Pullback $\mathcal{D}$-attractors of the three-dimensional non-autonomous micropolar equations with damping

1.   Introduction

In this paper, we study the following 3D non-autonomous micropolar equations with a damping term

where $(x, t)\in \mathbb{T}^{3}\times[\tau, +\infty)$ and $\tau\in \mathbb{R}$. $\mathbb{T}^{3}\subset \mathbb{R}^{3}$ is a periodic domain. In system (1.1), the fluid velocity and the micro-rotational velocity are represented by $u = u(x, t)$ and $w = w(x, t)$, respectively. $p = p(x, t)$ is the scalar pressure. $f = f(x, t)$ and $g = g(x, t)$ denote the given external forces. $\nu$, $\kappa$ and $\sigma$ denote kinematic viscosity, micro-rotational viscosity and damping coefficient, respectively, which are all positive constants. $\beta\geq1$ is a constant. $\gamma$ and $\mu$, representing the angular viscosities, are also positive constants. For convenience, let $\nu = \kappa = \gamma = \mu = \sigma = 1$.

Eringen firstly introduced microfluids in ^[4] and showed a complete theory for micropolar fluids in ^[5]. For physical background and mathematical theory, we can refer ^[12] and ^[18]. Galdi and Rionero proved the existence and uniqueness of weak solutions for the micropolar equations in ^[6]. In ^[24], the existence of strong solution for 3D micropolar equations was proved for $\beta = 3$ and $4\sigma(\nu+\kappa) > 1$ or $\beta > 3$. And there were also many works with magneto-micropolar equations, we can refer ^[9,10,11,20]. In ^[20], global well-posedness of a 3D MHD system was studied in porous media.

As $w = 0$, Eqs (1.1) are Navier-Stokes equations. Cai and Jiu ^[1] firstly considered 3D incompressible Navier-Stokes equations with damping $\alpha|u|^{\beta-1}u$ and they proved the existence of weak solution with $\beta\geq1$ as well as strong solution with $\beta\geq\frac{7}{2}$, and uniqueness for strong solution with $\frac{7}{2}\leq\beta\leq5$. In ^[25], the regularity and uniqueness for three-dimensional incompressible Navier-Stokes system with damping term were studied. The generalized Navier-Stokes equations with damping were researched in ^[13] and the existence of weak solution was proved in $R^{n}$, $n\geq2$. The uniform global attractor and trajectory attractor for 3D Navier-Stokes equations were considered in ^[3]. In ^[8], $L_{2}$ decay of weak solutions for $\beta > 2$ with $\alpha > 0$ and the asymptotic stability of the solution to incompressible Navier-Stokes equations with damping for $\beta > 3$ with $\alpha > 0$ or $\beta = 3$ with $\alpha\geq\frac{3}{2}$ were proved.

In this paper, we devote to research pullback attractors for three-dimensional non-autonomous micropolar equations (1.1) with damping term. Recently, attractors have been interested many authors ^{[2,7,10,17,19,21,22,23]}. Caraballo, Lukaszewicz and Real considered pullback attractors of two-dimensional Navier-Stokes system in ^[2]. In ^[7], the existence of pullback attractors for nonautonomous reaction-diffusion equation was proved in $R^{n}$, $n\geq3$. In ^[17], the existence of pullback attractors for 3D Navier-Stokes problem with damping was proved in $V$ and $H^{2}$ for $3 < \beta\leq5$. In ^[19], Sun and Li have studied global pullback attractors and pullback exponential attractors for the 2D non-autonomous micropolar fluid system. Global attractor of the 3D magnetohydrodynamics equations with damping was considered in ^[10]. However, the existence of pullback $\mathcal{D}$-attractors for Eqs (1.1) is not obtained in $V_{1}\times V_{2}$ and $H^{2}\times H^{2}$.

To obtain our main results, we need to deal with nonlinear terms $(u\cdot\nabla)u$, $(u\cdot\nabla)w$ and $\sigma|u|^{\beta-1}u$. Hence, we should show some estimates by using Sobolev and uniform Gronwall inequalities. To prove the existence of pullback $\mathcal{D}$-attractors, we should restrict $3 < \beta < 5$ from (3.20) and (3.47).

In this paper, the structure is organized as follows. In section 2, some definitions as well as notions are recalled and our main results are given. In section 3, we show some estimates to overcome the difficulties of nonlinear terms. In section 4, the existence of pullback $\mathcal{D}$-attractors for Eqs (1.1) is proved in $V_{1}\times V_{2}$ and $H^{2}\times H^{2}$ with $3 < \beta < 5$.

2.   Preliminaries

In this section, we will show some definitions and lemmas. We also give some notions and assumptions which we will use in the following. Finally, we give our main results.

Firstly, assumed $X$ is a complete metric space, $\mathcal{P}(X)$ represents the family of all nonempty subsets of $X$ and $\mathcal{D}$ is a nonempty class of parameterized sets $\hat{D} = \{D(t):t\in \mathbb{R}\}\subset\mathcal{P}(X)$. In the following, we give the definition of pullback $\mathcal{D}$-attractor which we can refer ^[17] to get.

Definition 2.1. A family $\hat{A} = \{A(t):t\in \mathbb{R}\}\subset\mathcal{P}(X)$ is called a pullback $\mathcal{D}$-attractor for the process $\{U(t, \tau)\}_{t\geq\tau}$ in $X$, if

(1) $A(t)$ is compact for every $t\in \mathbb{R}$,

(2) $\hat{A}$ is invariant, that is, $U(t, \tau)A(\tau) = A(t)$, for $-\infty < \tau\leq t < +\infty$,

(3) $\hat{A}$ is pullback $\mathcal{D}$-attracting, that is,

And $\hat{A}$ is said to be minimal if $A(t)\subset C(t)$ for any family $\hat{C} = \{C(t); t\in \mathbb{R}\}\subset\mathcal{P}(X)$ of closed sets such that for any $\hat{B}\in\mathcal{D}$,

In following, we give lemmas and definitions which we can refer ^[17], and these play important roles in the proof of our main results.

Definition 2.2. Assumed $X$ is a complete metric space, a two-parameter family $\{U(t, \tau):-\infty < \tau\leq t < +\infty\}$ of mapping $U(t, \tau): X\rightarrow X, t\geq\tau, \tau\in \mathbb{R}$ is called evolutionary process if

(1) $U(t, s)U(s, \tau) = U(t, \tau)$, for all $\tau\leq s\leq t$,

(2) $U(\tau, \tau) = Id$ is identity operator for all $\tau\in \mathbb{R}$.

Lemma 2.3. Assume $\{U(t, \tau)\}_{t\geq\tau}$ is a process in $X$ satisfying the following conditions

(1) $\{U(t, \tau)\}_{t\geq\tau}$ is norm-to-weak continuous in $X$,

(2) there exists a family $\hat{B}$ of pullback $\mathcal{D}$-absorbing sets $\{B(t); t\in \mathbb{R}\}$ in $X$,

(3) $\{U(t, \tau)\}_{t\geq\tau}$ is pullback $\mathcal{D}$-asymptotically compact.

Then there exists a minimal pullback $\mathcal{D}$-attractor $\hat{A} = \{A(t):t\in \mathbb{R}\}$ in $X$ given by

Definition 2.4. It is said that a process $U(t, \tau)$ is norm-to-weak continuous on $X$ if for all $t, \tau\in \mathbb{R}$ with $t\geq\tau$ and for any sequence $x_{n}\in X$,

where $\rightarrow$ and $\rightharpoonup$ represent strong convergence and weak convergence, respectively.

And we can easily get that it is norm-to-weak continuous process as it is a continuous process.

Next, we give the following lemma which can help us complete the proof of norm-to-weak continuous.

Lemma 2.5. Assume $X$, $Y$ are two Banach spaces. Let $X^{*}$, $Y^{*}$ be dual spaces of $X$ and $Y$, respectively. Suppose that $X$ is dense in $Y$, the injection $i:X\rightarrow Y$ is continuous, its adjoint $i^{*}:Y^{*}\rightarrow X^{*}$ is dense, and $\{U(t, \tau)\}_{t\geq\tau}$ is a norm-to-weak continuous process on $Y$. Then $\{U(t, \tau)\}_{t\geq\tau}$ is a norm-to-weak continuous process on $X$ if and only if $U(t, \tau)$ maps compact sets of $X$ into bounded sets of $X$, for every $\tau\in \mathbb{R}$ and $t\geq\tau$.

Definition 2.6. $\hat{B}\in\mathcal{D}$ is said to be pullback $\mathcal{D}$-absorbing for the process $\{U(t, \tau)\}_{t\geq\tau}$, if for every $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, there exists a $\tau_{0}(t, \hat{D})\leq t$ such that $U(t, \tau)D(\tau)\subset B(t)$ for every $\tau\leq\tau_{0}(t, \hat{D})$.

Definition 2.7. It is said that the process $\{U(t, \tau)\}_{t\geq\tau}$ is a pullback $\mathcal{D}$-asymptotically compact, if sequence $\{U(t, \tau_{n})x_{n}\}_{n = 1}^{\infty}$ is relatively compact in $X$ for all $t\in \mathbb{R}$, $\hat{D}\in\mathcal{D}$ and every sequence $\tau_{n}\rightarrow -\infty$ as well as $x_{n}\in D(\tau_{n})$.

In the following, we give notions of function spaces

Let the norm of the space $(L^{p}(\mathbb{T}^{3}))^{3}$ be represented by $\|\cdot\|_{p}$, particularly, $\|\cdot\|$ represents the norm of the space $H_{1}$ and the space $H_{2}$. $H^{s}$ means the usual Sobolev space and its norm $\|\cdot\|_{H^{s}} = \|A^{\frac{s}{2}}\cdot\|$, particularly, as $s = 2$, $\|\cdot\|_{H^{2}} = \|A\cdot\|$.

In the periodic space, we recall that

Lemma 2.8. ^[14] The Leray projector $\mathbb{P}$ on the torus and on the whole space commutes with any derivative:

for all $u\in \dot{H}^{1}$.

In the following, let

where $\mathbb{P}$ represents the Helmholtz-Leray orthogonal projection from $(L^{2}(\mathbb{T}^{3}))^{3}$ onto $H_{1}$ and $\mathbb{P}u = u$ on the torus. Then we can rewrite the Eqs (1.1) as following

where let $G(u) = \mathbb{P}|u|^{\beta-1}u$.

In this paper, to complete our proof, we should assume

and

where $L_{b}^{2}(\mathbb{R}; H_{1})$ represents the collection of functions that are translation bounded in $L^{2}_{loc}(\mathbb{R}; H_{1})$. Note that function $f(t)$ is translation bounded in $L^{2}_{loc}(\mathbb{R}; H_{1})$ if

And for $L_{b}^{2}(\mathbb{R}; H_{2})$, we have similar definition.

We also assume $f(x, t)$ is uniformly bounded in $H_{1}$, $g(x, t)$ is uniformly bounded in $H_{2}$, that is, there exists a positive constant $C$ such that

Please note that $C$ is a positive constant and it could mean different numbers in different places.

Further suppose that $f(x, t)$ and $g(x, t)$ satisfy following inequalities

for any $t\in\mathbb{R}$, where $\lambda$ is given in the following.

Next, let $\mathcal{D}$ be the class of all families $\hat{D} = \{D(t):t\in \mathbb{R}\}\subset\mathcal{P}((H^{1}(\mathbb{T}^{3}))^{3})$ such that

where $[D(t)] = \sup\{\|u(t)\|^{2}_{V_{1}}+\|w(t)\|^{2}_{V_{2}}:u, w\in D(t)\}$ and $\lambda > 0$ is given in the following.

And we can use the Poincar$\rm{\acute{e}}$ inequality, i.e., there exists a constant $\lambda > 0$ such that

where $\lambda$ is the minimum of the first eigenvalues of Stokes operators $Au$ and $Aw$.

Then, we give our main theorems.

Theorem 2.9. Suppose (2.3)-(2.5) hold, $f\in L^{2}_{loc}(\mathbb{R}; H_{1})$, $g\in L^{2}_{loc}(\mathbb{R}; H_{2})$, $f_{t}\in L^{2}_{b}(\mathbb{R}; H_{1})$ and $g_{t}\in L^{2}_{b}(\mathbb{R}; H_{2})$. Let $3 < \beta < 5$ and $\tau\in \mathbb{R}$, then there exists a pullback $\mathcal{D}$-attractor $\mathcal{A}_{1}$ of the process $\{U(t, \tau)\}_{t\geq\tau}$ for system (1.1) in $V_{1}\times V_{2}$.

Theorem 2.10. Suppose (2.3)-(2.5) hold, $f\in L^{2}_{loc}(\mathbb{R}; H_{1})$, $g\in L^{2}_{loc}(\mathbb{R}; H_{2})$, $f_{t}\in L^{2}_{b}(\mathbb{R}; H_{1})$ and $g_{t}\in L^{2}_{b}(\mathbb{R}; H_{2})$. Let $3 < \beta < 5$ and $\tau\in \mathbb{R}$, then $\{U(t, \tau)\}_{t\geq\tau}$ for system (1.1) has a pullback $\mathcal{D}$-attractor $\mathcal{A}_{2}$ in $H^{2}\times H^{2}$.

Now, let recall the existence of weak and strong solutions for Eqs (1.1).

Theorem 2.11. Suppose $f\in L^{2}_{b}(\mathbb{R}; H_{1})$, $g\in L^{2}_{b}(\mathbb{R}; H_{2})$, $u_{\tau}\in H_{1}$, $w_{\tau}\in H_{2}$ and $\beta\geq1$. Then for every given $T > \tau$, there exist at least one solution $(u, w)$ of (2.2),

Proof. In ^[1], the existence of weak solution for Navier-Stoke equations with damping has been proved, we can use the similar proof to get the existence of weak solutions for Eqs (2.2) and omit it.

We say that $(u, w)$ is a strong solution of (1.1), if it is a weak solution of (1.1), and satisfies

Theorem 2.12. Suppose $\beta > 3$, $f\in L^{2}_{b}(\mathbb{R}; H_{1})$, $g\in L^{2}_{b}(\mathbb{R}; H_{2})$, $u_{\tau}\in V_{1}\cap(L^{\beta+1}(\mathbb{T}^{3}))^{3}$ and $w_{\tau}\in V_{2}$. Then there exists a strong solution $(u, w)$ of Eqs (1.1),

Proof. Due to ^[24], we can take similar method to prove the existence of strong solution for Eqs (1.1) and omit it.

3.   Estimates of solutions

In this section, some estimates will be given. These estimates play an important role in the proof of our main results. In the following, we give some lemmas we will use.

Lemma 3.1. Assume (2.3)-(2.5) hold, $f\in L^{2}_{loc}(\mathbb{R}; H_{1})$ and $g\in L^{2}_{loc}(\mathbb{R}; H_{2})$. Let $3 < \beta < 5$ and $\tau\in \mathbb{R}$, and $(u, w)$ be the solution of system (1.1). For every $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, there exists a constant $\tau_{0} = \tau_{0}(t, \hat{D}) < t$ such that

and

for every $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$ and $\tau\leq\tau_{0}(t, \hat{D})$.

Proof. Multiplying the first equation and the second equation of (2.2) by $u$ and $w$, respectively, and integrating over $\mathbb{T}^{3}$, we can have

So, we easily get

and

Multiplying (3.5) by $e^{\lambda t}$ and integrating over $[\tau, t]$, then we obtain

Due to $u_{\tau}\in D(\tau)$ and $w_{\tau}\in D(\tau)$, for any $t\in \mathbb{R}$, we can have there exists a constant $\tau_{0}\leq t$ such that

where $\tau_{0} = \frac{1}{\lambda}\ln\frac{\int_{-\infty}^{t}e^{\lambda s}(\|f(s)\|^{2}+\|g(s)\|^{2})ds}{\lambda(\|u_{\tau}\|^{2}+\|w_{\tau}\|^{2})}$.

So we easily get

Integrating (3.6) over $[\tau, t]$, we have

Multiplying (3.4) by $e^{\lambda t}$ and integrating over $[\tau, t]$, then using (3.9), we can get

By (3.8) and (3.10), the proof of Lemma 3.1 is completed.

Lemma 3.2. Under the assumption of Lemma 3.1. For every $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, then there exists a constant $\tau_{1} = \tau_{1}(t, \hat{D})$ such that for every $\tau\leq\tau_{1}$ and $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$,

and

Proof. Multiplying (3.5) by $e^{\lambda t}$ and integrating over $[\tau, s]$, then we can have for any $s\in[t-1, t]$, there exists a constant $\tau_{1}\equiv\tau_{0}-1 < t-1$, such that for every $\tau\leq\tau_{1}$,

Integrating (3.13) over $[t-1, t]$ with respect to $s$, we can obtain

Multiplying (3.4) by $e^{\lambda t}$ and integrating over $[t-1, t]$, then using (3.14), we can have for every $\tau\leq\tau_{1}$,

This completes the proof of Lemma 3.2.

Lemma 3.3. Under the hypothesis of Lemma 3.2, for every $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, we have

and

for $\tau\leq\tau_{1}$ and $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$.

Proof. By using Lemma 3.2, we can directly get the result.

Lemma 3.4. Assume (2.3)-(2.5) hold, $f\in L^{2}_{loc}(\mathbb{R}; H_{1})$ and $g\in L^{2}_{loc}(\mathbb{R}; H_{2})$. Let $3 < \beta < 5$ and $\tau\in \mathbb{R}$, and $(u, w)$ be the solution of system (1.1). For every $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, then there exists a constant $\tau_{3} = \tau_{3}(t, \hat{D})$, such that for any $\tau\leq\tau_{3}$ and $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$,

and

Proof. Inspired by ^[24], we can get for $\beta > 3$,

then by using uniform Gronwall Lemma on $[t-1, t]$, we can obtain that there exists a constant $\tau_{2}\equiv\tau_{1}-1$, for any $\tau\leq\tau_{2}$,

where we use the following inequality

Multiplying the second equation of (2.2) by $Aw$ and integrating over $\mathbb{T}^{3}$, then we can get

Since

By using above inequalities, we easily get

Then by using uniform Gronwall inequality on $[t-1, t]$, we can obtain there has a constant $\tau_{3}\equiv\tau_{2}-1$, for any $\tau\leq\tau_{3}$,

Hence, we complete the Lemma 3.4.

Lemma 3.5. Under the hypothesis of Lemma 3.4. Then for any $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$,

and

for every $\tau\leq\tau_{3}$ and $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$.

Proof. Multiplying the first equation of (2.2) by $u_{t}$ and integrating over $\mathbb{T}^{3}$, then we can obtain

For $3 < \beta < 5$, we can have the following inequality

Taking (3.29) into (3.28), we can have

For (3.30), using uniform Gronwall Lemma, we get

Next, multiplying the second equation of (2.2) by $w_{t}$ and integrating over $\mathbb{T}^{3}$, we have

where we use inequality (3.23). Then applying uniform Gronwall Lemma, we obtain

So, the proof of Lemma 3.5 is finished.

Lemma 3.6. Assume (2.3)-(2.5) hold, $f\in L^{2}_{loc}(\mathbb{R}; H_{1})$, $g\in L^{2}_{loc}(\mathbb{R}; H_{2})$, $f_{t}\in L^{2}_{b}(\mathbb{R}; H_{1})$ and $g_{t}\in L^{2}_{b}(\mathbb{R}; H_{2})$. Let $3 < \beta < 5$ and $\tau\in \mathbb{R}$ and $(u, w)$ be the solution of Eqs (1.1). Then for every $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, there exists a constant $\tau_{4} = \tau_{4}(t, \hat{D})$, such that for any $\tau\leq\tau_{4}$ and $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$,

where $r_{1}(t)$ is a positive constant which is independent of the initial data.

Proof. According to (3.28), (3.29) and (3.32), we have

then integrating over $[t-1, t]$, we can get there has a constant $\tau_{4}\equiv\tau_{3}-1$, for every $\tau\leq\tau_{4}$,

where $r_{0}(t) = Ce^{-\lambda t}(G_{1}(t)+G_{2}(t))$.

Applying $\partial_{t}$ to the first and the second equations of system (2.2), and multiplying $u_{t}$ and $w_{t}$, respectively, we have

For $I_{1}$, according to the Lemma 2.4 of ^[16], we obtain that $I_{1}\leq0$.

For $I_{2}$, by using H$\rm{\ddot{o}}$lder and Young inequalities, we easily get

Similarly, we have

and

By using above inequalities, we can get

Then using uniform Gronwall Lemma, we can obtain

where we let $C(t) = C\int_{t-1}^{t}(1+\|\nabla u(s)\|^{4}+\|\nabla w(s)\|^{4})ds$.

By inequality (3.42), the proof of Lemma 3.6 is completed.

Lemma 3.7. Under the assumption of Lemma 3.6. Then for every $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, we obtain that for any $\tau\leq\tau_{4}$ and $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$,

and

where $r_{2}(t)$ and $r_{3}(t)$ are all positive constants which are independent of the initial data.

Proof. Here, applying Minkowshi inequality to the first equation of system (2.2), then we can obtain

For term $\|B(u)\|$, using similar method of (3.23), we have

For term $\||u|^{\beta-1}u\|$, using Sobolev Lemma, we get for $3 < \beta < 5$,

Taking (3.46) and (3.47) into (3.45), we obtain

Inspired by ^[15], we let

and we have

Hence, multiplying the second equation of (2.2) by $Aw$, we have

By using (3.23), we get

The proof of Lemma 3.7 is finished.

Lemma 3.8. Under the hypothesis of Lemma 3.7, for any $t\in \mathbb{R}$ and $\hat{D}\in\mathcal{D}$, then we get for every $\tau\leq\tau_{4}$ and $u_{\tau}\in D(\tau)$, $w_{\tau}\in D(\tau)$,

where $r_{5}(t)$ is a positive constant which is independent of the initial data.

Proof. Integrating (3.41) over $[t, t+1]$, we have

Then using Sobolev inequality and Lemma 3.7, we get

similarly,

Applying $\partial_{t}$ to the first and second equations of (2.2), then multiplying them by $Au_{t}$ and $Aw_{t}$, respectively, we obtain

Next, we estimate right terms of inequality.

For $J_{1}$, using Sobolev embedding Lemma, we have

For $J_{3}$, we can apply the similar method and obtain

For others, using H$\rm{\ddot{o}}$lder and Young inequalities, we get

and

Taking (3.57)-(3.63) into (3.56), we deduce

Then by using uniform Gronwall Lemma over $[t, t+1]$, we obtain

where let $C_{2}(t) = \int_{t}^{t+1}(1+\|\nabla u(s)\|^{4}+\|\nabla w(s)\|^{4}+\|u(s)\|_{\infty}^{2})ds$.

Hence, the proof of Lemma 3.8 is finished.

4.   Existence of pullback $\mathcal{D}$-attractors

In this section, we devote to prove the existence of pullback $\mathcal{D}$-attractors in $V_{1}\times V_{2}$ and $H^{2}\times H^{2}$ for Eqs (2.2).

Firstly, we give the following lemma.

Lemma 4.1. Assume $(u_{1}, w_{1})$ and $(u_{2}, w_{2})$ are two solutions of Eqs (2.2) with the initial data $(u_{i\tau}, w_{i\tau})\in V_{1}\times V_{2}$, $i = 1, 2$ and the external forces $(f_{i}, g_{i})$, where $(f_{i}, g_{i})\in L^{2}_{loc}(\mathbb{R}; H_{1})\times L^{2}_{loc}(\mathbb{R}; H_{2})$, $i = 1, 2$. Let $(\tilde{u}, \tilde{w}) = (u_{1}-u_{2}, w_{1}-w_{2})$, then for every $t\geq\tau$, it holds

Proof. By using (2.2), we can get

Inspired by ^[16], we can have for $3 < \beta < 5$,

Then for $K_{1}$, we can deduce

For other estimates, using H$\ddot{\rm{o}}$lder inequality and Young inequality, we get

Summing above inequalities, we can obtain

Then by using Gronwall Lemma on $[\tau, t]$, we can get

where we use Lemma 3.4 and Lemma 3.7. Hence the proof of Lemma 4.1 is finished.

According to Lemma 4.1, we know that $\{U(t, \tau)\}_{t\geq\tau}$ is continuous in $V_{1}\times V_{2}$. So, it is also norm-to-weak continuous in $V_{1}\times V_{2}$.

Proof of Theorem 2.9. By using Lemma 3.4 and Lemma 3.7, we obtain there exist pullback $\mathcal{D}$-absorbing sets in $V_{1}\times V_{2}$ and $H^{2}\times H^{2}$, respectively. According to compact embedding $H^{2}\times H^{2}\hookrightarrow V_{1}\times V_{2}$, we can get $\{U(t, \tau)\}_{t\geq\tau}$ is pullback $\mathcal{D}$-asymptotically compact in $V_{1}\times V_{2}$. Finally, due to Lemma 2.3 and Lemma 4.1, we obtain that $\{U(t, \tau)\}_{t\geq\tau}$ has a minimal pullback $\mathcal{D}$-attractor $\mathcal{A}_{1}$ in $V_{1}\times V_{2}$.

Lemma 4.2. The process $\{U(t, \tau)\}_{t\geq\tau}$ is norm-to-weak continuous in $H^{2}\times H^{2}$.

Proof. Firstly, let $i:H^{2}\times H^{2}\hookrightarrow V_{1}\times V_{2}$ and $i^{*}:V_{1}^{*}\times V_{2}^{*}\hookrightarrow(H^{2})^{*}\times (H^{2})^{*}$. $i$ and $i^{*}$ are dense. Then, $\{U(t, \tau)\}_{t\geq\tau}:V_{1}\times V_{2}\rightarrow V_{1}\times V_{2}$ is norm-to-weak continuous which we can get from Lemma 4.1. Next, Lemma 3.7 can show that the process $\{U(t, \tau)\}_{t\geq\tau}$ has a pullback $\mathcal{D}$-absorbing set in $H^{2}\times H^{2}$. In other words, $\{U(t, \tau)\}_{t\geq\tau}$ maps a bounded set in $V_{1}\times V_{2}$ into a bounded set in $H^{2}\times H^{2}$. Hence, $\{U(t, \tau)\}_{t\geq\tau}$ maps a compact set in $H^{2}\times H^{2}$ into a bounded set in $H^{2}\times H^{2}$. By using Lemma 2.5, we can finish Lemma 4.2.

Proof of Theorem 2.10. Firstly, according to Lemma 3.7, we can assume $B_{0} = \{B(t):t\in \mathbb{R}\}$ is a pullback $\mathcal{D}$-absorbing set in $H^{2}\times H^{2}$. Let

Then, we prove that for every $t\in \mathbb{R}$, every $\tau_{n}\rightarrow -\infty$ and $(u_{0n}, w_{0n})\in C(\tau_{n})$, $\{(u_{n}(\tau_{n}), w_{n}(\tau_{n}))\}_{n = 0}^{\infty}$ is precompact in $H^{2}\times H^{2}$.

By using the fact that $V_{1}\times V_{2}\hookrightarrow H_{1}\times H_{2}$ and $H^{2}\times H^{2}\hookrightarrow V_{1}\times V_{2}$ are compact and estimates in section 3, we can have $\{(u_{n}(\tau_{n}), w_{n}(\tau_{n}))\}_{n = 0}^{\infty}$ and $\{(\frac{\partial}{\partial t}u_{n}(\tau_{n}), \frac{\partial}{\partial t}w_{n}(\tau_{n}))\}_{n = 0}^{\infty}$ are precompact in $V_{1}\times V_{2}$ and $H_{1}\times H_{2}$, respectively.

Nextly, we will show that $\{(u_{n}(\tau_{n}), w_{n}(\tau_{n}))\}_{n = 1}^{\infty}$ is a Cauchy sequence in $H^{2}\times H^{2}$. Let

By using (2.2), we get

and

Multiplying (4.15) and (4.16) by $Au_{nk}(\tau_{nk})-Au_{nj}(\tau_{nj})$ and $Aw_{nk}(\tau_{nk})-Aw_{nj}(\tau_{nj})$, respectively, we obtain

where we use (3.13) in ^[15].

By using Lemma 3.7 and Sobolev inequality, we have

Inspired by ^[16], we get

By using Sobolev inequality, we have

and

By using above inequalities, we can obtain

Hence, $\{U(t, \tau)\}_{t\geq\tau}$ is pullback $\mathcal{D}$-asymptotically compact in $H^{2}\times H^{2}$. Finally, according to Lemma 2.3, Lemma 3.7 and Lemma 4.2, the proof of Theorem 2.10 is completed.

Acknowledgments

The authors are grateful to the anonymous referees for their useful suggestions which improve the contents of this article.

The second author is supported by the National Natural Science Foundation of China (No. 11901342), Postdoctoral Innovation Project of Shandong Province (No. 202003040) and China Postdoctoral Science Foundation (No. 2019M652350) and the Natural Science Foundation of Shandong Province (No. ZR2018QA002). The third author is supported by the National Natural Science Foundation of China (No. 12001276). The fourth author is supported by the National Natural Science Foundation of China (No. 11701269).

Conflict of interest

The authors declare there is no conflict of interest.