The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay

1.   Introduction

The micropolar fluid model is a qualitative generalization of the well-known Navier-Stokes model in the sense that it takes into account the microstructure of fluid [7]. The model was first derived in 1966 by Eringen [4] to describe the motion of a class of non-Newtonian fluid with micro-rotational effects and inertia involved. It can be expressed by the following equations:

where $u = (u_1, u_2, u_3)$ is the velocity, $\omega = (\omega_1, \omega_2, \omega_3)$ is the angular velocity field of rotation of particles, $p$ represents the pressure, $f = (f_1, f_2, f_3)$ and $\tilde{f} = (\tilde{f}_1, \tilde{f}_2, \tilde{f}_3)$ stand for the external force and moment, respectively. The positive parameters $\nu, \nu_r, c_0, c_a$ and $c_d$ are viscous coefficients. Actually, $\nu$ represents the usual Newtonian viscosity and $\nu_r$ is called microrotation viscosity.

Micropolar fluid models play an important role in the fields of applied and computational mathematics. There is a rich literature on the mathematical theory of micropolar fluid model. Particularly, the existence, uniqueness and regularity of solutions for the micropolar fluid flows have been investigated in [6]. Extensive studies on long time behavior of solutions for the micropolar fluid flows have also been done. For example, in the case of 2D bounded domains: Łukaszewicz [7] established the existence of $L^2$-global attractors and its Hausdorff dimension and fractal dimension estimation. Chen, Chen and Dong proved the existence of $H^2$-global attractor and uniform attractor in [1] and [2], respectively. Łukaszewicz and Tarasińska [9] investigated the existence of $H^1$-pullback attractor. Zhao, Sun and Hsu [18] established the existence of $L^2$-pullback attractor and $H^1$-pullback attractor of solutions for a universe given by a tempered condition, respectively. For the case of 2D unbounded domains: Dong and Chen [3] investigated the existence and regularity of global attractors. Zhao, Zhou and Lian [19] established the existence of $H^1$-uniform attractor and further gave the inclusion relation between $L^2$-uniform attractor and the $H^1$-uniform attractor. Sun and Li [15] verified the existence of pullback attractor and further investigated the tempered behavior and upper semicontinuity of the pullback attractor. More recently, Sun, Cheng and Han [14] investigated the existence of random attractors for 2D stochastic micropolar fluid flows.

As we know, in the real world, delay terms appear naturally, for instance as effects in wind tunnel experiments (see [10]). Also the delay situations may occur when we want to control the system via applying a force which considers not only the present state but also the history state of the system. The delay of partial differential equations (PDE) includes finite delays (constant, variable, distributed, etc) and infinite delays. Different types of delays need to be treated by different approaches.

In this paper, we consider the situation 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, that is $u = (u_1, u_2, 0)$, $\omega = (0, 0, \omega_3)$, $f = (f_1, f_2, 0)$, $\tilde{f} = (0, 0, \tilde{f}_3)$. Let $\Omega\subseteq\mathbb{R}^2$ be an open set with boundary $\Gamma$ that is not necessarily bounded but satisfies the following Poincaré inequality:

Then we discuss the following 2D non-autonomous incompressible micropolar fluid flows with finite delay:

where $\bar{\alpha}: = c_0+2c_d>0$, $x : = (x_1, x_2)\in\Omega\subseteq \mathbb{R}^2$, $u : = (u_1, u_2)$, $g$ and $\tilde{g}$ stand for the external force containing some hereditary characteristics $u_t$ and $\omega_t$, which are defined on $(-h, 0)$ as follows

where $h$ is a positive fixed number, and

To complete the formulation of the initial boundary value problem to system (3), we give the following initial boundary conditions:

For problem (3)-(5), Sun and Liu established the existence of pullback attractor in [16], recently.

The first purpose of this work is to investigate the boundedness of the pullback attractor obtained in [16]. We remark that García-Luengo, Marín-Rubio and Real [5] proved the $H^2$-boundedness of the pullback attractors of the 2D Navier-Stokes equations in bounded domains. Motivated by [5] and following its main idea, we generalize their results to the 2D micropolar fluid flows with finite delay in unbounded domains. Compared with the Navier-Stokes equations ($\omega = 0, \nu_{r} = 0$), the micropolar fluid flow consists of the angular velocity field $\omega$, which leads to a different nonlinear term $B(u, w)$ and an additional term $N(u)$ in the abstract equations (13). In addition, the time-delay term considered in this work also increases the difficulty.

The second purpose of this work is to investigate the upper semicontinuity of the pullback attractor with respect to the domain $\Omega$. To this end, using the arguments in [15,17], we first let $\{\Omega_m\}_{m = 1}^{\infty}$ be an expanding sequence of simply connected, bounded and smooth subdomains of $\Omega$ such that $\bigcup\limits_{m = 1}^{\infty}\Omega_m = \Omega$. Then we consider the Cauchy problem (3)-(5) in $\Omega_m$. We will conclude that there exists a pullback attractor $\widehat{\mathcal{A}}_{\widehat{H}(\Omega_m)}$ for the problem (3)-(5) in each $\Omega_m$. Finally, we establish the upper semicontinuity by showing $\lim\limits_{m\rightarrow\infty} {\rm dist}_{E^2_{\widehat{H}}} (\mathcal{A}_{\widehat{H}(\Omega_m)}(t), \mathcal{A}_{\widehat{H}}(t)) = 0, \ \forall\, t\in\mathbb{R}$.

Throughout this paper, we denote the usual Lebesgue space and Sobolev space by $L^p(\Omega)$ and $W^{m, p}(\Omega)$ endowed with norms $\|\cdot\|_p$ and $\|\cdot\|_{m, p}$, respectively. Especially, we denote $H^m(\Omega): = W^{m, 2}(\Omega)$.

$(\cdot, \cdot) -$ the inner product in $L^2(\Omega), H$ or $\widehat{H}$, $\langle \cdot, \cdot \rangle -$ the dual pairing between $V$ and $V^*$ or between $\widehat{V}$ and $\widehat{V}^*$. Throughout this article, we simplify the notations $\|\cdot\|_{2}$, $\|\cdot\|_{H}$ and $\|\cdot\|_{\widehat{H}}$ by the same notation $\|\cdot\|$ if there is no confusion. Furthermore, we denote

Following the above notations, we additionally denote

The norm $\|\cdot\|_{X}$ for $X\in \{E_{\widehat{H}}^2, \, E_{\widehat{V}}^2, \, E_{\widehat{H}\times L_{\widehat{V}}^2}^2\}$ is defined as

The rest of this paper is organized as follows. In section 2, we make some preliminaries. In section 3, we investigate the boundedness of the pullback attractor. In section 4, we prove the upper semicontinuity of the pullback attractor with respect to the domains.

2.   Preliminaries

In this section, for the sake of discussion, we first introduce some useful operators and put problem (3)-(5) into an abstract form. Then we recall some important known results about the non-autonomous micropolar fluid flows.

To begin with, we define the operators $A, B(\cdot, \cdot)$ and $N(\cdot)$ by

What follows are some useful estimates and properties for the operators $A$, $B(\cdot, \cdot)$ and $N(\cdot)$, which have been established in works [11,13].

Lemma 2.1. ${\rm (1)}$ The operator $A$ is linear continuous both from $\widehat{V}$ to $\widehat{V}^*$ and from $D(A)$ to $\widehat{H}$, and so is for the operator $N(\cdot)$ from $\widehat{V}$ to $\widehat{H}$, where $D(A): = \widehat{V}\cap \left(H^2(\Omega)\right)^3$.

${\rm (2)}$ The operator $B(\cdot, \cdot)$ is continuous from $V\times\widehat{V}$ to $\widehat{V}^*$. Moreover, for any $u\in V$ and $w\in\widehat{V}$, there holds

Lemma 2.2. ${\rm (1)}$ There are two positive constants $c_1$ and $c_2$ such that

${\rm (2)}$ There exists a positive constant $\alpha_0$ which depends only on $\Omega$, such that for any $(u, \psi, \varphi)\in V\times\widehat{V}\times \widehat{V}$ there holds

Moreover, if $(u, \psi, \varphi)\in V\times D(A)\times D(A)$, then

${\rm (3)}$ There exists a positive constant $c(\nu_r)$ such that

In addition,

where $\delta_1 : = \mathrm{min}\{\nu, \bar{\alpha}\}$.

According to the definitions of operators $A, B(\cdot, \cdot)$ and $N(\cdot)$, equations (3)-(5) can be formulated into the following abstract form:

where

Before recalling the known results for problem (13), we need to make the following assumptions with respect to $F$ and $G$.

Assumption 2.1. Assume that $G: \mathbb{R}\times L^2(-h, 0;\widehat{H}) \mapsto (L^2(\Omega))^3$ satisfies:

(ⅰ) For any $\xi\in L^2(-h, 0;\widehat{H})$, the mapping $\mathbb{R} \ni t \mapsto G(t, \xi)\in (L^2(\Omega))^3$ is measurable.

(ⅱ) $G(\cdot, 0) = (0, 0, 0)$.

(ⅲ) There exists a constant $L_G>0$ such that for any $t\in\mathbb{R}$ and any $\xi, \eta \in L^2(-h, 0; \widehat{H})$,

(ⅳ) There exists $C_G\in (0, \delta_1)$ such that, for any $t\geq\tau$ and any $w, v \in L^2(\tau-h, t; \widehat{H})$,

Assumption 2.2. Assume that $F(t, x) \in L_{loc}^2(\mathbb{R}; \widehat{H}), \ \forall\, t\geq\tau, \, \tau\in \mathbb{R}$, and there exists a $\gamma\in(0, 2\delta_1-2C_G)$ such that

In order to facilitate the discussion, we denote by $\mathcal{P}(X)$ the family of all nonempty subsets of $X$. Let $\mathcal{D}$ be a nonempty class of families parameterized in time $\widehat{D} = \{D(t): t\in\mathbb{R}\}\subseteq \mathcal{P}(X)$, which will be called a universe in $\mathcal{P}(X)$. Based on these notations, we can construct the universe $\mathcal{D}_{\gamma}$ in the following.

Definition 2.3. (Definition of universe $\mathcal{D}_{\gamma}$) Set

We denote by $\mathcal{D}_{\gamma}$ the class of all families $\widehat{D} = \{D(t) \ | \ t\in\mathbb{R} \} \subseteq \mathcal{P}(E_{\widehat{H}}^2)$ such that

where $\bar{B}_{E_{\widehat{H}}^2} (0, \rho_{\widehat{D}}(t))$ represents the closed ball in $E_{\widehat{H}}^2$ centered at zero with radius $\rho_{\widehat{D}}(t)$.

Based on the above assumptions, we can recall the global well-posedness of solutions and the existence of pullback attractor of problem (13).

Proposition 2.1. (Existence and uniqueness of solution, see [13,16]) Let Assumption 2.1 and Assumption 2.2 hold. Then for any $(w^{in}, \phi^{in}(s))\in E_{\widehat{H}}^2$, there exists a unique weak solution $w(\cdot): = w(\cdot;\tau, w^{in}, \phi^{in}(s))$ for system (13), which satisfies

Remark 2.1. According to Proposition 2.1, the biparametric mapping defined by

generates a continuous process in $E_{\widehat{H}}^2$ and $E_{\widehat{V}}^2$, respectively, which satisfies the following properties:

Proposition 2.2. (Existence of pullback attractor, see [16]) Under the Assumption 2.1 and Assumption 2.2, there exists a pullback attractor $\widehat{\mathcal{A}}_{\widehat{H}} = \big\{\mathcal{A}_{\widehat{H}}(t) \, \big| \, t\in\mathbb{R} \big\}$ for the process $\{U(t, \tau)\}_{t \geq \tau}$ that satisfies the following properties:

$\circ$ Compactness: for any $t\in {\mathbb R}, \mathcal{A}_{\widehat{H}}(t)$ is a nonempty compact subset of $E_{\widehat{H}}^2$;

$\circ$ Invariance: $U(t, \tau) \mathcal{A}_{\widehat{H}}(\tau) = \mathcal {A}_{\widehat{H}}(t), \, \forall \, t \geq \tau$;

$\circ$ Pullback attracting: for any $\widehat{B} = \{ B(\theta)|\, \theta \in {\mathbb R}\} \in \mathcal{D}_{\gamma}$, it holds that

$\circ$ Minimality: the family of sets $\widehat{\mathcal{A}}_{\widehat{H}}$ is the minimal in the sense that if

$\widehat {O} = \{ O(t)|\, t\in {\mathbb R}\} \subseteq \mathcal {P}(E_{\widehat{H}}^2)$ is another family of closed sets such that

then $\mathcal {A}_{\widehat{H}}(t)\subseteq O(t)$ for any $t\in \mathbb{R}$.

Finally, we introduce a useful lemma, which plays an important role in the proof of higher regularity of the pullback attractor.

Lemma 2.4. (see [12]) Let X, Y be Banach spaces such that X is reflexive, and the inclusion $X\subset Y$ is continuous. Assume that $\{w_n\}_{n\geq 1}$ is a bounded sequence in $L^{\infty}(\tau, t; X)$ such that $w_n\rightharpoonup w$ weakly in $L^q(\tau, t; X)$ for some $q\in[1, +\infty)$ and $w\in\mathcal{C}([\tau, t]; Y)$. Then $w(t)\in X$ and

3.   Boundedness of the pullback attractor for the universe $\mathcal{D}_{\gamma}$

This section is devoted to investigating the boundedness of the pullback attractor for the universe $\mathcal{D}_{\gamma}$ given by a tempered condition in space $E_{\widehat{H}}^2$. To this end, we consider the Galerkin approximation of the solution $w(t)$ of system (13), which is denoted by

where the sequence $\{e_j\}_{j = 1}^{\infty}$ is an orthonormal basis of $\widehat{H}$ and formed by eigenvectors of the operator $A$, that is, for all $j\geq 1$,

where the eigenvalues $\{\lambda_j\}_{j\geq 1}$ of $A$ are real number that we can order in such a way

It is not difficult to check that $\xi_{n, j}(t)$ is the solution of the following ordinary differential equations:

Next we verify the following estimates of the Galerkin approximate solutions defined by (16).

Lemma 3.1. Let Assumption (2.1) and Assumption (2.2) hold. Then for any $\tau\in\mathbb{R}, \epsilon > 0, t > \tau+h+\epsilon$ and $(w^{in}, \phi^{in}(s)) \in \mathcal{A}_{\widehat{H}}(\tau)$, we have

(ⅰ) the set $\big\{ w_n(r;\tau, w^{in}, \phi^{in}(s)) \, \big| \, r\in [\tau+\epsilon, t]\big\}_{n\geq 1}$ is bounded in $\widehat{V}$;

(ⅱ) the set $\big\{ w_{nr}(\cdot;\tau, w^{in}, \phi^{in}(s)) \, \big| \, r\in [\tau+h+\epsilon, t]\big\}_{n\geq 1}$ is bounded in $L_{\widehat{V}}^2$;

(ⅲ) the set $\big\{ w_n(\cdot;\tau, w^{in}, \phi^{in}(s)) \big\}_{n\geq 1}$ is bounded in $L^2(\tau+\epsilon, t; D(A))$;

(ⅳ) the set $\big\{ w_n'(\cdot;\tau, w^{in}, \phi^{in}(s)) \big\}_{n\geq 1}$ is bounded in $L^2(\tau+\epsilon, t; \widehat{H})$.

Proof. Multiplying (18) by $\beta_{nj}(t)$ and summing them for $j = 1$ to $n$, then using (7), (12) and Young's inequality, we obtain

Then integrating the above inequality over $[\tau, t], t {\geq} \tau,$ leads to

which implies

Thanks to (17), multiplying (18) by $\lambda_j\beta_{nj}(t)$ and summing the resultant equation for $j = 1$ to $n$ yields that

Observe that $\|u_n(t)\| {\leq} \|w_n(t)\|$, $\|\nabla w_n(t)\| {\leq} \|w_n(t)\|_{\widehat V}$, using (10), (11) and Young's inequality, it is easy to see that

and

Therefore

Set

then we get

By Gronwall inequality, (21) yields

Integrating the above inequality for $\tilde{r}$ from $r-\epsilon$ to $r$, we obtain

Since

we can conclude that

which together with (19) and Assumption 2.2 implies the assertion (i).

Now, integrating (20) over $[\tau+\epsilon, t]$, we obtain

which together with (19), Assumption 2.2 and the assertion (ⅰ) gives the assertion (ⅲ).

In addition,

which together with the assertion (ⅰ) yields the assertion (ⅱ).

Finally, multiplying (18) by $\beta_{nj}'(t)$ and summing the resultant equation for $j = 1$ to $n$, we obtain

From Assumption 2.1, it follows that

By Lemma 2.2,

and

Taking (23)-(26) into account, we obtain

Integrating the above inequality, yields

which together with (20), Assumption 2.2 and the assertions (ⅰ)-(ⅲ) gives the assertion (ⅳ). The proof is complete.

With the above lemma, we are ready to conclude this section with the following $H^1$-boundedness of the pullback attractor $\widehat{\mathcal{A}}_{\widehat{H}}$ for the universe $\mathcal{D}_{\gamma}$.

Theorem 3.2. Let assumptions 2.1-2.2 hold and $\widehat{\mathcal{A}}_{\widehat{H}} = \big\{ \mathcal{A}_{\widehat{H}}(t) \, \big| \, t\in\mathbb{R} \big\}$ be the pullback attractor of system (13). Then for any $\tau\in\mathbb{R}, \epsilon>0$, $t > \tau+h+\epsilon$ and $(w^{in}, \phi^{in})\in E_{\widehat{H}}^2$, the set $\bigcup\limits_{r\in[\tau+h+\epsilon, t]} U(r, \tau)\mathcal{A}_{\widehat{H}}(\tau)$ is bounded in $E_{\widehat{V}}^2$.

Proof. Based on Lemma 3.1, following the standard diagonal procedure, there exist a subsequence (denoted still by) $\{w_n\}_{n{\geq} 1}$ and a function $w(\cdot) \in L^{\infty}(\tau+\epsilon, t; \widehat{V}) \cap L^2(\tau+\epsilon, t;D(A))$ with $w'(\cdot) \in L^2(\tau+\epsilon, t; \widehat{H})$ such that, as $n\rightarrow \infty$,

Furthermore, it follows from the uniqueness of the limit function that $w(\cdot)$ is a weak solution of system (13). According to compact embedding theorem, (28) and (29) implies $w(\cdot) \in \mathcal{C}([\tau+\epsilon, t];\widehat{V})$. Then Theorem 3.1 follows from (27), Lemma 2.4 and Lemma 3.1.

Remark 3.1. We here point out that the boundedness of pullback attractor $\widehat{\mathcal{A}}$ in $E_{D(A)}^2$ can be proved by using similar proof as that in $E_{\widehat V}^2$ if we improve the regularity of $F(t)$ and $G(t, w_{nt})$ with respect to $t$, where $D(A): = \widehat{V}\cap(H^2)^3$. Exactly, assume that

$\bf{\mathrm{(I)}}$ $F(t, x) \in W_{loc}^{1, 2}(\mathbb{R}; \widehat{H}), \ \forall\, t{\geq}\tau, \, \tau\in \mathbb{R}$, and $\int_{-\infty}^t e^{\gamma\theta} \|F'(\theta, x)\|_{\widehat{V}^*}^2 {\rm d}\theta < + \infty$.

$\bf{\mathrm{(II)}}$ $\big(G(t, \xi)\big)' = \frac{{\rm d}G}{{\rm d}t}$: $\mathbb{R}\times L^2(-h, 0;\widehat{H}) \mapsto (L^2(\Omega))^3$ satisfies:

$\circ$ For any $\xi\in L^2(-h, 0;\widehat{H})$, the mapping $\mathbb{R} \ni t \mapsto G(t, \xi)\in (L^2(\Omega))^3$ is measurable.

$\circ$ $\big(G(\cdot, 0)\big)' = (0, 0, 0)$.

$\circ$ There exists a constant $\tilde{L}_G>0$ such that for any $t\in\mathbb{R}$ and any $\xi, \eta \in L^2(-h, 0; \widehat{H})$,

$\circ$ There exists $\tilde{C}_G\in (0, \delta_1)$ such that, for any $t{\geq}\tau$ and any $w, v \in L^2(\tau-h, t; \widehat{H})$,

Then we can deduce that the Galerkin approximate solutions $\{w_n(\cdot)\}_{n{\geq} 1}$ is bounded in $D(A) = \widehat{V}\cap(H^2)^3$. Moreover, $\{w_n'(\cdot)\}_{n{\geq} 1}$ is bounded in $\widehat{H}$. Further, we can conclude the $H^2$-boundedness of the pullback attractor $\widehat{\mathcal{A}}$.

4.   Upper semicontinuity of the pullback attractor

In this section, we concentrate on verifying the upper semicontinuity of the pullback attractor $\widehat{\mathcal A}_{\widehat H}$ obtained in Propositon 2.2 with respect to the spatial domain. To this end, first we let $\{\Omega_m\}_{m = 1}^{\infty}$ be an expanding sequence of simply connected, bounded and smooth subdomains of $\Omega$ such that $\bigcup\limits_{m = 1}^{\infty}\Omega_m = \Omega$. Then we consider the system (3) in each $\Omega_m$ and define the operators $A, B(\cdot, \cdot)$ and $N(\cdot)$ as previous (in (6)) with the spatial domain $\Omega$ replaced by $\Omega_m$. Further we can formulate the weak version of problem (3)-(5) as follows:

On each bounded domain $\Omega_m$, the well-posedness of solution can be established by Galerkin method and energy method, one can refer to [7].

Lemma 4.1. Suppose Assumption 2.1 and Assumption 2.2 hold, then for any given $(w_m^{in}, \phi_m^{in})\in E_{\widehat{H}(\Omega_m)}^2$, system (30) has a unique weak solution $w_m$ satisfying

Moreover, the solution $w_m(\cdot)$ depends continuously on the initial value $w_m^{in}$ with respect to $\widehat{H}(\Omega_m)$ norm.

According to Lemma 4.1, the map defined by

generates a continuous process $\{U(t, \tau)\}_{t{\geq}\tau}$ in $\widehat{H}(\Omega_m)$. Moreover, on any smooth bounded domain $\Omega_m$, with similar proof as those of Lemma 3.2, Lemma 3.3 and Lemma 3.6 in [16], we can obtain the existence of pullback $\mathcal{D}_{\gamma}^{\widehat{H}(\Omega_m)}$-absorbing for the process $\{U_m(t, \tau)\}_{t{\geq}\tau}$ and the pullback $\mathcal{D}_{\gamma}^{\widehat{H}(\Omega_m)}$-asymptotic compactness of the process in $\widehat{H}(\Omega_m)$. That is,

Lemma 4.2. Under the assumptions 2.1 and 2.2, it holds that

$\mathrm{(1)}$ for any $(w_m^{in}, \phi_m^{in}(s))\in E_{\widehat{H}(\Omega_m)}^2$, the family $\widehat{\mathcal{B}}^{\widehat{H}(\Omega_m)} : = \big\{ \mathcal{B}^{\widehat{H}(\Omega_m)}(t) \, \big| \, t\in \mathbb{R} \big\}$ given by

is pullback $\mathcal{D}_{\gamma}^{\widehat{H}(\Omega_m)}$-absorbing for the process $\{U_m(t, \tau)\}$, where $R_1(t)$ is bounded for all $t \in \mathbb{R}$.

$\mathrm{(2)}$ for any $\epsilon > 0$, there exist $r_m: = r_m(\epsilon, t, \widehat{\mathcal{B}}^{\widehat{H}(\Omega_m)}) > 0$, $\tau_m : = \tau_m(\epsilon, t, \widehat{\mathcal{B}}^{\widehat{H}(\Omega_m)})<t$ such that for any $r \in [r_m, m], \tau {\leq} \tau_m$,

$\mathrm{(3)}$ the process $\{U_m(t, \tau)\}_{t{\geq}\tau}$ is pullback $\mathcal{D}_{\gamma}^{\widehat{H}(\Omega_m)}$-asymptotically compact in $\widehat{H}(\Omega_m)$.

Then based on the Remark 2.1 in [16], we conclude that

Theorem 4.3. Let assumptions 2.1 and 2.2 hold. Then there exists a pullback attractor $\widehat{\mathcal{A}}_{\widehat{H}(\Omega_m)} = \big\{ \mathcal{A}_{\widehat{H}(\Omega_m)}(t) \, \big| \, t\in\mathbb{R} \big\}$ for the system (30) in $\widehat{H}(\Omega_m)$.

In the following, we investigate the relationship between the solutions of system (30) and (13). Indeed, we devoted to proving the solutions $w_m$ of system (30) converges to the solution of system (13) as $m\rightarrow\infty$. To this end, for $w_m\in \widehat{H}(\Omega_m)$, we extend its domain from $\Omega_m$ to $\Omega$ by setting

then it holds that

Next, using the same proof as that of Lemma 8.1 in [8], we have

Lemma 4.4. Let assumptions 2.1-2.2 hold and $\{(w_m^{in}, \phi_m^{in}(s))\}_{m{\geq} 1}$ be a sequence in $E^2_{\widehat{H}(\Omega_m)\times L^2_{\widehat{V}(\Omega_m)}}$ converging weakly to an element $(w^{in}, \phi^{in}(s))\in E^2_{\widehat{H}\times L^2_{\widehat{V}}}$ as $m\rightarrow\infty$. Then for any $t {\geq} \tau$,

Based on Lemma 4.4, we set out to prove the following important lemma.

Lemma 4.5. Let assumptions 2.1-2.2 hold, then for any $t\in\mathbb{R}$, any sequence $\{(w_m(t), w_{mt}(s))\}_{m{\geq} 1}$ with $(w_m(\tau), w_{m\tau}(s)) = (w_m^{in}, \phi_m^{in}(s))\in\mathcal{A}_{\widehat{H}(\Omega_m)}(\tau), m = 1, 2, \cdots$, there exists $(w(t), w_{t}(s))\in\mathcal{A}_{\widehat{H}}(t)$ such that

Proof. From the compactness of pullback attractor, it follows that the sequence $\{(w_m^{in}, \phi_m^{in}(s))\}_{m{\geq} 1}$ is bounded in $E_{\widehat H}^2$. Hence, there exist a subsequence (denoted still by) $\{(w_m^{in}, \phi_m^{in}(s))\}_{m{\geq} 1}$ and a $(w^{in}, \phi^{in}(s))\in \mathcal{A}_{\widehat{H}}(\tau)$ such that

Further, according to Lemma 4.4 and the invariance of the pullback attractor, we can conclude that for any $t\in\mathbb{R}$, there exist a $(w_m(t), w_{mt}(s))\in \mathcal{A}_{\widehat{H}(\Omega_m)}(t)$ with $(w_m(\tau), w_{m\tau}(s))\in \mathcal{A}_{\widehat{H}(\Omega_m)}(\tau)$ and a $(w(t), w_t(s))\in \mathcal{A}_{\widehat{H}}(t)$ with $(w(\tau), w_{\tau}(s))\in \mathcal{A}_{\widehat{H}}(\tau)$ such that

Then, using the same way of proof as Lemma 3.6 in [16], we can obtain that the convergence relation of (37) is strong. The proof is complete. With the above lemma, we are ready to state the main result of this section.

Theorem 4.6. Let Assumption 2.1 and Assumption 2.2 hold, then for any $t\in\mathbb{R}$, it holds that

where $\mathcal{A}_{\widehat{H}}(t)$ and $\mathcal{A}_{\widehat{H}(\Omega_m)}(t)$ are the pullback attractor of system (13) and system (30), respectively.

Proof. Suppose the assertion (38) is false, then for any $m = 1, 2, \cdots$, there exist $t_0\in\mathbb{R}, \epsilon_0 > 0$ and a sequence $(w_m(t_0), w_{mt_0}(s))\in\mathcal{A}_{\widehat{H}(\Omega_m)}(t_0)$ such that

However, it follows from Lemma 4.5 that there exists a subsequence

such that

which is in contradiction to (39). Therefore, (38) is true. The proof is complete.