_{0, n}$
$\in {\tilde{D}} (\tau -t_n, \theta_{2, -t_n} \omega)$,
where
$\tilde{D} \in \mathcal{D}$ is  the family defined by
\eqref{plematt1_2}.
Then it follows   from Lemma \ref{lem4}   that
the sequence
$v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0,n} )$
of solutions   of problem \eqref{v1}-\eqref{v4}
has a convergent subsequence in $H$.
By \eqref{vu}    we have
$$
u(\tau, \tau -t_n, \theta_{2, -\tau} \omega, u_{0,n} )
= {\frac 1{z(\tau, \theta_{2, -\tau} \omega )}}
v(\tau, \tau -t_n, \theta_{2, -\tau} \omega, v_{0,n} ),
$$
and hence  the sequence
$u(\tau, \tau -t_n, \theta_{2, -\tau} \omega, u_{0,n} )$
has a convergent subsequence in $H$.   This implies
$\Phi(t_n, \tau -t_n,  \theta_{2, -t_n} \omega,   u_{0,n}  ) $   has a
   convergent subsequence in $H $.
\end{proof}

 We  are now   in a position to  present
  the main result of the paper, that is,   the existence of
  tempered pullback attractors for
  the stochastic  Navier-Stokes equations.

  \begin{theorem} \label{thmnse1}
 Suppose  \eqref{poincare}  and   \eqref{fcond2}  hold.
 Then the continuous cocycle $\Phi$ associated with
 problem \eqref{nse1}-\eqref{nse4}
   has a unique $\mathcal{D}$-pullback attractor $\mathcal{A}
   =\{\mathcal{A}(\tau, \omega):
      \tau \in \mathbb{R}, \ \omega \in \Omega \} \in \mathcal{D}$
 in $H$.  Moreover,
 for each $\tau  \in \mathbb{R}$   and
$\omega \in \Omega$,
\begin{align}\label{thm1_1}
\mathcal{A} (\tau, \omega) &=\Omega(K, \tau, \omega)
=\cup_{B \in \mathcal{D}} \Omega(B, \tau, \omega) \\
\label{thm1_2}
& =\{\psi(0, \tau, \omega): \psi \text{ is any  $\mathcal{D}$-complete orbit of } \Phi\} .
\end{align}
\end{theorem}

\begin{proof}
By Lemma \ref{lematt2} we know   that
$\Phi$ has a closed measurable $\mathcal{D}$-pullback absorbing set
in $H$.  On the other   hand,  by Lemma \ref{lematt3} we know that
  $\Phi$ is   $\mathcal{D}$-pullback
asymptotically  compact.
Then it follows    from Proposition \ref{att}
 that  $\Phi$   has a unique
 $\mathcal{D}$-pullback attractor
 $\mathcal{A}$ in $H$
 and the structure of
 $\mathcal{A}$  is given by
 \eqref{thm1_1}-\eqref{thm1_2}.
\end{proof}



    We now  discuss    the existence of periodic
    pullback   attractors   for problem \eqref{nse1}-\eqref{nse4}.
    Suppose   $f: \mathbb{R}\to V^*$   is  a  periodic   function
with period $T>0$.
If, in addition,  $f \in L^2_{loc} (\mathbb{R}, V^*)$,   then
  one   can verify    that $f$  satisfies
  \eqref{fcond2}
  for  any $\delta>0$.
  In this case,
for every ${\tilde{u}} \in H$,
 $t \ge 0$, $\tau \in \mathbb{R}$ and $\omega \in \Omega$,
   we  have    that
\begin{align*}
\Phi (t, \tau +T, \omega, {\tilde{u}} )
&= u(t+ \tau +T, \tau +T,  \theta_{2, -\tau -T} \omega, {\tilde{u}})\\
&=u(t +\tau, \tau, \theta_{2, -\tau} \omega, {\tilde{u}} ).
= \Phi (t, \tau,  \omega, {\tilde{u}} ). 
\end{align*}
By  Definition \ref{ds1}, we find   that
  $\Phi$ is  periodic with period  $T$.
Let $D \in \mathcal{D}$ and $D_T$  be the $T$-translation
of $D$.  Then for every
$c>0$,  $s  \in \mathbb{R}$   and $\omega \in \Omega$,
\begin{equation}\label{tranrde1}
\lim_{r \to \infty}e^{-cr } \| D(s -r , \theta_{2, -r} \omega )\|^2 =0.
\end{equation}
In particular, for $s  = \tau +T$   with $\tau \in \mathbb{R}$,  we get
from \eqref{tranrde1}  that
\begin{equation}\label{tranrde2}
\lim_{r \to  \infty}
e^{-cr } \| D_T(\tau -r, \theta_{2, -r} \omega )\|^2
=
\lim_{r \to \infty}
e^{-cr } \| D(\tau +T-r, \theta_{2, -r} \omega )\|^2 =0.
\end{equation}
From \eqref{tranrde2}  we see   that
$D_T \in \mathcal{D}$,    and hence
$\mathcal{D}$  is $T$-translation closed.
Similarly, one may check  that
$\mathcal{D}$  is also $-T$-translation closed.
Therefore,   we find that  $\mathcal{D}$  is $T$-translation invariant.
By   Proposition \ref{periodatt},
   the periodicity of the $\mathcal{D}$-pullback attractor of
problem \eqref{nse1}-\eqref{nse4}   follows.


\begin{theorem} \label{thmnse2}
Let $f: \mathbb{R}\to V^*$  be a periodic function with period $T>0$ and
   $ f \in  L^2 ((0,T), V^*)$.
   If    \eqref{poincare} holds,
 then the continuous cocycle $\Phi$ associated with
 problem \eqref{nse1}-\eqref{nse4}
   has a unique   $\mathcal{D}$-pullback attractor $\mathcal{A} \in \mathcal{D}$
 in $H$, which is periodic with period $T$.
\end{theorem}

In the present article,   we  have  discussed   the  pullback
attractors of the two-dimensional stochastic Navier-Stokes
equations with  non-autonomous deterministic force.  It  is
also interesting to consider the same problem for  the three-dimensional
Navier-Stokes equations,  where the uniqueness of solutions does not hold
anymore.
In this case,   the author believes  that the idea of multivalued 
 dynamical systems developed in \cite{car6} can be
extended to study the pullback attractors of  the three-dimensional equations
with non-autonomous deterministic force.
The author will pursue this  line of   research in the future.

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\end{document}