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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 03, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/03\hfil Asymptotic behavior of pullback attractors]
{Asymptotic behavior of pullback attractors for non-autonomous
 micropolar fluid flows in 2D unbounded domains}

\author[Wenlong Sun, Yeping Li \hfil EJDE-2018/03\hfilneg]
{Wenlong Sun, Yeping Li}

\address{Wenlong Sun \newline
Department of Mathematics,
East China University of Science and Technology,
Shanghai 200237, China}
\email{wenlongsun1988@163.com}

\address{Yeping Li (corresponding author) \newline
Department of Mathematics,
East China University of Science and Technology,
Shanghai 200237, China}
\email{yplee@ecust.edu.cn}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted March 6, 2017. Published January 4, 2018.}
\subjclass[2010]{35B40, 35B41, 35Q30}
\keywords{ Micropolar fluid flow; pullback attractor; truncation function;
\hfill\break\indent tempered behavior; upper semicontinuity}

\begin{abstract}
 In this article, we investigate the pullback asymptotic behavior of
 solutions for a non-autonomous micropolar fluid flows in 2D unbounded
 channel-like domains. First, applying the technique of truncation
 functions, decomposition of spatial domain, and the energy method,
 we show the existence of the pullback attractor in the space
 $\widehat{H}(\Omega)$ (has $L^2$-regularity). In fact,
 we can deduce the existence of pullback attractor in space
 $\widehat{V}(\Omega)$ (has $H^1$-regularity). Also the tempered behavior
 of the pullback attractor is verified. Moreover, when the spatial
 domain varies from $\Omega_m$($\{\Omega_m\}_{m=1}^{\infty}$ be an
 expanding sequence of simply connected, bounded and smooth subdomains
 of $\Omega$ such that $\cup_{m=1}^{\infty}\Omega_m = \Omega$)
 to $\Omega$, the upper semicontinuity of the pullback attractor is discussed.
\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}

The micropolar fluid model is firstly derived by Eringen \cite{E66} in 1966,
which is used to describe fluids consisting of randomly oriented particles
suspended in a viscous medium. The model has the form:
\begin{equation}
\begin{gathered}
  \frac{\partial u}{\partial t}-(\nu+\nu_{\theta})\Delta u
  -2\nu_{\theta}\operatorname{rot} \omega
   +(u \cdot \nabla)u   +\nabla p
  =f,  \\
  \nabla\cdot u =0, \\
\begin{aligned}
&\frac{\partial \omega}{\partial t}
 -(c_a+c_d)\Delta \omega
  +4\nu_{\theta} \omega
  +(u \cdot \nabla)\omega \\
&-(c_0+c_d-c_a)\nabla \operatorname{div}\omega
  -2\nu_{\theta} \operatorname{rot} u
  =\tilde{f},
\end{aligned}
\end{gathered} \label{1.1}
\end{equation}
where $u=(u_1, u_2,u_3)$ is velocity,
$\omega=(\omega_1,\omega_2,\omega_3)$ represents the angular
velocity field of rotation of particles, $f=(f_1, f_2, f_3)$ and
$\tilde{f}=(\tilde{f}_1, \tilde{f}_2, \tilde{f}_3)$ stand for the
external force and moments, respectively.  $p$ is pressure.
The positive parameters $\nu, \nu_{\theta}, c_0, c_a$ and $c_d$
are viscosity coefficients. Indeed, $\nu$ is the usual Newtonian
viscosity and $\nu_{\theta}$ is called microrotation viscosity.
Note that when the gyration is neglected, the micropolar fluid equations
are reduce to the classical Navier-Stokes equations.

Micropolar fluid model plays important role in the fields of
applied and computational mathematics, there are lots of literatures
on the mathematical theory of micropolar fluid model \eqref{1.1}.
The existence and uniqueness of solutions for the micropolar fluids
has been investigated in \cite{DZ10,L99,L01}. At the same time, lots
of works are devoted to the long time behavior of solutions for the
micropolar fluids. More precisely, Chen, Chen and Dong proved the
existence of $H^2$-compact global attractors in a bounded domain in
\cite{CCD06} and verified the existence of uniform attractors in
non-smooth domains in \cite{CCD07}. Chen\cite{C09} showed the
existence of $L^2$-pullback attractor for the micropolar fluid flows
in a Lipschitz bounded domain with non-homogeneous boundary
conditions. \L{u}kaszewicz\cite{L01} verified the estimates of
Hausdorff and fractal dimension of the $L^2$-global attractor.
Later, \L{u}kaszewicz and Tarasi\'nska\cite{LT09} proved the
existence of $H^1$-pullback attractor for non-autonomous micropolar
fluid equation in a bounded domain. As for the long time behavior of
solutions for the micropolar fluid flows on unbounded domain, Dong
and Chen \cite{DC06} investigated the existence and regularity of
the global attractors in 2D unbounded domains. Later, Zhao, Zhou and
Lian \cite{ZZL08} established the existence of $H^1$-uniform
attractor and further proved the $L^2$-uniform attractor belongs to
the $H^1$-uniform attractor in 2D unbounded domains. Nowakowski
\cite{N13} investigated the existence of $H^1$-uniform attractor and
long time behavior of solutions in 3D cylindrical domains. So far,
to our knowledge, there is no results about pullback attractors of
the micropolar fluid model in 2D unbounded domains. Here, we will
give a positive answer for this problem.

Since we investigate the pullback asymptotic behavior of solutions for the micropolar fluid model in 2D unbounded domains, we assume that the velocity
component $u_3$ in the $x_3$ direction is zero and the axes of
rotation of particles are parallel to the $x_3$ axis. Then,
the form of $u, \omega, f, \tilde{f}$ are that $u=(u_1, u_2,0)$,
$\omega=(0,0,\omega_3)$, $f=(f_1, f_2, 0)$,
$\tilde{f}=(0,0,\tilde{f}_3)$. Further, the equations \eqref{1.1} can
be reduced to the following 2D non-autonomous dynamical
system:
\begin{equation}
\begin{gathered}
  \frac{\partial u}{\partial t}
   -(\nu+\nu_{\theta})\Delta u
   -2\nu_{\theta} \nabla\times \omega
   +(u \cdot \nabla)u+\nabla p
   =f(t, x), \\
 \frac{\partial \omega}{\partial t}
   -\alpha\Delta \omega+4\nu_{\theta} \omega
   -2\nu_{\theta} \nabla\times u+(u \cdot \nabla)\omega
  =\tilde{f}(t, x), \\
\nabla\cdot u=0, \quad \text{in } (\tau, T)\times \Omega,
\end{gathered} \label{1.2}
\end{equation}
where $\Omega := \mathbb{R}\times (-L, L)$ for some $L>0$,
$\alpha :=c_a+c_d$, and $x := (x_1,x_2)\in\Omega\subseteq
\mathbb{R}^2$, $u :=(u_1,u_2)$, $f:=(f_1, f_2)$. $\omega$ and
$\tilde{f}$ are scalar functions,
$$
  \nabla\times u:=\frac{\partial u_2}{\partial x_1}
-\frac{\partial u_1}{\partial x_2}
\quad\text{and}\quad
   \nabla\times \omega:=(\frac{\partial \omega}{\partial x_2},
-\frac{\partial \omega}{\partial x_1}).
$$
To complete the formulation of the initial boundary value problem to
the system \eqref{1.2}, We give the following initial data and
boundary conditions:
\begin{gather}
w(\tau, x)=(u(\tau,x), \omega(\tau,x))=(u_0(x),\omega_0(x)),
  \quad x\in\Omega, \; \tau\in\mathbb{R}, \label{1.3}  \\
u=0,\quad \omega=0, \quad \text{on }  (\tau, T)
   \times\partial\Omega. \label{1.4}
\end{gather}

In fact, there have been some papers in the literature on the
pullback asymptotic behaviors of solutions for the Navier-Stokes
equations and the non-Newtonian fluid in some unbounded domains. For
example, Caraballo, \L{u}kaszewicz and Real \cite{CLR2006}
established the existence of a pullback attractor for a
non-autonomous Navier-Stokes. Langa, \L{u}kaszewicz and Real
\cite{LLR07} proved the existence of pullback attractor for the
non-autonomous Navier-Stokes equations in 2D unbounded domains. In
particular, they gave sufficient conditions for their pullback
attractor to have finite fractal dimension. Wang and Li \cite{WL09}
studied the existence of a pullback attractor for a non-autonomous
2D Navier-Stokes equation with linear dampness. Zhao \cite{Z12}
showed the existence of pullback attractor and their upper
semicontinuity for the non-autonomous non-Newtonian fluid in 2D
unbounded domain. Here, borrowing the ideas and argument in
\cite{Z12}, we discuss the existence of pullback attractor and their
tempered behavior and upper semicontinuity in 2D unbounded
channel-like domains.

For the sake of convenience, we introduce the following useful operators:
\begin{equation} \label{15}
\begin{gathered}
 \langle Aw, \phi \rangle
  := (\nu+\nu_{\theta})(\nabla u, \nabla \Phi)
 +\alpha(\nabla \omega, \nabla \phi_3), \\
 \forall w =(u,\omega), \; \varphi =(\Phi, \phi_3)\in\widehat{V},  \\
 \langle B(u, w), \phi \rangle
 := ((u \cdot \nabla)w, \phi), \quad
  \forall u\in V, \; w=(u,\omega)\in \widehat{V},
  \;\forall\phi\in\widehat{V},  \\
 N(w) :=(-2\nu_{\theta} \nabla\times \omega, -2\nu_{\theta} \nabla\times
u+4\nu_{\theta}\omega), \quad
  \forall w=(u, \omega)\in\widehat{V}.
 \end{gathered}
\end{equation}
Then, equations \eqref{1.2}-\eqref{1.4} can be represented into the
 abstract form
\begin{equation} \label{1.6}
\begin{gathered}
  \frac{\partial w}{\partial t}
   + Aw +B(u,w) + N(w) = F(t,x), \quad \text{in }(\tau, +\infty)\times\Omega,  \\
 \nabla\cdot u = 0, \quad \text{in }(\tau, +\infty)\times \Omega,  \\
 w = (u,\omega) = 0, \quad \text{on }(\tau, +\infty)\times \partial\Omega,  \\
 w(\tau,x) = (u(\tau,x), \omega(\tau,x)) = w_{\tau}(x),\quad x\in\Omega, \;
 \tau\in\mathbb{R},
 \end{gathered}
\end{equation}
where $F(t,x)=(f(t,x), \tilde{f}(t,x))$.

Before stating our results, we first give some notation. We denote
by $L^p(\Omega)$ and $W^{m,p}(\Omega)$ the usual Lebesgue space and
Sobolev space (see \cite{A75}) endowed with norms $\|\cdot\|_p$ and
$\|\cdot\|_{m,p}$, respectively. For example, $ \|\varphi\|_{L^p}=
  (\int_\Omega |\varphi|^p{\mathrm d}x)^{1/p}$ and $
\|\varphi\|_{m,p}:=(\sum_{|\beta|\leqslant m}
  \int_\Omega |D^\beta \varphi|^p{\mathrm d}x)^{1/p}$.
Especially, we denote $H^m(\Omega):=W^{m,2}(\Omega)$ and
$H_0^1(\Omega)$ the closure of $\{
\varphi\in\mathcal{C}_{0}^\infty(\Omega)\}$ with respect to
$H^1(\Omega)$ norm. Then, we introduce the following function
spaces:
\begin{itemize}
\item $\mathcal{V} :=\{ \varphi \in \mathcal{C}_0^\infty(\Omega)$

\item  $\mathcal{C}_0^\infty(\Omega)|\, \varphi=(\varphi_1, \varphi_2),
  \nabla \cdot \varphi=0 \}$,

\item  $H$ is the closure  of $\mathcal{V}$  in
$L^2(\Omega)\times L^2(\Omega)$  with the norm
$ \|\cdot\|_{H}$ and   dual  space $H^*$,

\item $V$ is the  closure  of $\mathcal{V}$
  in $H^1(\Omega)\times H^1(\Omega)$ with the norm
  $\|\cdot\|_V$  and dual space $V^*$,

\item $\widehat{H} := H \times L^2(\Omega)$
   with the norm $\|\cdot\|_{\widehat{H}}$  and  dual  space $\widehat{H}^*$,

\item $\widehat{V} := V \times H_0^1(\Omega)$
  with the norm $\|\cdot\|_{\widehat{V}}$   and dual space $\widehat{V}^*$,

\item $\mathcal {O}_\sigma(B) :=\{w\in\widehat{V} :
\inf_{v\in B} \|w-v\|_{\widehat V}<\sigma \}$.
\end{itemize}
Using the above notation, we further denote
\begin{itemize}
\item $L^p(I; X)$ is the space of strongly measurable functions on
the closed interval $I$, with values in a Banach space $X$, endowed with norm
\[
\|\varphi\|_{L^p(I; X)}
  :=(\int_I \|\varphi\|^p_X {\mathrm d}t)^{1/p},\quad \text{for }1\leqslant
p<\infty;
\]

\item $\mathcal{C}(I; X)$ is the space of continuous functions on the interval
$I$, with values in the Banach space $X$, endowed with
  the usual norm;

\item $L_{\mathrm{loc}}^2(I; X)$ is the space of locally square integrable
functions on the interval $I$, with values
in the Banach space $X$, endowed with the usual norm;


\item $L_{b}^2(I; X)$ is the set of functions
$F\in L_{\mathrm{loc}}^2(I;X)$ satisfying
\[
 \|F\|_{L_b^2(I;X)}=\sup_{\tau\in\mathbb{R}}\int_{\tau}^{\tau+1}
\|F(\zeta)\|^2d\zeta<+\infty.
\]
Here
\begin{gather*}
\|(u,v)\|_{H}:=(\|u\|_2^2+\|v\|_2^2)^{1/2},\quad
\|(u,v)\|_{V}:=(\|u\|^2_{H^1}+\|v\|^2_{H^1})^{1/2},\\
\|(u,v,w)\|_{\widehat H}:=(\|(u,v)\|^2_H+\|w\|_2^2)^{1/2},\quad
\|(u,v,w)\|_{\widehat V}:=(\|(u,v)\|^2_{V}+\|w\|^2_{H^1})^{1/2}.
\end{gather*}
\end{itemize}
Subsequently, we simplify  $\|\cdot\|_{2}$,
$\|\cdot\|_{H}$ and $\|\cdot\|_{\widehat{H}}$ by the same notation
$\|\cdot\|$ if there is no confusion. In addition, we denote by
$(\cdot,\cdot)$ the inner product in $L^2(\Omega), H$ or
$\widehat{H}$, and $\langle \cdot, \cdot \rangle $ the dual pairing
between $V$ and $V^*$ or between $\widehat{V}$ and $\widehat{V}^*$.
We also denote the compact embedding between spaces by
$\hookrightarrow\hookrightarrow$, and use $\rm{dist}_M(X,Y)$ to
represent the Hausdorff semidistance between $X \subseteq M $ and $Y
\subseteq M $ with $\rm{dist}_M(X,Y)=\sup_{x\in X}
\inf_{y\in Y} \operatorname{dist}_M(x,y)$.

Now, we state the first results of this paper in the following theorem.

\begin{theorem}\label{T1.1}
Assume $ F(t,x)\in L_b^2(\mathbb{R}; \widehat{H})$,
$\int_{-\infty}^t e^{\frac{\delta_1}{2}s} \|F(s)\|^2 {\rm d}s < \infty$, for all
$ t\in \mathbb{R}$ and
\begin{align*}
\lim_{r\to +\infty} \int_{-\infty}^{t} \int_{|x|\geqslant r}
 e^{\frac{\delta_1}{2}s} |F(s,x)|^2 {\rm d}x{\rm d}s =0.
\end{align*}
Then  system \eqref{1.6} possesses a unique pullback $\mathcal{D}$-attractor
 $\mathcal{A}_{\widehat H}(t)$ in $\widehat{H}$.
\end{theorem}

\begin{remark}\label{R1.1} \rm
In fact, we point out that the pullback $\mathcal{D}$-attractor
$\mathcal{A}_{\widehat V}(t)$ in $\widehat{V}$ can be obtained by
using similar proof as that in $\widehat{H}$. Specifically, based on
Lemma \ref{L2.3}(2), there exists a continuous process
$\{U(t,\tau)\}_{t\geqslant\tau}$ in $\widehat{V}$. Applying energy
method, we can prove the existence of pullback
$\mathcal{D}$-absorbing set for the process
$\{U(t,\tau)\}_{t\geqslant\tau}$ in $\widehat{V}$. Then, the
technique of truncation functions and decomposition of spatial
domain enable us to obtain the uniform a priori estimates for the
far-field values of solutions. Further, we can show the pullback
asymptotic compactness of the process in $\widehat{V}$. Finally, we
can obtain the existence of pullback $\mathcal{D}$-attractor
$\mathcal{A}_{\widehat V}(t)$ in $\widehat{V}$.
\end{remark}

Based on Theorem \ref{T1.1}, we further verify the tempered behavior
and upper semicontinuity of the pullback attractor obtained in
Theorem \ref{T1.1}. That is the following two theorems.

\begin{theorem}\label{T1.2}
Under the conditions of Theorem \ref{T1.1}, it holds that
\begin{gather}
\lim_{t\to -\infty}
 \big(e^{\frac{\delta_1}{2}t}
  \sup_{w\in\mathcal{A}_{\widehat H}(t)}\|w\|^2\big)
= 0, \label{1.7} \\
\lim_{t\to -\infty}
 \big(e^{\frac{\delta_1}{2}t}
  \sup_{w\in\mathcal{A}_{\widehat H}(t)}\|w\|_{\widehat{V}(\Omega)}^2\big)
= 0. \label{1.8}
\end{gather}
\end{theorem}

\begin{theorem}\label{T1.3}
Assume the conditions of Theorem \ref{T1.1} hold. Then for any
$t\in\mathbb{R}$, we have
\begin{equation} \label{1.9}
\lim_{m\to\infty}\operatorname{dist}_{\widehat{H}(\Omega)}
 (\mathcal{A}_{\widehat{H}(\Omega_m)}(t),\mathcal{A}_{\widehat{H}(\Omega)}(t))
= 0,
\end{equation}
where $\mathcal{A}_{\widehat{H}(\Omega)}(t)$ and
$\mathcal{A}_{\widehat{H}(\Omega_m)}(t)$ are the pullback attractors
of system \eqref{1.6} and system \eqref{4.1}, respectively.
\end{theorem}

\begin{remark} \rm
In \cite{ZZL08}, the authors proved the existence of $H^1$-uniform
attractor and pointed out that the $L^2$-uniform attractor belongs
to the $H^1$-uniform attractor, under the proper conditions of
$F=(f, \tilde{f})$. Here, we established the existence of pullback
attractor in $\widehat{H}(\Omega)$ (has $L^2$-regularity),
similarly, in $\widehat{V}(\Omega)$ (has $H^1$-regularity). Further,
we investigated the tempered behavior and upper semicontinuity (in
the sense of (1.9)) of the pullback attractors. About the
differences and more details  between pullback attractor and uniform
attractor, we can refer to \cite{LRSV07} and some references
therein.
\end{remark}

The outline of proofs for Theorem \ref{T1.1}--\ref{T1.3} is as follows.
First, we construct the continuous process $\{U(t,\tau)\}_{t\geqslant\tau}$ by
solution maps in the space $\widehat{H}(\Omega)$, then show the
asymptotic compactness of the process
$\{U(t,\tau)\}_{t\geqslant\tau}$ in space $\widehat{H}(\Omega)$. The
main difficulty comes from two aspects. The first one is that the
usual Sobolev embedding is no longer compact in unbounded domain
$\Omega$. The other is from the angular velocity field $\omega$ of
the micropolar particles in the micropolar fluid flows, which leads
to a different nonlinear term $B(u,w)$ and an additional term $N(u)$
in the abstract equation (see \eqref{15}). To overcome the first
difficulty, borrowing the arguments and ideas in \cite{Z12}, we show
the existence of the pullback absorbing set by establishing some a
priori estimates of the solutions. Further, we use the technique of
truncation function and the decomposition of spatial domain, to
prove the asymptotic compactness of the process
$\{U(t,\tau)\}_{t\geqslant\tau}$ in space $\widehat{H}(\Omega)$. To
deal with the second difficulty, more delicate estimates and
analysis for the solutions are required in our study. Next, using
the arguments in \cite{HXR88,LW05,ZD12,Z12}, we can show the
tempered behavior and upper semicontinuity of the pullback attractor
$\mathcal{A}_{\widehat{H}(\Omega)}$. The tempered behavior of the
pullback attractor $\mathcal{A}_{\widehat{H}(\Omega)}$ is relatively
easy to obtain. To prove the upper semicontinuity of the attractor,
we first let $\{\Omega_m\}_{m=1}^{\infty}$ be an expanding sequence
of simply connected, bounded and smooth subdomains of $\Omega$ such
that $\cup_{m=1}^{\infty}\Omega_m = \Omega$. Then we
consider the Cauchy problem \eqref{1.2}-\eqref{1.4} in $\Omega_m$.
We will conclude that there exists a pullback attractor
$\mathcal{A}_{\widehat{H}(\Omega_m)}$ for the problem
\eqref{1.2}-\eqref{1.4} in each $\Omega_m$. Finally, we establish
the upper semicontinuity by showing
$\lim_{m\to\infty} \operatorname{dist}_{\widehat{H}(\Omega)}
(\mathcal{A}_{\widehat{H}(\Omega_m)}(t),
\mathcal{A}_{\widehat{H}(\Omega)}(t))=0, \ \forall\,
t\in\mathbb{R}$.

The rest of this paper is organized as follows. In section 2, we
make some preliminaries. That is, we introduce several important
definitions and recall some known results of non-autonomous
micropolar fluid flows. Section 3 is committed to the proof of
Theorem \ref{T1.1}, that is to prove the existence of pullback
attractors in $\widehat{H}(\Omega)$. Further, the tempered behavior
and upper semicontinuity of the pullback attractors, i.e. Theorem
\ref{T1.2} and Theorem \ref{1.3}, will be verified in section 4.

\section{Preliminaries}

In this section, we first make some necessary preliminaries. That
is, we first give some useful properties and estimates about those
operators \eqref{15}. Then, we give some definitions and recall some
key results for the non-autonomous micropolar fluid model. To begin
with, we have

\begin{lemma} \label{L2.1}
 The operator $A$ is a linear continuous operator both from $V$ to $V^*$
and from $D(A):=V\cap \left(H^2(\Omega)\right)^3$ to $H$. Indeed,
$A=-\mathbb{P}\Delta$, where $\mathbb{P}$ is the Leray projector from
$\mathbb{L}^2(\Omega)$ to $H$.  The operator $B(\cdot,\cdot)$ is continuous from
$V\times V$ to $V^*$. Moreover, for any $u \in V, w \in V$, it holds
\begin{equation} \label{2.1}
\langle B(u,w),\varphi \rangle = - \langle B(u,\varphi),w\rangle.
\end{equation}
\end{lemma}

\begin{proof}
The linearity and continuity of the operator $A$ can be deduced directly
from its definition. Similarly, the continuity of the operator
$B(\cdot,\cdot)$ can be obtained easily from its definition. We only need
to verify \eqref{2.1}. In fact, for any $u \in V, w \in \widehat{V}$, we have
\begin{equation} \label{2.2}
\begin{aligned}
\langle B(u,w),w \rangle &=((u \cdot \nabla)w, w) \\
&= \int_{\Omega}(u_1 \frac{\partial}{\partial x_1} + u_2
  \frac{\partial}{\partial x_2}
  (w_1, w_2, w_3)(w_1, w_2, w_3)\mathrm{d}x   \\
&=\sum_{j=1}^{3} \sum_{i=1}^{2}
  \int_{\Omega}u_i\frac{\partial w_j}{\partial x_i}w_j \mathrm{d}x\\
&=\sum_{j=1}^{3} \sum_{i=1}^{2}
  \frac 12 \int_{\Omega}u_i\frac{\partial w_j^2}{\partial x_i} \mathrm{d}x \\
&=\frac 12 \sum_{j=1}^{3}
\sum_{i=1}^{2}
  (u_i w_j^2|_{\partial \Omega}-\int_{\Omega}w_j^2 D_i u_i \mathrm{d}x)
      \\
&=- \frac 12 \sum_{j=1}^{3}
\sum_{i=1}^{2}
  \int_{\Omega}w_j^2 D_i u_i \mathrm{d}x \\
&=-\frac 12 \sum_{j=1}^{3}
  \int_{\Omega}w_j^2(\nabla\cdot u) \mathrm{d}x =0.
\end{aligned}
\end{equation}
Hence, \eqref{2.1} is valid as a consequence of \eqref{2.2}.
The proof is complete.
\end{proof}

About the operators $A(w)$ and $N(w)$, we have the following result.

\begin{lemma}[see \cite{L01,ZZL08,ZSH15}] \label{L2.2}
{\rm (1)}
   There are two positive constants $c_1$ and $c_2$    such that
\begin{equation}  \label{2.3}
  c_1\langle Aw, w \rangle
\leqslant    \|w\|_{\widehat V}^2
 \leqslant    c_2\langle Aw, w \rangle, \quad  \forall w\in {\widehat V}.
\end{equation}
{\rm (2)} There exists a positive constant $c(\nu_{\theta})$ such that
\begin{gather}
   \|N(w)\| \leqslant
   c(\nu_{\theta})\|w\|_{\widehat {V}}, \enskip \forall w\in\widehat {V},
   \label{2.4}\\
 \langle Aw,w \rangle
  + \langle N(w),w \rangle
\geqslant
    \delta_1 \|w\|_{\widehat V}^2, \enskip \forall w\in\widehat {V},\label{2.5}
\end{gather}
where $\delta_1 :=\min \{ \nu,\alpha \}$.
\end{lemma}

Next, we give the definition and existence result of weak solutions
for \eqref{1.6}.

\begin{definition}\label{D2.1} \rm
For each $T>\tau, \tau\in\mathbb{R}$, function $w$ is called a weak solution
of \eqref{1.6} if,
$w=(u,\omega)\in L^2(\tau,T; \widehat{V})\cap L^{\infty}(\tau,T; \widehat{H})$
such that for $t\in (\tau, T)$ and any $\varphi\in\widehat{V}$,
\begin{equation} \label{2.6}
\begin{gathered}
  \frac{\rm d}{{\rm d}t} (w(t), \varphi)
   + \langle Aw(t), \varphi \rangle
   + \langle B(u(t),w(t)), \varphi \rangle
   + \langle N(w(t)), \varphi \rangle
   = \langle F(t,x), \varphi \rangle,  \\
 w\big|_{t=\tau} = w_{\tau} = (u_{\tau}, \omega_{\tau}) = w_0
 \end{gathered}
\end{equation}
holds in the sense of $\mathcal{D}'(\tau,T)$.
\end{definition}

Using decomposition of spatial domain and the Galerkin method, we immediately
 have the following well-posedness of solutions to \eqref{1.6} on unbounded
domain $\Omega$.

\begin{lemma}\label{L2.3}
Assume $F(t,x)=(f(t,x),\tilde{f}(t,x)) \in L_b^2(\mathbb {R}; \widehat {H}(\Omega))$.
\begin{enumerate}
\item[(1)]
  If $w_\tau \in \widehat {H}$, then system \eqref{1.6}
has a unique solution $w=(u,\omega)$ satisfying
$$
  w\in L^\infty( \tau, +\infty; \widehat{H} )
  \cap \mathcal {C}( [\tau, +\infty);\widehat{H})
  \cap L_{\rm loc}^2( \tau,+\infty;\widehat{V} ), \,
  w'\in L_{\rm loc}^2
  ( \tau, +\infty; \widehat{V}^* ).
$$

\item[(2)] If $w_\tau \in \widehat {V}$, then problem
\eqref{1.6} has a unique solution $w=(u,\omega)$ satisfying
$$
  w\in L^\infty ( \tau, +\infty; \widehat{V} )
  \cap\mathcal {C}( [\tau, +\infty);\widehat{V} )
  \cap L_{\rm loc}^2( \tau, +\infty; D(A) ), \,
  w' \in L_{\rm loc}^2( \tau,+\infty; \widehat{H}).
$$
\end{enumerate}
\end{lemma}

Since the proof is standard, we omit it.
The interesting readers are referred to \cite{LS04}.

On the basis of Lemma \ref{L2.3}, the biparametric map defined by
\begin{align}\label{2.7}
U(t,\tau) : w_{\tau}\mapsto U(t,\tau)w_{\tau}=w(t), \ t\geqslant\tau,\,
 w_{\tau}\in\widehat{H}(\Omega)\text{ or } w_{\tau}\in\widehat{V}(\Omega),
\end{align}
generates a continuous process $\{U(t,\tau)\}_{t\geqslant\tau}$ in
$\widehat{H}(\Omega)$ or $\widehat{V}(\Omega)$, which satisfies the
following properties:
\begin{itemize}
\item[(i)] $U(\tau, \tau)w_{\tau} = w_{\tau}$,

\item[(ii)]  $U(t,s)U(s,\tau)w_{\tau} = U(t,\tau)w_{\tau} = w(t)$.
\end{itemize}
Finally, we introduce some definitions related to the pullback attractor
(see \cite{GMR12,SW07,MR09,ZLW14}). For convenience, we denote by $X$ the
space $\widehat{H}$ or $\widehat{V}$ and by $\mathcal{P}(X)$ the family
of all nonempty subsets of $X$. A universe $\mathcal{D}(X)$ in
$\mathcal{P}(X)$ represents the class of families parameterized in time
$\widehat{B}(X)= \{B(t)\big| t\in\mathbb{R}\}\subseteq \mathcal{P}(X)$.

\begin{definition}\label{D2.2} \rm
  A family of sets $\widehat{B}_0=\{B_0(t): t\in {\mathbb R}\} \subseteq \mathcal
 {P}(X)$ is called pullback $\mathcal{D}$-absorbing for the process
 $\{U(t,\tau)\}_{t\geqslant \tau}$
 in $X$ if for any $t\in{\mathbb R}$ and any
 $\widehat{B}=\{B(t): t\in {\mathbb R}\}\in \mathcal
 {D}$, there exists a $\tau_0(t, \widehat {B})\leqslant t$ such that
  $U(t,\tau)B(\tau)\subseteq B_0(t)$ for all $\tau\leqslant
  \tau_0(t,\widehat {B})$.
\end{definition}

\begin{definition}\label{D2.3} \rm
 The process $\{U(t,\tau)\}_{t\geqslant \tau}$ is said to be pullback
$\widehat{B}_0$-asymptotically compact in $X$ if for any
$t\in {\mathbb R}$, any sequences $\{\tau_n\}\subseteq(-\infty, t]$ and
$\{x_n\}\subseteq X$ satisfying $\tau_n\to -\infty$ as
$n\to \infty $ and $x_n\in B_0(\tau_n)$ for all $n$, the
sequence $\{ U(t, \tau_n;x_n)\}$ is relatively compact in $X$.
$\{U(t,\tau)\}_{t\geqslant \tau}$ is called pullback
$\mathcal {D}$-asymptotically compact in $X$ if it is pullback
$\widehat{B}$-asymptotically compact for any $\widehat{B} \in
\mathcal {D} $.
\end{definition}

\begin{definition}\label{D2.4} \rm
 A family of sets $\mathcal {A}_X =\{ \mathcal {A}_X(t):
 t \in{\mathbb R} \} \subseteq \mathcal {P} (X)$ is called a pullback
$\mathcal {D}$-attractor for the process $\{U(t,\tau)\}_{t\geqslant \tau}$ on
$X$ if it has the following properties:
\begin{itemize}
\item   Compactness: for any $t\in {\mathbb R}, \mathcal {A}_X(t)$ is a nonempty
compact subset of $X$;

\item  Invariance: $U(t, \tau)\mathcal {A}_X(\tau)
  =\mathcal {A}_X(t), \, \forall\,t\geqslant \tau$;

\item  Pullback attracting: $\mathcal {A}_X$ is pullback
  $\mathcal {D}$-attracting in the following sense:
$$
  \lim_{\tau\to -\infty} \operatorname{dist}_X\left( U(t, \tau)B(\tau),
  \mathcal {A}_X(t) \right)=0, \quad
   \forall \widehat{B}=\{ B(s)|\; s\in {\mathbb R}\} \in \mathcal {D},t\in{\mathbb
   R};
$$

\item  Minimality: the family of sets $\mathcal{A}_X$ is the minimal in
the sense that if
 $\widehat {O}=\{ O(t): t\in  {\mathbb R}\} \subseteq \mathcal
{P}(X)$ is another family of closed sets such that
$$
 \lim_{\tau\to -\infty} \operatorname{dist}_X(U(t, \tau)B(\tau),
  O(t))=0 ,\ \ \text{for any }\widehat {B}=
\{ B(t)|\, t\in {\mathbb R}\} \in \mathcal{D},
$$
then $\mathcal {A}_X(t)\subseteq O(t)$ for $t\in
\mathbb{R}$.
\end{itemize}
\end{definition}

\section{Existence of the pullback $\mathcal{D}$-attractor in $\widehat{H}$}

In this section, we are devoted to proving the existence of the pullback
attractor in $\widehat{H}$, i.e. Theorem \ref{T1.1}. Throughout this section,
we simply denote $w(t; \tau,w_{\tau})$ by $w(t)$ if there is no confusion.
First, we have

\begin{lemma}\label{L3.1}
Assume $F(t,x)\in L_b^2(\mathbb{R}; \widehat{H})$, then for any
$w_{\tau}=(u_{\tau},\omega_{\tau})\in \widehat{H}$, it holds that
\begin{gather}
\|w(t;\tau,w_{\tau})\|^2
\leqslant  e^{\frac{\delta_1}{2}(\tau-t)} \|w_{\tau}\|^2
 + \frac{e^{-\frac{\delta_1}{2}t}}{\delta_1}
\int_{\tau}^t e^{\frac{\delta_1}{2}s} \|F(s)\|^2 {\rm d}s, \label{3.1} \\
 \int_{\tau}^t
 e^{\frac{\delta_1}{2}s} \|w(s)\|_{\widehat V}^2 {\rm d}s
 \leqslant
  \frac{2}{\delta_1}e^{\frac{\delta_1}{2}\tau}\|w_{\tau}\|^2
  + \frac{2}{\delta_1^2} \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|F(s)\|^2 {\rm d}s.
\label{3.2}
\end{gather}
\end{lemma}

\begin{proof}
Multiplying \eqref{1.6}$_{1}$ by $w(t)$, we obtain from \eqref{2.2} and \eqref{2.5}
that
\begin{align*}
\frac 12 \frac{\rm d}{{\rm d}t}\|w(t)\|^2
 + \delta_1 \|w(t)\|_{\widehat V}^2
 \leqslant   (F(t,x),w(t))
 \leqslant   \frac{1}{2 \delta_1} \|F(t)\|^2
  + \frac{\delta_1}{2} \|w(t)\|_{\widehat V}^2,
\end{align*}
which implies
\begin{align}\label{3.3}
\frac{\rm d}{{\rm d}t}\|w(t)\|^2 + \delta_1\|w(t)\|^2
\leqslant
 \frac{\rm d}{{\rm d}t} \|w(t)\|^2
 + \delta_1 \|w(t)\|_{\widehat V}^2
\leqslant
 \frac{1}{\delta_1}\|F(t)\|^2.
\end{align}
Further, we have
\[
\frac{\rm d}{{\rm d}t} (e^{\frac{\delta_1t}{2}}\|w(t)\|^2)
\leqslant
 \frac{e^{\frac{\delta_1t}{2}}}{\delta_1} \|F(t,x)\|^2.
\]
Changing the time variable $t$ by $s$ and integrating it over $[\tau, t]$, we obtain
\begin{align*}
e^{\frac{\delta_1t}{2}} \|w(t)\|^2
\leqslant
 e^{\frac{\delta_1\tau}{2}} \|w_{\tau}\|^2
 + \frac{1}{\delta_1} \int_{\tau}^t
   e^{\frac{\delta_1}{2}s} \|F(s)\|^2 {\rm d}s,
\end{align*}
hence
\begin{align}\label{3.4}
\|w(t)\|^2
\leqslant  e^{\frac{\delta_1(\tau-t)}{2}} \|w_{\tau}\|^2
 + \frac{e^{-\frac{\delta_1t}{2}}}{\delta_1}
   \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|F(s)\|^2 {\rm d}s.
\end{align}

From \eqref{3.3}, we can also deduce that
\begin{align*}
\frac{\rm d}{{\rm d}t} (e^{\frac{\delta_1}{2}t}\|w(t)\|^2)
 + \frac{\delta_1e^{\frac{\delta_1}{2}t}}{2} \|w(t)\|_{\widehat V}^2
\leqslant
 \frac{e^{\frac{\delta_1}{2}t}}{\delta_1} \|F(t)\|^2.
\end{align*}
Further, one have
\[
e^{\frac{\delta_1}{2}t} \|w(t)\|^2
 + \frac{\delta_1}{2} \int_{\tau}^t
   e^{\frac{\delta_1}{2}s}\|w(s)\|_{\widehat V}^2 {\rm d}s
\leqslant
 \frac{1}{\delta_1}\int_{\tau}^t e^{\frac{\delta_1}{2}s}\|F(s)\|^2 {\rm d}s
 + e^{\frac{\delta_1}{2}\tau} \|w_{\tau}\|^2,
\]
which implies
\begin{align*}
\int_{\tau}^t e^{\frac{\delta_1}{2}s}\|w(s)\|_{\widehat V}^2 {\rm d}s
\leqslant
 \frac{2}{\delta_1}e^{\frac{\delta_1}{2}\tau} \|w_{\tau}\|^2
 + \frac{2}{\delta_1^2} \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|F(s)\|^2
   {\rm d}s.
\end{align*}
This completes the proof.
\end{proof}
Set
\begin{equation} \label{3.5}
\mathcal{R}_{\frac{\delta_1}{2}}
 := \{\rho(s): \mathbb{R}\to\mathbb{R}_+ :
 \lim_{s\to{-\infty}} e^{\frac{\delta_1}{2}s}\rho^2(s)=0\}.
\end{equation}
$\mathcal{D}_{\frac{\delta_1}{2}}(\widehat H)$ denotes the class of families
$\widehat{B} = \{B(s)\big| s\in\mathbb{R}\}\subseteq \mathcal{P}(\widehat{H})$
such that
\begin{align*}
 B(s)\subseteq {\bar{\mathcal{B}}(0,\rho_{\widehat B}(s))} \text{ for some }
 \rho_{\widehat B}(s)\in \mathcal{R}_{\frac{\delta_1}{2}},
\end{align*}
where $\bar{\mathcal{B}}(0, \rho_{\widehat B}(s))$ is the closed ball in
$\widehat{H}$ with center 0 and radius $\rho_{\widehat B}(s)$.
Then, based on Lemma \ref{L3.1}, there exists a pullback $\mathcal{D}$-absorbing
set in $\widehat{H}$.

\begin{lemma}\label{L3.2}
Under the condition of Lemma \ref{L3.1}. For any $t\in\mathbb{R}$,
$\widehat{B} = \{B(s)\big| s\in\mathbb{R}\}\in
\mathcal{D}_{\frac{\delta_1}{2}}(\widehat H)$ and $w_{\tau} \in B(\tau)$,
the family $\widehat{\mathcal{B}} = \{\mathcal{B}(t)\big| t\in\mathbb{R}\}$
defined by
\begin{align}\label{3.6}
\mathcal{B}(t) = \{ w\in\widehat{H}: \|w\|\leqslant \rho(t)\}
\end{align}
is pullback $\mathcal{D}$-absorbing in $\widehat{H}$, where
\begin{align}\label{3.7}
\rho^2(t) := \frac{2e^{-\frac{\delta_1}{2}t}}{\delta_1}
 \int_{-\infty}^t e^{\frac{\delta_1}{2}s} \|F(s)\|^2 {\rm d}s.
\end{align}
\end{lemma}

In the following, we focus on proving the pullback $\mathcal{D}$-asymptotical
compactness of the process $\{U(t,\tau)\}_{t\geqslant\tau}$. First, we have

\begin{lemma}\label{L3.3}
Under the conditions of Theorem \ref{T1.1}. For any $\epsilon>0$, $t\in\mathbb{R}$
and $\widehat{B} = \{B(s) \big| s\in \mathbb{R}\}\in
\mathcal{D}_{\frac{\delta_1}{2}}(\widehat{H})$, there exist
 $r_0:=r_0(\epsilon, t, \widehat{B})> 0$ and
$\tau_0 := \tau_0(\epsilon, t, \widehat{B}) < t$ such that for any
$r\geqslant r_0, \tau\leqslant \tau_0$ and $w_{\tau}\in B(\tau)$, it holds
\begin{equation} \label{3.8}
\|w(t; \tau,w_{\tau})\|_{\mathbb{L}^2(\Omega\setminus\Omega_r)}^2
\leqslant \epsilon,
\end{equation}
where $\Omega_r = \{x\in\Omega \big| |x|<r\}$.
\end{lemma}

\begin{proof}
First, we take a function $\chi(\cdot)\in C^2(\mathbb{R}^2)$,
$\chi(x)\in[0,1]$  for all $x\in\mathbb{R}^2$ such that
\[
\chi(x) =
\begin{cases}
 0, & |x|\leqslant 1,  \\
 1, & |x|\geqslant 2,
 \end{cases}
\]
and
\[
\|\nabla \chi(x)\|_{\mathbb{L}^{\infty}(\mathbb{R}^2)}
\leqslant
 c_0, \quad
\|D^2 \chi(x)\|_{\mathbb{L}^{\infty}(\mathbb{R}^2)}
\leqslant
 c_0,
\]
where $c_0$ is a constant.
In particular, set $\chi_r(x)=\chi(\frac{x}{r})$ with $r\geqslant 1$, we have
\begin{equation} \label{3.9}
\|\nabla \chi_r(x)\|_{\mathbb{L}^{\infty}(\mathbb{R}^2)}
\leqslant
\frac{c_0}{r}, \quad
\|D^2 \chi_r(x)\|_{\mathbb{L}^{\infty}(\mathbb{R}^2)}
\leqslant
\frac{c_0}{r^2}.
\end{equation}

Then, taking the inner product of $\eqref{1.6}_{1}$ with
$\chi_r^2 w = (\chi_r^2u, \chi_r^2\omega)$ and considering the term
$\nabla p$ yield
\begin{align}
&\frac 12 \frac{\rm d}{{\rm d}t} \|\chi_r w\|^2
 + (\nu+\nu_{\theta})(\nabla u, \nabla (\chi_r^2 u))
 + \alpha(\nabla\omega, \nabla(\chi_r^2\omega))
 + ((u\cdot\nabla)w, \chi_r^2 w) \nonumber \\
&  + ((-2\nu_{\theta}\nabla\times\omega, -2\nu_{\theta}\nabla\times u
     +4\nu_{\theta} \omega), (\chi_r^2 u, \chi_r^2\omega))
 + (\nabla p, \chi_r^2 u)  \label{3.10}\\
&=  \frac 12 \frac{\rm d}{{\rm d}t} \|\chi_r w\|^2
  + \langle A(\chi_r w), \chi_r w \rangle
  - (\nu+\nu_{\theta}) \int_{\Omega} |u \nabla\chi_r|^2 {\rm d}x
  - \alpha \int_{\Omega} |\omega \nabla\chi_r|^2 {\rm d}x \nonumber \\
&\quad + ((u\cdot \nabla)w, \chi_r^2 w)
  + \langle N(\chi_r w), \chi_r w \rangle
  + (\nabla p, \chi_r^2 u) \nonumber\\
&=  (F(t,x), \chi_r^2w). \nonumber
\end{align}
Let us estimate the terms in the above equality one by one.
From \eqref{3.9} and H\"older inequality, we have
\begin{equation} \label{3.11}
\begin{aligned}
(\nu+\nu_{\theta})\int_{\Omega} |u \nabla\chi_r|^2 {\rm d}x
&\leqslant
  (\nu+\nu_{\theta})\|\nabla\chi_r\|_{\mathbb{L}^{\infty}(\Omega)}^2 \|u\|^2\\
&\leqslant
  c_0^2(\nu+\nu_{\theta})r^{-2} \|u\|^2.
\end{aligned}
\end{equation}
Similarly, we have
\begin{equation} \label{3.12}
\alpha\int_{\Omega} |\omega \nabla\chi_r|^2 {\rm d}x
 \leqslant
  \alpha\|\nabla\chi_r\|_{\mathbb{L}^{\infty}(\Omega)}^2 \|\omega\|^2
 \leqslant
  c_0^2\alpha r^{-2} \|\omega\|^2.
\end{equation}
Integrating by parts and using that $\nabla\cdot u=0$, we obtain
\begin{align*}
&((u\cdot\nabla)w, \chi_r^2w)
= \sum_{i,j=1}^2
  \int_{\Omega} u_i\frac{\partial u_j}{\partial x_i}\chi_r^2u_j {\rm d}x
  + \sum_{i=1}^2
    \int_{\Omega} u_i\frac{\partial\omega}{\partial x_i}\chi_r^2\omega{\rm d}x
    \\
&= -\sum_{i,j=1}^2
 (\int_{\Omega} u_i u_j\chi_r^2\frac{\partial u_j}{\partial x_i}{\rm d}x
  + 2\int_{\Omega}u_i u_j^2\chi_r\frac{\partial\chi_r}{\partial x_i}{\rm d}x)
    \\
&\quad - \sum_{i=1}^2
 (\int_{\Omega} u_i \omega\chi_r^2\frac{\partial\omega}{\partial x_i}{\rm d}x
  +2\int_{\Omega}u_i\omega^2\chi_r\frac{\partial\chi_r}{\partial x_i}{\rm d}x)
    \\
&= -((u\cdot\nabla)w,\chi_r^2 w)
  - 2\sum_{i,j=1}^2
   \int_{\Omega} u_i u_j^2\chi_r\frac{\partial\chi_r}{\partial x_i}{\rm d}x
  - 2\sum_{i=1}^2
   \int_{\Omega} u_i\omega^2\chi_r\frac{\partial\chi_r}{\partial x_i}{\rm d}x,
\end{align*}
which combine with \eqref{3.9}, H\"older inequality, Gagliardo-Nirenberg
inequality and Young's inequality yields
\begin{equation} \label{3.13}
\begin{aligned}
&|((u\cdot\nabla)w, \chi_r^2 w)|
 = \big| \sum_{i,j=1}^2
    \int_{\Omega} u_i u_j^2\chi_r \frac{\partial\chi_r}{\partial x_i}{\rm d}x
    + \sum_{i=1}^2 \int_{\Omega}
      u_i\omega^2\chi_r\frac{\partial\chi_r}{\partial x_i}{\rm d}x \big|
       \\
 \leqslant &
  \|\nabla\chi_r\|_{\mathbb{L}^{\infty}(\Omega)} \|u\|\|w\|_{\mathbb{L}^4(\Omega)}^2
 \leqslant
  c_0r^{-1}\|u\|\|w\|\|w\|_{\widehat V}
 \leqslant
  \frac{c_0}{r}(\|w\|^4 + \|w\|_{\widehat V}^2).
\end{aligned}
\end{equation}
Since $F(t,x)\in L_b^2(\mathbb{R}; \widehat{H})$, it follows from Lemma
\ref{L3.2} that for any $t\in\mathbb{R}, \tau\leqslant\tau_0$,
\begin{equation} \label{3.14}
\begin{aligned}
& \|w(t;\tau,w_{\tau})\|^2
\leqslant \rho^2(t) \\
& \leqslant \frac{2e^{-\frac{\delta_1}{2}t}}{\delta_1}
  \int_{-\infty}^t e^{\frac{\delta_1}{2}s}\|F(s)\|^2 {\rm d}s \\
&=  \frac{2}{\delta_1} \Big(\int_{t-1}^t
     e^{-\frac{\delta_1}{2}(t-s)}\|F(s)\|^2{\rm d}s
  + \int_{t-2}^{t-1} e^{-\frac{\delta_1}{2}(t-s)}\|F(s)\|^2 {\rm d}s
  + \dots\Big) \\
\leqslant &
 \frac{2}{\delta_1}\Big(\int_{t-1}^t \|F(s)\|^2{\rm d}s
  + e^{-\frac{\delta_1}{2}} \int_{t-2}^{t-1}\|F(s)\|^2{\rm d}s \\
 &\quad + e^{-\frac{2\delta_1}{2}} \int_{t-3}^{t-2}\|F(s)\|^2{\rm d}s
  + \dots\Big) \\
&\leqslant
 \frac{2}{\delta_1}(1+e^{-\frac{\delta_1}{2}}+e^{-\frac{2\delta_1}{2}}+\dots)
  \|F\|_{L_b^2(\mathbb{R}; \widehat{H})}^2
= \frac{2}{\delta_1(1-e^{-\frac{\delta_1}{2}})}
  \|F\|_{L_b^2(\mathbb{R};\widehat{H})}^2 \\
&\leqslant
  \frac{2}{\delta_1}(1+\frac{2}{\delta_1}) \|F\|_{L_b^2(\mathbb{R};\widehat{H})}^2.
\end{aligned}
\end{equation}
By \eqref{1.6}$_{3}$, \eqref{3.9} and the fact $\nabla\cdot u = 0$, we also have
\begin{equation} \label{3.15}
\begin{aligned}
|(\nabla p, \chi_r^2u)|
& = |\sum_{i=1}^2
  \int_{\Omega} \frac{\partial p}{\partial x_i}\chi_r^2 u_i{\rm d}x|
  = |\sum_{i=1}^2
     \int_{\Omega} 2p \chi_r\frac{\partial \chi_r}{\partial x_i}u_i {\rm d}x|
      \\
& \leqslant
  2\|p\| \|\nabla\chi_r\|_{L^{\infty}(\Omega)} \|\chi_r u\|
\leqslant
  2c_0r^{-1} \|p\| \| \chi_r u \|.
\end{aligned}
\end{equation}
Taking \eqref{2.5} and \eqref{3.10}-\eqref{3.15} into account, we obtain
\begin{equation} \label{3.16}
\begin{aligned}
&\frac 12 \frac{\rm d}{{\rm d}t} \|\chi_r w\|^2
 + \delta_1 \|\chi_r w\|_{\widehat V}^2 \\
&\leqslant
 \frac 12 \frac{\rm d}{{\rm d}t} \|\chi_r w\|^2
 + \langle A(\chi_r w), \chi_r w \rangle
 + \langle N(\chi_r w), \chi_r w \rangle  \\
&=  (\nu+\nu_{\theta}) \int_{\Omega} |u \nabla\chi_r|^2 {\rm d}x
 + \alpha \int_{\Omega} | \omega \nabla\chi_r|^2 {\rm d}x
 - ((u\cdot\nabla)w, \chi_r^2 w)\\
&\quad + (F(t,x), \chi_r^2 w)
 - (\nabla p, \chi_r^2 u)
  \\
&\leqslant   \frac{c_0^2(\nu+\nu_{\theta})}{r^2}\|u\|^2
 + \frac{c_0^2\alpha}{r^2}\|\omega\|^2
 + \frac{c_0}{r} (\|w\|^4 + \|w\|_{\widehat V}^2)
 + \|\chi_r F\| \|\chi_r w\| \\
&\quad + \frac{2c_0}{r}\|p\| \|\chi_r u\|   \\
&\leqslant
 \frac{c_0^2\cdot\max\{\nu+\nu_{\theta},\alpha\}}{r^2}\|w\|^2
 + \frac{4c_0}{r\delta_1^2}
   (1+\frac{2}{\delta_1})^2 \|F\|_{L_b^2(\mathbb{R};\widehat{H})}^4
 + \frac{c_0}{r}\|w\|_{\widehat V}^2 \\
&\quad + \frac{1}{\delta_1}\|\chi_r F\|^2
  + \frac{4c_0^2}{\delta_1r^2}\|p\|^2
 + \frac{\delta_1}{2}\|\chi_r w\|^2.
\end{aligned}
\end{equation}
Further, by Lemma \ref{L3.2} and that
$\|u\| \leqslant \|w\|$  and $\|w\| \leqslant \|w\|_{\widehat V}$,
one can deduce that there exist constants $c_3, c_4, c_5$ such that
\begin{align*}
&\frac{\rm d}{{\rm d}t}\|\chi_r w\|^2
 + \frac{\delta_1}{2} \|\chi_r w\|^2
 + \frac{\delta_1}{2} \|\chi_r w\|_{\widehat V}^2  \\
&\leqslant
 \frac{c_3}{r^2}\rho^2(t)
 + \frac{c_4}{r} \|F\|_{L_b^2(\mathbb{R};\widehat{H})}^4
 + \frac{c_0}{r} \|w\|_{\widehat V}^2
 + \frac{1}{\delta_1} \|\chi_r F\|^2
 + \frac{c_5}{r^2} \|p\|^2,
\end{align*}
which yields
\begin{align*}
\frac{\rm d}{{\rm d}t} (e^{\frac{\delta_1}{2}t}\|\chi_r w\|^2)
&\leqslant
 \frac{c_3}{r^2} e^{\frac{\delta_1}{2}t}\rho^2(t)
 + \frac{c_4}{r}
   e^{\frac{\delta_1}{2}t}\|F\|_{L_b^2(\mathbb{R};\widehat{H})}^4
 + \frac{c_0}{r} e^{\frac{\delta_1}{2}t}\|w\|_{\widehat V}^2 \\
&\quad + \frac{c_5}{r^2} e^{\frac{\delta_1}{2}t}\|p\|^2
 + \frac{1}{\delta_1} e^{\frac{\delta_1}{2}t}\|\chi_r F\|^2.
\end{align*}
Hence,
\begin{equation}\label{3.17}
\begin{aligned}
&\|\chi_r w(t)\|^2 \\
&\leqslant
 e^{-\frac{\delta_1}{2}(t-\tau)} \|\chi_r w_{\tau}\|^2
 + \frac{c_3}{r^2} e^{-\frac{\delta_1}{2}t}
   \int_{\tau}^t e^{\frac{\delta_1}{2}s}\rho^2(s) {\rm d}s  \\
&\quad + \frac{c_4}{r} e^{-\frac{\delta_1}{2}t} \int_{\tau}^t
   e^{\frac{\delta_1}{2}s}\|F\|_{L_b^2(\mathbb{R};\widehat{H})}^4 {\rm d}s
 + \frac{c_0}{r} e^{-\frac{\delta_1}{2}t}
   \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|w\|_{\widehat V}^2 {\rm d}s
     \\
&\quad + \frac{c_5}{r^2} e^{-\frac{\delta_1}{2}t}
   \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|p\|^2 {\rm d}s
 + \frac{1}{\delta_1} e^{-\frac{\delta_1}{2}t}
   \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|\chi_r F\|^2 {\rm d}s.
\end{aligned}
\end{equation}
In the following, we make a more detailed detailed estimate for each
term in \eqref{3.17}. First, for any $\epsilon>0$, there exists a
$\tau_1=\tau_1(\epsilon,t,\widehat{D})$ such that
\begin{equation} \label{3.18}
e^{-\frac{\delta_1}{2}(t-\tau)} \|\chi_r w_{\tau}\|^2
\leqslant
 e^{-\frac{\delta_1}{2}(t-\tau)} \|w_{\tau}\|^2
< \frac{\epsilon}{6} \quad \text{for all }  \tau\leqslant\tau_1.
\end{equation}
Then, from the condition
\[
\int_{-\infty}^t e^{\frac{\delta_1}{2}s} \|F(s)\|^2 {\rm d}s < \infty,
\quad \forall t\in \mathbb{R},
\]
it is not difficult to check that there exists a
$r_1=r_1(\epsilon,t,\widehat{D})$ such that for any $r\geqslant r_1$,
\begin{gather}
\frac{c_3}{r^2} e^{-\frac{\delta_1}{2}t}
 \int_{\tau}^t e^{\frac{\delta_1}{2}s} \rho^2(s) {\rm d}s
< \frac{\epsilon}{6},  \label{3.19} \\
\frac{c_4}{r} e^{-\frac{\delta_1}{2}t}
 \int_{\tau}^t e^{\frac{\delta_1}{2}s}
    \|F\|_{L_b^2(\mathbb{R}; \widehat{H})}^4 {\rm d}s
< \frac{\epsilon}{6},  \label{3.20} \\
\label{3.21}
\frac{1}{\delta_1} e^{-\frac{\delta_1}{2}t}
 \int_{\tau}^t e^{\frac{\delta_1}{2}s} \|\chi_r F\|^2 {\rm d}s
\leqslant
 \frac{1}{\delta_1} e^{-\frac{\delta_1}{2}t}
 \int_{-\infty}^t \int_{|x|\geqslant r} e^{\frac{\delta_1}{2}s} |F(s,x)|^2
  {\rm d}x{\rm d}s
< \frac{\epsilon}{6}.
\end{gather}
Moreover, it follows from \eqref{1.2} that
$\nabla p\in L_{\rm loc}^2(\tau,+\infty; H^{-1}(\Omega))$,
which implies $p\in L_{\rm loc}^2(\tau,+\infty; L^2(\Omega))$.
In addition, noting that
\[
\int_{\tau}^t e^{\frac{\delta_1}{2}s} \|p(s)\|^2 {\rm d}s
\leqslant
 c \int_{\tau}^t e^{\frac{\delta_1}{2}s} \|w(s)\|_{\widehat V}^2 {\rm d}s,
\]
and using \eqref{3.2} and the condition
\begin{align*}
\lim_{r\to +\infty} \int_{-\infty}^{t} \int_{|x|\geqslant r}
 e^{\frac{\delta_1}{2}s} |F(s,x)|^2 {\rm d}x{\rm d}s =0,
\end{align*}
we obtain that there exists
$r_2= r_2(\epsilon,t,\widehat{D})$ such that for any $r\geqslant r_2$,
it holds that
\begin{equation} \label{3.22}
\frac{c_5}{r^2} e^{-\frac{\delta_1}{2}t}
 \int_{\tau}^t  e^{\frac{\delta_1}{2}s}\|p\|^2 {\rm d}s
\leqslant
 \frac{cc_5}{r^2} e^{-\frac{\delta_1}{2}t}
 \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|w(s)\|_{\widehat V}^2 {\rm d}s
 < \frac{\epsilon}{6},
\end{equation}
and
\begin{equation} \label{3.23}
\frac{c_0}{r} e^{-\frac{\delta_1}{2}t}
 \int_{\tau}^t e^{\frac{\delta_1}{2}s}\|w(s)\|_{\widehat V}^2 {\rm d}s
 < \frac{\epsilon}{6}.
\end{equation}
Substituting \eqref{3.18}-\eqref{3.23} into \eqref{3.17}, we immediately
have \eqref{3.8}. This completes the proof.
\end{proof}

\begin{lemma}\label{L3.4}
Assume the conditions of Theorem \ref{T1.1} hold, then for any
$t\in\mathbb{R}$ and
$\widehat{B}=\{B(s) | s\in\mathbb{R}\}\in\mathcal{D}(\widehat{H})$,
there exists a $\tau^*(\widehat{B},t)$ such that the weak solution
$w(t) := w(t;\tau,w_{\tau})$ of \eqref{1.6} with initial value
$w_{\tau}\in B(\tau)$ is bounded in $\widehat{V}$.
\end{lemma}

\begin{proof}
To complete the proof, we need a higher regularity of the
solutions. Hence, we consider the Galerkin approximate solutions.
For each integer $n\geqslant 1$, we denote by
\begin{equation} \label{3.24}
w_n(t)=w_n(t;\tau,w_{\tau}) :=\sum_{i=1}^n\xi_{ni}(t)e_i,
\end{equation}
the Galerkin approximation of the solution $w(t)$ of system \eqref{1.6},
 where $\xi_{ni}(t)$ is the solution of the
following Cauchy problem of ODEs:
\begin{equation}\label{3.25}
\begin{gathered}
  \frac{\mathrm d}{{\mathrm d}t}(w_n(t), e_i)
     + \langle Aw_n(t)+B(u_n(t), w_n(t)) + N(w_n(t)), e_i \rangle
    =(F(t), e_i),  \\
 (w_n(\tau), e_i)
 =(w_{\tau}, e_i), \enskip i=1,2,\dots, n,
 \end{gathered}
\end{equation}
here $\{e_i: i\geqslant 1\}\subseteq D(A)$, which forms a Hilbert
basis of ${\widehat V}$ and is orthonormal in ${\widehat H}$.
Multiplying equation \eqref{3.25} by $A \xi_{ni}(t)$ and summing
them for $i=1$ to $n$, we obtain
\begin{equation} \label{3.26}
\begin{aligned}
&\frac 12\frac{\mathrm d}{{\mathrm d}t}\langle Aw_n(t), w_n(t) \rangle
  +\| Aw_n(t) \|^2 + \langle B(u_n(t), w_n(t)), Aw_n(t) \rangle \\
& +\langle N(w_n(t)), Aw_n(t) \rangle\\
& = (F(t), Aw_n(t)).
\end{aligned}
\end{equation}
Now, we give a further estimate for the above equation.
According to the definition of $B(\cdot,\cdot)$ and the facts
\begin{align*}
 \|u_n\|^2\leqslant \|w_n\|^2, \quad
 \|\nabla u_n\|^2 \leqslant \|w_n\|_{\widehat{V}}^2,
\end{align*}
and using the H$\rm{\ddot{o}}$lder inequality, Gagliardo-Nirenberg
inequality and Young inequality, we conclude that there exists a
constant $c_6$ such that
\begin{align*}
-\langle B(u_n, w_n), Aw_n \rangle
& \leqslant
   |\langle B(u_n, w_n), Aw_n\rangle| \\
&\leqslant c_6\|u_n\|^{1/2}\|\nabla u_n\|^{1/2}
   \|\nabla w_n\|^{1/2}\|Aw_n\|^{3/2}  \\
& \leqslant
  \frac 14\|Aw_n\|^2 + c_6^4\|w_n\|^2\|w_n\|_{\widehat V}^4,
\end{align*}
which together with \eqref{2.4} and \eqref{3.26} implies that
\begin{align*}
&\frac 12\frac{\mathrm d}{{\mathrm d}t}\langle Aw_n, w_n \rangle \\
&= -\|Aw_n\|^2
   +\langle F(t),Aw_n \rangle
   -\langle B(u_n, w_n), Aw_n \rangle
   -\langle N(w_n), Aw_n \rangle
    \\
&\leqslant
   -\|Aw_n\|^2
   +\frac 14\|Aw_n\|^2
   +\| F(t)\|^2
   +\frac 14\|Aw_n\|^2
   +c_6^4\|w_n\|^2\|w_n\|_{\widehat V}^4
    \\
&  \quad   +\frac{c^2(\nu_{\theta})}{2}\|w_n\|_{\widehat V}^2
   +\frac 12\|Aw_n\|^2
    \\
& =\|F(t)\|^2
   +\|w_n\|_{\widehat{V}}^2\Big(c_6^4\|w_n\|^2\|w_n\|_{\widehat V}^2
   +\frac{c^2(\nu_{\theta})}{2} \Big).
\end{align*}
Further, from \eqref{2.3} and the above inequality, we have
\begin{equation} \label{3.27}
\begin{aligned}
&\frac{\mathrm d}{{\mathrm d}t}\langle Aw_n(t), w_n(t) \rangle\\
&\leqslant 2\|F(t)\|^2 + \langle Aw_n(t), w_n(t) \rangle
   \left(2c_2c_6^4\|w_n(t)\|^2\|w_n(t)\|_{\widehat V}^2
    + c_2c^2(\nu_{\theta}) \right).
\end{aligned}
\end{equation}
Let us set
\begin{gather*}
 H_n(\theta)
 :=  \langle Aw_n(\theta), w_n(\theta) \rangle, \
  I(\theta)
:=  2\|F(\theta)\|^2, \\
  K_n(\theta)
 :=2c_2c_6^4\|w_n(\theta)\|^2\|w_n(\theta)\|_{\widehat V}^2
    + c_2c^2(\nu_{\theta}).
\end{gather*}
Replacing the variable $t$ with $\theta$ in \eqref{3.27}, we obtain
\begin{equation} \label{3.28}
   \frac{\mathrm d}{{\mathrm d}\theta}H_n(\theta)
\leqslant    K_n(\theta)H_n(\theta)+I(\theta).
\end{equation}
Using Gronwall inequality to \eqref{3.28}, for all
$\tau \leqslant t-1 \leqslant s \leqslant t$, we have
\begin{equation} \label{3.29}
   H_n(t)\leqslant \big (H_n(s)+\int_{t-1}^t I(\theta){\mathrm d}\theta \big )
   \exp \Big\{ \int_{t-1}^t K_n(\theta){\mathrm d}\theta \Big\}.
\end{equation}
Integrating \eqref{3.29} from $s=t-1$ to $s=t$,
we obtain
\begin{equation} \label{3.30}
   H_n(t)\leqslant \Big(\int_{t-1}^t H_n(s){\mathrm d}s
   + \int_{t-1}^t I(\theta){\mathrm d}\theta \Big)
   \exp \Big\{ \int_{t-1}^t K_n(\theta){\mathrm d}\theta \Big\}.
\end{equation}
In addition, it follows from \eqref{2.3} and \eqref{3.2} that
\begin{align*}
\int_{t-1}^t H_n(s){\mathrm d}s
   +\int_{t-1}^t I(\theta){\mathrm d}\theta
&= \int_{t-1}^t \langle Aw_n(s), w_n(s) \rangle{\mathrm d}s
   +\int_{t-1}^t 2\|F(\theta)\|^2{\mathrm d}\theta
   \\
& \leqslant   {c_1}^{-1}\int_{t-1}^t\| w_n(s) \|_{\widehat V}^2{\mathrm d}s
 +2\int_{t-1}^t \|F(\theta)\|^2{\mathrm d}\theta
   \\
& \leqslant  c_7\big(\|w_n(t-1)\|^2
   +\int_{t-1}^t\|F(\theta)\|^2{\mathrm d}\theta \big),
\end{align*}
where $ c_7:=\max\{ 2c_1^{-1}\delta_1^{-1},
2+2c_1^{-1}\delta_1^{-2}e^{\frac{\delta_1}{2}} \}$.
From \eqref{3.1}, it holds
\begin{align*}
&\int_{t-1}^t K_n(\theta){\mathrm d}\theta\\
&=    \int_{t-1}^t\big(2c_2
   c_6^4\|w_n(\theta)\|^2\|w_n(\theta)\|_{\widehat V}^2
  +c_2c^2(\nu_{\theta})\big){\mathrm d}\theta
   \\
&\leqslant
   2c_2c_6^4 \big(e^{-\frac{\delta_1}{2}}\|w_n(t-1) \|^2
  +\frac{1}{\delta_1}
   \int_{t-1}^t \|F(\theta)\|^2 {\mathrm d}\theta\big)
   \int_{t-1}^t \|w_n(\theta)\|_{\widehat V}^2
    {\mathrm d}\theta + c_2c^2(\nu_{\theta})
   \\
&\leqslant
   2c_2c_6^4 \big(e^{-\frac{\delta_1}{2}}\|w_n(t-1)\|^2
   +\frac{1}{\delta_1}\int_{t-1}^t \|F(\theta)\|^2 {\mathrm d}\theta \big)
   \\
& \quad \times\big(\frac{2\|w_n(t-1)\|^2}{\delta_1}
  +2{\delta_1}^{-2}e^{\frac{\delta_1}{2}} \int_{t-1}^t \|F(\theta)\|^2
    {\mathrm d}\theta\big) + c_2c^2(\nu_{\theta})
   \\
&\leqslant    c_8\Big[\big(\|w_n(t-1)\|^2
  +\int_{t-1}^t \|F(\theta)\|^2{\mathrm d}\theta
   \big)^2+1\Big],
\end{align*}
where
\[
 c_8 :=\max\big\{ 2c_2c_6^4
 \max \big\{e^{-\frac{\delta_1}{2}}, \delta_1^{-1} \big\}
 \max\big\{2\delta_1^{-1}, 2\delta_1^{-2}e^{\frac{\delta_1}{2}} \big\},\;
 c_2c^2(\nu_{\theta}) \big\}.
\]
With the aid of \eqref{2.3}, substituting the above two inequalities
into \eqref{3.30}, yields
\begin{equation} \label{3.31}
\begin{aligned}
  \|w_n(t)\|_{\widehat V}^2
&\leqslant   c_2H_n(t) \\
&\leqslant c_2c_7\big( \|w_n(t-1)\|^2
  +\int_{t-1}^t \|F(\theta)\|^2{\mathrm d}\theta \big)
   \\
&\quad \times \exp \big\{c_8\big[\big(\|w_n(t-1)\|^2
  +\int_{t-1}^t \|F(\theta)\|^2{\mathrm d}\theta\big)^2
  +1\big] \big\}.
\end{aligned}
\end{equation}
Observe that $w_n(t;\tau,w_{\tau})\in L^{\infty}(\tau,t;\widehat{V})
\cap \mathcal{C}([\tau,t];\widehat{V})$ and
$w(t; \tau,w_{\tau})\in \mathcal{C}([\tau,t];\widehat{H})$,
 by the lower semicontinuity of the norm, we can pass to the limit in
\eqref{3.31} and obtain that
\begin{equation} \label{3.32}
\begin{aligned}
  \|w(t)\|_{\widehat V}^2
&\leqslant
  c_2c_7\big( \|w(t-1)\|^2
  +\int_{t-1}^t \|F(\theta)\|^2{\mathrm d}\theta \big)
   \\
& \quad\times \exp \big\{c_8\big[\big(\|w(t-1)\|^2
  +\int_{t-1}^t \|F(\theta)\|^2{\mathrm d}\theta\big)^2
  +1\big] \big\},
\end{aligned}
\end{equation}
which together with Lemma \ref{L3.2} implies the result of Lemma \ref{L3.4}.
This completes the proof.
\end{proof}

On the basis of the above results, we can prove the pullback asymptotical
compactness of the process
$\{U(t,\tau)\}_{t\geqslant\tau}$, that is the following Lemma.

\begin{lemma}\label{L3.5}
Under the conditions of Theorem \ref{T1.1}, the process
$\{U(t,\tau)\}_{t\geqslant\tau}$ generated by \eqref{2.7} is
pullback $\mathcal{D}$-asymptotically compact in $\widehat{H}$.
\end{lemma}

\begin{proof}
For any fixed $t\in\mathbb{R}$, any family
$\widehat{B} =\{B(s)| s\in\mathbb{R}\}\in\mathcal{D}_{\frac{\delta_1}{2}}
(\widehat{H})$,
any sequences $\{\tau_n\}\subseteq (-\infty, t]$ satisfying
$\tau_n\to -\infty$ as $n\to +\infty$ and
$\{w_{\tau_n}\}\in \mathcal{D}(\tau_n)$, it suffice to show the sequence
$\{w^n(t)\}_{n\geqslant 1}$ defined by
\begin{align*}
w^n(\cdot) := w^n(\cdot; \tau_n,w_{\tau_n})=U(\cdot,\tau_n; w_{\tau_n})
\end{align*}
is relatively compact in $\widehat{H}$.

In fact, by Lemma \ref{L3.2}, there exists a time $\tau_0(\widehat{B},t)<t$
such that the sequence $\{w^n(t) | \tau_n\leqslant \tau_0(\widehat{B},t)\}$
is uniformly bounded in $\widehat{H}$. Since $\widehat{H}$ is a reflect
Banach space, it follows from the diagonal procedure that there exists a
 function $w(t)$ such that (by extracting a subsequence if necessary)
\[
w^n(t)\rightharpoonup w(t) \quad \text{weakly  in $\widehat{H}$
  as }  n\to \infty.
\]
Moreover, from Lemma \ref{L3.3}, for any $\epsilon>0$, there exist
$\tau_2 := \tau_2(\epsilon,t,\widehat{B})$,
$r_3 := r_3(\epsilon,t,\widehat{B})>0$ such that
\begin{align}\label{3.33}
\|w^n(t; \tau_n,w_{\tau_n})\|_{\mathbb{L}^2(\Omega\setminus\Omega_r)}
\leqslant
 \frac{\epsilon}{3}, \quad \forall\tau_n\leqslant\tau_2, r\geqslant r_3.
\end{align}
Observe that, for any fixed $t\in\mathbb{R}, w(t)\in \widehat{H}$ is fixed.
Hence for the above $\epsilon>0$, there exists a $r_4>0$ such that
\begin{equation} \label{3.34}
\|w(t)\|_{\mathbb{L}^2(\Omega\setminus\Omega_r)}
\leqslant
 \frac{\epsilon}{3}, \quad \forall r\geqslant r_4.
\end{equation}
Now, we define respectively the restrictions of $w^n$ and $w$ in
$\Omega_r$ by
\begin{gather*}
w^n(t)|_{\Omega_r}
= w^n(t; \tau_n,w_{\tau_n})|_{\Omega_r}
 := \begin{cases}
 w^n(t), & x\in\Omega_r,  \\
 0, & x\in\Omega\setminus\Omega_r,
 \end{cases} \\
w(t)|_{\Omega_r}
 := \begin{cases}
 w(t), & x\in\Omega_r,  \\
 0, & x\in\Omega\setminus\Omega_r.
 \end{cases}
\end{gather*}
From Lemma \ref{L3.4}, it follows that, for any $r>0$, the sequence
$\{w^n(t)|_{\Omega_r}\}_{n\geqslant 1}$ is bounded in $\widehat{V}(\Omega_r)$.
Since the embedding
$\widehat{V}(\Omega_r)\hookrightarrow\widehat{H}(\Omega_r)$ is compact,
 there exists a subsequence (denoting by the same symbol)
$\{w^n(t)|_{\Omega_r}\}_{n\geqslant 1}$ satisfying
\begin{equation} \label{3.35}
\|w^n(t)-w(t)\|_{\widehat{H}(\Omega_r)}
\to 0 \quad \text{as } n\to\infty,
\end{equation}
which together with \eqref{3.33}-\eqref{3.34} implies that there exists a
$N_0\in\mathbb{N}$ such that for any $n\geqslant N_0$,
\begin{equation} \label{3.36}
\begin{aligned}
\|w^n(t)-w(t)\|_{\widehat{H}}
&= \|w^n(t)-w(t)\|_{\widehat{H}(\Omega_r)}
  + \|w^n(t)-w(t)\|_{\mathbb{L}^2(\Omega\setminus\Omega_r)} \\
&\leqslant  \|w^n(t)-w(t)\|_{\widehat{H}(\Omega_r)}
  + \|w^n(t)\|_{\mathbb{L}^2(\Omega\setminus\Omega_r)} \\
&\quad  + \|w(t)\|_{\mathbb{L}^2(\Omega\setminus\Omega_r)}
\leqslant
  \epsilon.
\end{aligned}
\end{equation}
Therefore, the sequence $\{w^n(t)\}_{n\geqslant 1}$ is relatively compact in
$\widehat{H}$. This completes the proof.
\end{proof}


\begin{proof}[Proof of Theorem \ref{T1.1}]
According to definitions \ref{D2.2}-\ref{D2.4}, the family
$\widehat{\mathcal{B}}= \{\mathcal{B}(t) | t\in\mathbb{R}\}$ is pullback
$\mathcal{D}$-absorbing which can be obtained directly from Lemma \ref{L3.2}.
Further, it follows from Lemma \ref{L3.5} that the continuous process
 $\{U(t,\tau)\}_{t\geqslant\tau}$ is pullback
$\mathcal{D}$-asymptotically compact in $\widehat{H}$.
Then, using \cite[Theorem 7]{CLR06}, we can show the existence and uniqueness
of the pullback $\mathcal{D}$-attractor
$\mathcal{A}_{\widehat H}(t)$ for $\{U(t,\tau)\}_{t\geqslant\tau}$ in $\widehat{H}$.
\end{proof}

\section{Tempered behavior and upper semicontinuity of the pullback attractor}

In this section, we will show the tempered behavior and upper semicontinuity
of the pullback attractor $\mathcal{A}_{\widehat H}(t)$, which is obtained
in section 3.

\begin{proof}[Proof of  Theorem  \ref{T1.2}]
According to Theorem \ref{T1.1}, we know that
$\mathcal{A}_{\widehat H}(t)\in\mathcal{D}_{\frac{\delta_1}{2}}(\widehat H)$.
Therefore, \eqref{1.7} holds. Since $F(t,x)\in L_b^2(\mathbb{R};\widehat{H})$,
\eqref{1.8} is a consequence of \eqref{1.7} and \eqref{3.32}.
This completes the proof.
\end{proof}

The rest of this section is devoted to verifying the upper semicontinuity of
the pullback attractors with respect to the spatial domain, that is,
we give the proof of  Theorem \ref{T1.3}. Followed the arguments
in \cite{Z12}, let $\{\Omega_m\}_{m=1}^{\infty}$ be an expanding sequence
of simply connected, bounded and smooth subdomains of $\Omega$ such that
$\cup_{m=1}^{\infty}\Omega_m = \Omega$. We will prove the upper semicontinuity
of the pullback attractor $\mathcal{A}_{\widehat H}$ in $\Omega$ from the
pullback attractor $\mathcal{A}_{\widehat{H}(\Omega_m)}$ in $\Omega_m$.

First, we consider  equations \eqref{1.2}-\eqref{1.4} in each $\Omega_m$ and
define the operators $A, B(\cdot, \cdot)$ and $N(\cdot)$ as before with the
spatial domain $\Omega$ replaced by $\Omega_m$. Then we can rewrite
\eqref{1.2}-\eqref{1.4} in the abstract form
\begin{equation} \label{4.1}
\begin{gathered}
  \frac{\partial w_m}{\partial t}
   + Aw_m +B(u_m,w_m) + N(w_m) = F(t,x), \quad
 \text{in }(\tau, +\infty)\times\Omega_m,  \\
 \nabla\cdot u_m = 0, \quad \text{in }(\tau, +\infty)\times \Omega_m,
   \\
 w_m = (u_m,\omega_m) = 0, \quad \text{on }(\tau, +\infty)\times
  \partial\Omega_m,   \\
 w_m(\tau,x) = (u_m(\tau,x), \omega_m(\tau,x)) = {w_m}_{\tau}(x),\quad
  x\in\Omega_m, \; \tau\in\mathbb{R}.
 \end{gathered}
 \end{equation}
For each bounded domain $\Omega_m$, the global existence and uniqueness of
the weak solutions of system \eqref{4.1} hold. That is,

\begin{lemma}\label{L4.1}[\cite{L01}]
Assume the conditions of Theorem \ref{T1.3} hold and
$w_\tau \in \widehat {H}(\Omega_m)$, then system \eqref{4.1} has a unique
solution $w_m$ satisfying
\begin{gather*}
 w_m\in L^\infty( \tau, +\infty; \widehat{H}(\Omega_m) )
  \cap \mathcal {C}( [\tau, +\infty);\widehat{H}(\Omega_m))
  \cap L_{\rm loc}^2( \tau,+\infty;\widehat{V}(\Omega_m) ),  \\
  w_m'\in L_{\rm loc}^2
  ( \tau, +\infty; \widehat{V}^*(\Omega_m) ).
 \end{gather*}
Moreover, the solution $w_m$ depends continuously on the initial value
$w_{\tau}$ with respect to the $\widehat{H}(\Omega_m)$ norm.
\end{lemma}

According to Lemma \ref{L4.1}, the maps of solution operators defined by
\begin{align}\label{4.2}
U_m(t,\tau): {w_m}_{\tau} \mapsto U_m(t,\tau; {w_m}_{\tau})=w_m(t), \quad
 t\geqslant \tau
\end{align}
generates a continuous process $\{U_m(t,\tau)\}_{t\geqslant\tau}$ in
$\widehat{H}(\Omega_m)$. Moreover, on any smooth bounded domain
$\Omega_m$, we have the following result.

\begin{lemma}\label{L4.2}
Under the conditions of Theorem \ref{T1.3}.

{\rm (1)}
For any $t\in\mathbb{R}$, $\widehat{B}^{\widehat{H}(\Omega_m)}
= \{B^{\widehat{H}(\Omega_m)}(s) | s\in\mathbb{R}\}
\in\mathcal{D}_{\frac{\delta_1}{2}}(\widehat{H}(\Omega_m))$ and
${w_m}_{\tau}\in B^{\widehat{H}(\Omega_m)}(\tau)$, the family
$\widehat{\mathcal{B}}^{\widehat{H}(\Omega_m)}
 = \{\mathcal{B}^{\widehat{H}(\Omega_m)}(t) \big| t\in\mathbb{R}\}$ given by
\[
\mathcal{B}^{\widehat{H}(\Omega_m)}(t)
= \{ w_m\in \widehat{H}(\Omega_m) \big| \|w_m\|_{\widehat{H}(\Omega_m)}
\leqslant   \rho(t) \}
\]
is pullback $\mathcal{D}^{\widehat{H}(\Omega_m)}$-absorbing in
$\widehat{H}(\Omega_m)$, where $\rho(t)$ is defined by \eqref{3.8}.

{\rm (2)}
For any $\epsilon>0, t\in\mathbb{R}, \widehat{B}^{\widehat{H}(\Omega_m)}
= \{B^{\widehat{H}(\Omega_m)}(s) | s\in\mathbb{R}\}
\in\mathcal{D}_{\frac{\delta_1}{2}}(\widehat{H}(\Omega_m))$ and
${w_m}_{\tau}\in B^{\widehat{H}(\Omega_m)}(\tau)$, there exists
${r_0}_m={r_0}_m(\epsilon,t,\widehat{B}^{\widehat{H}(\Omega_m)})>0$
and a time ${\tau_0}_m={\tau_0}_m(\epsilon,t,\widehat{B}^{\widehat{H}(\Omega_m)})<t$
such that for any $r\in [{r_0}_m, m]$ and $\tau\leqslant{\tau_0}_m$, it holds
\begin{align*}
\|w_m(t; \tau,{w_m}_{\tau})\|_{\mathbb{L}^2(\Omega_m\setminus\Omega_r)}
\leqslant
 \epsilon.
\end{align*}

{\rm (3)}
The process $\{U_m(t,\tau)\}_{t\geqslant\tau}$ is pullback
$\mathcal{D}^{\widehat{H}(\Omega_m)}$-asymptotically compact in
$\widehat{H}(\Omega_m)$.
\end{lemma}

Since the proof is similar to those of Lemma \ref{L3.2}, Lemma \ref{L3.3} and
Lemma \ref{L3.5}, we can omit it here.
As a consequence of Lemma \ref{L4.2}, we have the following  result.

\begin{proposition}\label{T4.2}
Assume the conditions of Theorem \ref{T1.3} hold, then system \eqref{4.1}
has a unique pullback $\mathcal{D}^{\widehat{H}(\Omega_m)}$-attractor
$\mathcal{A}_{\widehat{H}(\Omega_m)}
 =\{\mathcal{A}_{\widehat{H}(\Omega_m)}(t)\}_{t\in\mathbb{R}}$
in $\widehat{H}(\Omega_m)$.
\end{proposition}

Next, let us consider the convergence of solutions for \eqref{4.1} with $m$.
That is, we show that the sequence $\{w_m\}_{m\geqslant 1}$ of solutions
to system \eqref{4.1} converges to the solution of system \eqref{1.6}
as $m\to\infty$.

For $w_m\in\widehat{H}(\Omega_m)$, we extend its domain from $\Omega_m$
to $\Omega$ by setting
\[
\tilde{w}_m=
\begin{cases}
 w_m, & x\in\Omega_m,  \\
 0, & x\in\Omega\setminus\Omega_m,
 \end{cases}
\]
then
\[
\|w_m\|_{\widehat{H}(\Omega)}
= \|\tilde{w}_m\|_{\widehat{H}(\Omega)}
= \|\tilde{w}_m\|_{\widehat{H}(\Omega_m)}
= \|w_m\|_{\widehat{H}(\Omega_m)}.
\]
By Lemma \ref{L3.2} and Lemma \ref{L4.2} (1), there exists a
$\tau(t, \widehat{B}^{\widehat{H}(\Omega)})$ (independent of $m$) such that
\begin{equation} \label{4.3}
\begin{aligned}
 U(t,\tau)B(\tau)\subseteq \mathcal{B}(t), \quad
 \forall\tau\leqslant\tau(t,\widehat{B}^{\widehat{H}(\Omega)}),  \\
 U_m(t,\tau)B^{\widehat{H}(\Omega_m)}(\tau)\subseteq
\mathcal{B}^{\widehat{H}(\Omega_m)}(t), \quad
 \forall\tau\leqslant\tau(t,\widehat{B}^{\widehat{H}(\Omega)}).
 \end{aligned}
\end{equation}
The following result can be obtained by using the same proof as
that of \cite[Lemma 8.1]{LS04}.

\begin{lemma}\label{L4.3}
Under the conditions of Theorem \ref{T1.3}, let $\{{w_m}_{\tau}\}_{m\geqslant 1}$
be a sequence in $\widehat{H}(\Omega_m)$ satisfying
\[
  {w_m}_{\tau}\rightharpoonup w_{\tau} \quad\text{weakly  in }
 \widehat{H}(\Omega) \text{ as } m\to\infty,
\]
then
\[
w_m(t;\tau,{w_m}_{\tau})\rightharpoonup w(t;\tau,w_{\tau}) \quad
  \text{weakly  in }  \widehat{H}(\Omega),\; \forall  t\geqslant\tau,
\]
and
\begin{equation} \label{4.4}
w_m(\cdot;\tau,{w_m}_{\tau})\rightharpoonup w(\cdot;\tau,w_{\tau}) \quad
  \text{weakly  in}  L^2(\tau,T;\widehat{V}(\Omega)), \; \forall T>\tau.
\end{equation}
\end{lemma}

\begin{lemma}\label{L4.4}
Assume the conditions of Lemma \ref{L4.3} hold, then for any $t\in\mathbb{R}$
and any sequence $\{w_m\}_{m\geqslant 1}$ with
$w_m(\tau)\in\mathcal{A}_{\widehat{H}(\Omega_m)}(\tau)$, $m=1,2,\dots$, there
exists $w(t)\in\mathcal{A}_{\widehat{H}(\Omega)}(t)$ such that
\begin{equation} \label{4.5}
w_m(\cdot) \to w(\cdot) \quad \text{strongly in } \widehat{H}(\Omega) \text{ as }
m\to\infty.
\end{equation}
\end{lemma}

\begin{proof}
First, it follows from \eqref{4.3} and the invariant of the pullback attractor
that the sequence $\{w_m(\tau)\}_{m\geqslant 1}$ is bounded in
$\widehat{H}(\Omega)$. Hence, one can deduce that there exists a
$w_{\tau}\in\mathcal{A}_{\widehat{H}(\Omega)}(\tau)$ and a subsequence
$\{w_m(\tau)\}_{m\geqslant 1}$ (denoted by the same) such that
\begin{equation} \label{4.6}
w_m(\tau)\rightharpoonup w_{\tau} \quad \text{weakly in }\widehat{H}(\Omega)
\text{ as } m\to \infty.
\end{equation}
By Lemma \ref{L4.3} and the invariant of the pullback attractor, we see that
for any $t\in\mathbb{R}$ and
$w_m(t;\tau,w_m(\tau))\in\mathcal{A}_{\widehat{H}(\Omega_m)}(t)$ with
${w_m}(\tau)\in\mathcal{A}_{\widehat{H}(\Omega_m)}(\tau)$, there exists a
$w(t;\tau,w_{\tau})\in\mathcal{A}_{\widehat{H}(\Omega)}(t)$ with
$w_{\tau}\in\mathcal{A}_{\widehat{H}(\Omega)}(\tau)$ such that
\begin{align*}
w_m(t)\rightharpoonup w(t) \quad \text{weakly in } \widehat{H}(\Omega)
\text{ as } m\to \infty,
\end{align*}
which together with the lower semicontinuity of the norm implies
\begin{equation} \label{4.7}
\|w(t)\| \leqslant  \liminf_{m\to\infty} \|w_m(t)\|.
\end{equation}

Next, we shall prove
\begin{equation}\label{4.8}
\|w(t)\|^2
\geqslant  \limsup_{m\to\infty} \|w_m(t)\|^2.
\end{equation}
In fact, multiplying $\eqref{4.1}_1$ by $w_m(t)$ and integrating the resultant
equality over $\Omega$, we obtain with the aid of \eqref{2.2} that
\begin{equation}\label{4.9}
\frac{\rm d}{{\rm d}t} \|w_m(t)\|^2
 + 2\langle Aw_m(t), w_m(t) \rangle
 + 2\langle N(w_m(t)), w_m(t) \rangle
= 2(F(t), w_m(t)).
\end{equation}
Then, from \eqref{2.3}, \eqref{2.4} and \eqref{2.5}, we see that
\begin{equation} \label{4.10}
\begin{aligned}
(c_1^{-1}+c(\nu_{\theta})) \|w\|_{\widehat{V}(\Omega)}^2
&\geqslant
 \ll w,w \gg \\
&= \langle Aw,w \rangle
  + \langle N(w),w \rangle
  - \frac{\delta_1}{4}\|w\|^2  \\
&\geqslant
 \delta_1\|w\|_{\widehat{V}(\Omega)}^2
 - \frac{\delta_1\|w\|^2}{4}
\geqslant
 \frac{3\delta_1}{4} \|w\|_{\widehat{V}(\Omega)}^2,
\end{aligned}
\end{equation}
where the bilinear mapping $\ll \cdot, \cdot \gg$ is defined by
\[
\ll \varphi,\phi \gg
= \langle A\varphi,\phi \rangle
  + \langle N(\varphi),\phi \rangle
  - \frac{\delta_1(\varphi,\phi)}{4}, \quad
     \forall\varphi, \phi \in \widehat{V}(\Omega).
\]
By \eqref{4.9} and \eqref{4.10}, we have
\begin{align*}
\frac{\rm d}{{\rm d}t} \|w_m(t)\|^2
 + \frac{\delta_1}{2} \|w_m(t)\|^2
= 2(F(t), w_m(t))
 - 2\ll w_m(t),w_m(t) \gg,
\end{align*}
which yields
\begin{equation} \label{4.11}
\begin{aligned}
 \|w_m(t)\|^2
&= e^{-\frac{\delta_1}{2}(t-\tau)}\|w_m(\tau)\|^2  \\
&\quad  + 2\int_{\tau}^t e^{-\frac{\delta_1}{2}(t-s)}
  \big[ (F(s),w_m(s))-\ll w_m(s),w_m(s) \gg \big] {\rm d}s,
\end{aligned}
\end{equation}
for all $t\geqslant\tau$.

Next, we estimate the terms on the right-half side of \eqref{4.11}
one by one. First, since $w_m(\tau)\in\mathcal{A}_{\widehat{H}(\Omega_m)}(\tau)$,
it follows from \eqref{3.7} and \eqref{4.3} that
\begin{equation} \label{4.12}
\begin{aligned}
e^{-\frac{\delta_1}{2}(t-\tau)} \|w_m(\tau)\|^2
& \leqslant
   e^{-\frac{\delta_1}{2}(t-\tau)} \rho^2(\tau)  \\
& = \frac{2}{\delta_1}e^{-\frac{\delta_1}{2}t}
     \int_{-\infty}^{\tau} e^{\frac{\delta_1}{2}s} \|F(s)\|^2 {\rm d}s
\to 0 \quad\text{as } \tau\to -\infty.
\end{aligned}
\end{equation}
Next, from \eqref{4.4} and \eqref{4.6}, we have
\begin{equation} \label{4.13}
\lim_{m\to\infty}
 \int_{\tau}^t e^{-\frac{\delta_1}{2}(t-s)} (F(s),w_m(s)) {\rm d}s
= \int_{\tau}^t e^{-\frac{\delta_1}{2}(t-s)} (F(s),w(s)) {\rm d}s.
\end{equation}
Finally, we see from \eqref{4.10} that $\ll w,w \gg$ is equivalent to
$\|w\|_{\widehat{V}(\Omega)}^2$, which combines with \eqref{4.4} and
\eqref{4.6} implies
\begin{align*}
\ll w_m(s),w_m(s) \gg
 \rightharpoonup
 \ll w(s),w(s) \gg \quad
 \text{weakly in }  L^2(\tau,t; \widehat{V}(\Omega)), \; \forall t>\tau.
\end{align*}
It follows from the lower semicontinuity of the norm that
\begin{align}\label{4.14}
\int_{\tau}^t \ll w(s),w(s) \gg {\rm d}s
\leqslant
 \liminf_{m\to\infty}
 \int_{\tau}^t \ll w_m(s),w_m(s) \gg {\rm d}s.
\end{align}
Inserting \eqref{4.13}-\eqref{4.14} into \eqref{4.11} leads to
\begin{equation} \label{4.15}
\begin{aligned}
\limsup_{m\to\infty} \|w_m(t)\|^2
&\leqslant
 e^{-\frac{\delta_1}{2}(t-\tau)} \|w_m(\tau)\|^2
+ 2\int_{\tau}^t e^{-\frac{\delta_1}{2}(t-s)}(F(s),w(s)) {\rm d}s \\
&\quad - 2\int_{\tau}^t e^{-\frac{\delta_1}{2}(t-s)} \ll w(s),w(s) \gg {\rm d}s.
\end{aligned}
\end{equation}
Similar to \eqref{4.11}, the following energy equality for $w(t)$ hold:
\begin{equation} \label{4.16}
\begin{aligned}
\|w(t)\|^2
&= e^{-\frac{\delta_1}{2}(t-\tau)} \|w(t)\|^2
 + 2\int_{\tau}^t e^{-\frac{\delta_1}{2}(t-s)}(F(s),w(s)) {\rm d}s
    \\
&\quad - 2\int_{\tau}^t e^{-\frac{\delta_1}{2}(t-s)} \ll w(s),w(s) \gg {\rm d}s.
\end{aligned}
\end{equation}
From the above two inequalities, we obtain
\begin{align*}
\limsup_{m\to\infty} \|w_m(t)\|^2
\leqslant
 e^{-\frac{\delta_1}{2}(t-\tau)} \|w_m(\tau)\|^2
 + \|w(t)\|^2,
\end{align*}
which together with \eqref{4.12} implies \eqref{4.8} when $\tau$ is small enough.
This completes the proof.
\end{proof}


\begin{proof}[Proof of Theorem \ref{T1.3}]
Suppose that \eqref{1.9} is false, then there exist
$t_0\in\mathbb{R}, \epsilon_0>0$ and
$w_m \in \mathcal{A}_{\widehat{H}(\Omega_m)}(t_0)$, $m=1,2,\dots$, such that
\begin{align}\label{4.17}
\operatorname{dist}_{\widehat{H}(\Omega)}
 (w_m,\mathcal{A}_{\widehat{H}(\Omega)}(t_0))
\geqslant
 \epsilon_0 > 0, \quad  \ m=1, 2, \dots.
\end{align}
However, by Lemma \ref{L4.4}, we see that there exists a subsequence
$\{w_{m_k}\}_{k\geqslant 1} \subseteq \{w_m\}_{m\geqslant 1}$ such that
\begin{align*}
\lim_{k\to\infty}\operatorname{dist}_{\widehat{H}(\Omega)}
 (w_{m_k},\mathcal{A}_{\widehat{H}(\Omega)}(t_0))
= 0,
\end{align*}
which leads to a contradiction with \eqref{4.17}.
Therefore, \eqref{1.9} follows. This completes the proof.
\end{proof}

\subsection*{Acknowledgements}  We are grateful to two anonymous
referees for valuable comments which greatly improved our original
manuscript. The research is supported in partial by the National
Science Foundation of China (Grant No. 11671134).


\begin{thebibliography}{00}

\bibitem{A75} R. A. Adams;
{\it Sobolev Spaces}, Academic Press, New York, 1975.

\bibitem{CLR2006} T. Caraballo, G. \L{u}kaszewicz, J. Real;
 Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some
unbounded domains, \textit{Comptes Rendus Mathematique}, \textbf{342} (2006), 263-268.

\bibitem{CLR06} T. Caraballo, G. \L{u}kaszewicz, J. Real;
 Pullback attractors for asymptotically compact non-autonomous dynamical systems,
{\it Nonlinear Anal.}, \textbf{64} (2006), 484-498.

\bibitem{CCD06} J. Chen, Z. Chen, B. Dong;
 Existence of $H^2$-global attractors of two-dimensional micropolar fluid flows, {\it J. Math. Anal.
Appl.}, {\bf 322} (2006), 512-522.

\bibitem{CCD07} J. Chen, B. Dong, Z. Chen;
Uniform attractors of non-homogeneous micropolar fluid flows in
non-smooth domains, {\it Nonlinearity}, \textbf{20} (2007), 1619-1635.

\bibitem{C09} G. Chen;
 Pullback attractor for non-homogeneous micropolar fluid flows in non-smooth
domains, {\it Nonlinear Anal.}, {\bf 10} (2009), 3018-3027.

\bibitem{DC06} B. Dong, Z. Chen;
Global attractors of two-dimensional micropolar fluid flows in some unbounded
domains, {\it Appl. Math. Comp.},
{\bf 182} (2006), 610-620.

\bibitem{DZ10} B. Dong, Z. Zhang;
 Global regularity of the 2D micropolar fluid flows with zero angular viscosity,
{\it J. Differential Equations}, {\bf 249} (2010), 200-213.

\bibitem{E66} A. C. Eringen;
Theory of micropolar fluids, {\it J.
Math. Mech.}, {\bf 16} (1966), 1-18.

\bibitem{GMR12} J. Garc\'ia-Luengo, P. Mar\'in-Rubio, J. Real;
 Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations
and their tempered behavior, {\it J. Differential Equations},
{\bf 252} (2012), 4333-4356.

\bibitem{HXR88} J. K. Hale, L. Xin, G. Raugel;
 Upper semicontinuity of attractors for approximations of semigroups and
partial differential equations, {\it Math. Comput.}, \textbf{5} (1988), 89-123.

\bibitem{L99} G. \L{u}kaszewicz;
 {\it Micropolar Fluids: Theory and Applications}, Birkh\"auser, Boston, 1999.

\bibitem{L01} G. \L{u}kaszewicz;
 Long time behavior of 2D micropolar
fluid flows, {\it Math. Comput. Modelling}, \textbf{34} (2001),
487-509.

\bibitem{LS04} G. \L{u}kaszewicz, W. Sadowski;
Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains,
{\it Z. Angew. Math. Phys.}, \textbf{55} (2004), 247-257.

\bibitem{LW05} K. Lu, B. Wang;
 Upper semicontinuity of attractors for the Klein-Gordon-Schr\"odinger equation,
{\it Inter. J. Bifur. Chaos}, \textbf{15} (2005), 157-168.

\bibitem{LLR07} J. A. Langa, G. \L{u}kaszewicz, J. Real;
 Finite fractal dimension of pullback attractors for non-autonomous 2D
Navier-Stokes equations in some unbounded domains, \textit{Nonlinear Anal.},
\textbf{66} (2007), 735-749.

\bibitem{LRSV07} J. A. Langa, J. C. Robinson, A. Rodr\'{i}guez-Bernal,
 A. Su\'{a}rez, A. Vidal-L\'{o}pez;
 Existence and nonexistence of unbounded forward attractor for a class of
non-autonomous reaction diffusion equations, \textit{Discrete Contin. Dyn. Sys.},
\textbf{18} (2007), 483-497.

\bibitem{LT09} G. \L{u}kaszewicz, A. Tarasi\'nska;
 On $H^1$-pullback attractors for nonautonomous micropolar fluid equations in a
bounded domain, \textit{ Nonlinear Anal.}, \textbf{71} (2009), 782-788.

\bibitem{MR09} P. Mar\'in-Rubio, J. Real;
 On the relation between two different concepts of pullback attractors for
non-autonomous dynamical systems, {\it Nonlinear Anal.}, \textbf{71} (2009),
3956-3963.

\bibitem{M10} A. M. M\'arquez;
Existence and uniqueness of solutions, and pullback attractor for a system
of globally modified 3D-Navier-Stokes equations with finite delay,
\textit{Bol. Soc. Esp. Mat. Apl.}, \textbf{51} (2010), 117-124.

\bibitem{N13} B. Nowakowski;
 Long-time behavior of micropolar fluid equations in cylindrical domains,
{\it Nonlinear Anal.: RWA}, \textbf{14} (2013), 2166-2179.

\bibitem{SW07} H. Song, H. Wu;
 Pullback attractors of nonautonomous reaction-diffusion equations,
{\it J. Math. Anal. Appl.}, \textbf{325} (2007), 1200-1215.

\bibitem{WL09} X. H. Wang, S. Y. Li;
 Pullback attractors for non-autonomous 2D Navier-Stokes equations with
linear damping in some unbounded domains,
{\it Acta Mathematica Sinica-English Series}, \textbf{4} (2009), 873-881.

\bibitem{YH11} P. J. Yeoul, P. S. Hye;
 Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation
in unbounded domains, {\it Sci. China Math.}, \textbf{54} (2011), 741-752.

\bibitem{ZZL08} C. Zhao, S. Zhou, X. Lian;
 $H^1$-uniform attractor and
asymptotic smoothing effect of solutions for a nonautonomous
micropolar fluid flow in 2D unbounded domains, {\it Nonlinear Anal.:
RWA}, {\bf 9} (2008), 608-627.

\bibitem{ZD12} C. Zhao, J. Duan;
 Upper semicontinuity of global attractors for 2D Navier-Stokes equations,
{\it Inter. J. Bifur. Chaos}, \textbf{22} (2012), 1-7.

\bibitem{Z12} C. Zhao;
 Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian
fluid on two-dimensional unbounded domains, {\it J. Math. Phys.},
\textbf{53} (2012), 122702-1-23.

\bibitem{ZLW14} C. Zhao, G. Liu, W. Wang;
 Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their
tempered behavior, {\it J. Math. Fluid Mech.}, \textbf{16} (2013), 243-262.

\bibitem{ZSH15} C. Zhao, W. Sun, C. Hsu;
 Pullback dynamical behaviors of the non-autonomous micropolar fluid flows,
 \textit{Dynamics of PDE}, \textbf{12} (2015), 265-288.

\end{thebibliography}

\end{document}