Refinements of Jensen’s and McShane’s inequalities with applications

1.   Introduction

In this paper, we consider the 3D nonlinear damped micropolar equation

where $(x, t)\in\Omega\times [\tau, +\infty)$, $\tau\in \mathbb{R}$, $\Omega\subseteq \mathbb{R}^{3}$ is a bounded domain, $u = u(x, t)$ is the fluid velocity, $\omega = \omega(x, t)$ is the angular velocity, $\sigma$ is the damping coefficient, which is a positive constant, $f_{1} = f_{1}(x, t)$ and $f_{2} = f_{2}(x, t)$ represent external forces, $\nu$, $\kappa$, $\gamma$, $\mu$ are all positive constants, $\gamma$ and $\mu$ represent the angular viscosities.

Micropolar flow can describe a fluid with microstructure, that is, a fluid composed of randomly oriented particles suspended in a viscous medium without considering the deformation of fluid particles. Since Eringen first published his paper on the model equation of micropolar fluids in 1966 ^[5], the formation of modern theory of micropolar fluid dynamics has experienced more than 40 years of development. For the 2D case, many researchers have discussed the long time behavior of micropolar equations (such as ^[2,4,10,24]). It should be mentioned that some conclusions in the 2D case no longer hold for the 3D case due to different structures of the system. In the 3D case, the work of micropolar equations (1.1) with $\sigma = 0$, $f_{1} = 0$, and $f_{2} = 0$ has attracted a lot of attention (see ^[6,14,19]). Galdi and Rionero ^[6] proved the existence and uniqueness of solutions of 3D incompressible micropolar equations. In a 3D bounded domain, for small initial data Yamaguchi ^[19] investigated the existence of a global solution to the initial boundary problem for the micropolar system. In ^[14], Silva and Cruz et al. studied the $L^2$-decay of weak solutions for 3D micropolar equations in the whole space $\mathbb{R}^3$. When $f_1 = f_2 = 0$, for the Cauchy problem of the 3D incompressible nonlinear damped micropolar equations, Ye ^[22] discussed the existence and uniqueness of global strong solutions when $\beta = 3$ and $4\sigma(\nu+\kappa) > 1$ or $\beta > 3$. In ^[18], Wang and Long showed that strong solutions exist globally for any $1\leq\beta\leq 3$ when initial data satisfies some certain conditions. Based on ^[22], Yang and Liu ^[20] obtained uniform estimates of the solutions for 3D incompressible micropolar equations with damping, and then they proved the existence of global attractors for $3 < \beta < 5$. In ^[7], Li and Xiao investigated the large time decay of the $L^2$-norm of weak solutions when $\beta > \frac{14}{5}$, and considered the upper bounds of the derivatives of the strong solution when $\beta > 3$. In ^[21], for $1\leq\beta < \frac{7}{3}$, Yang, Liu, and Sun proved the existence of trajectory attractors for 3D nonlinear damped micropolar fluids.

To the best of our knowledge, there are few results on uniform attractors for the three-dimensional micropolar equation with nonlinear damping term. The purpose of this paper is to consider the existence of uniform attractors of system (1.1). When $\omega = 0, \kappa = 0$, system (1.1) is reduced to the Navier-Stokes equations with damping. In recent years, some scholars have studied the three-dimensional nonlinear damped Navier-Stokes equations (see ^{[1,13,15,16,23,25]}). In order to obtain the desired conclusion, we will use some proof techniques which have been used in the 3D nonlinear damped Navier Stokes equations. Note that, in ^[20], for the convenience of discussion the authors choose $\kappa, \mu = \frac{1}{2}, \gamma = 1$, and $\nu = \frac{3}{2}$. In this work, we do not specify these parameters, but only require them to be positive real numbers. More importantly, we obtain the existence of uniform attractors in the case of $\beta > 3$, which undoubtedly expands the range of $\beta$ when the global attractor exists in ^[20], i.e., $3 < \beta < 5$. For the convenience of discussion, similar to ^{[3,8,9,11,16]}, we make some translational compactness assumption on the external forces term in this paper.

The organizational structure of this article is as follows: In Section 2, we give some basic definitions and properties of function spaces and process theory which will be used in this paper. In Section 3, using various Sobolev inequalities and Gronwall inequalities, we make some uniform estimates from the space with low regularity to high regularity on the solution of the equation. Based on these uniform estimates, in Section 4 we prove that the family of processes $\{U_{(f_1, f_2)}(t, \tau)\}_{t\geq\tau}$ corresponding to (1.1) has uniform attractors $\mathcal{A}_1$ in $V_1\times V_2$ and $\mathcal{A}_2$ in $\mathbf{H}^{2}(\Omega)\times\mathbf{H}^2(\Omega)$, respectively. Furthermore, we prove $\mathcal{A}_1 = \mathcal{A}_2$.

2.   Preliminaries

We define the usual functional spaces as follows:

For $H_1$ and $H_2$ we have the inner product

and norm $\|\cdot\|^{2} = \|\cdot\|^{2}_{2} = (\cdot, \cdot).$ In this paper, $\mathbf{L}^{p}(\Omega) = (L^{p}(\Omega))^{3}$, and $\|\cdot\|_{p}$ represents the norm in $\mathbf{L}^{p}(\Omega)$.

We define operators

where $P$ is the orthogonal projection from $\mathbf{L}^2(\Omega)$ onto $H_{1}$. $\mathbf{H}^s(\Omega) = (H^s(\Omega))^3$ is the usual Sobolev space, and its norm is defined by $\parallel \cdot\parallel_{\mathbf{H}^s} = \parallel A^{\frac{s}{2}}\cdot\parallel$; as $s = 2$, $\parallel\cdot\parallel_{\mathbf{H}^2} = \parallel A\cdot\parallel$.

Let us rewrite system (1.1) as

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

The Poincar$\acute{e}$ inequality ^[17] gives

where $\lambda_1$ is the first eigenvalue of $Au$, and $\lambda_2$ is the first eigenvalue of $A\omega$. Let $\lambda = \min\{\lambda_1, \lambda_2\}$. Then, we have

Agmon's inequality ^[17] gives

The trilinear inequalities ^[12] give

Recall that a function $f(t)$ is translation bounded (tr.b.) in $L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$ if

where $L_b^2(\mathbb{R}; \mathbf{L}^2(\Omega))$ represents the collection of functions that are tr.b. in $L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$. We say that $\mathcal{H}(f_0) = \overline{\{f_0(\cdot+t):t\in\mathbb{R}\}}$ is the shell of $f_0$ in $L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$. If $\mathcal{H}(f_0)$ is compact in $L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$, then we say that $f_0(x, t)\in L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$ is translation compact (tr.c.). We use $L_c^2(\mathbb{R}; \mathbf{L}^2(\Omega))$ to express the collection of all translation compact functions in $L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$.

Next, we will provide the existence and uniqueness theorems of the solution of Eq (2.1).

Definition 2.1. A function pair $(u, \omega)$ is said to be a global strong solution to system (2.1) if it satisfies

for any given $T > \tau$.

Theorem 2.1. Suppose $(u_\tau, \omega_\tau)\in V_1\times V_2$ with $\nabla\cdot u_\tau = 0, f_1, f_2\in L_b^2(\mathbb{R}; \mathbf{L}^2(\Omega))$. If $\beta = 3$ and $4\sigma(\nu+\kappa) > 1$ or $\beta > 3$, then there exists a unique global strong solution of (2.1).

Proof. Since the proof method is similar to that of Theorem 1.2 in ^[22], we omit it here. □

Let $\Sigma$ be a metric space. $X$, $Y$ are two Banach spaces, and $Y\subset X$ is continuous. $\{U_{\sigma}(t, \tau)\}_{t\geq\tau}$, $\sigma\in\Sigma$ is a family of processes in Banach space $X$, i.e., $u(t) = U_\sigma(t, \tau)u_\tau,$ $U_\sigma(t, s)U_\sigma(s, \tau) = U_\sigma(t, \tau), \forall t\geq s\geq\tau, \tau\in\mathbb{R}, U_\sigma(\tau, \tau) = I$, where $\sigma\in\Sigma$ is a time symbol space. $\mathcal{B}(X)$ is the set of all bounded subsets of $X$. $\mathbb{R}^{\tau} = [\tau, +\infty)$.

For the basic concepts of bi-space uniform absorbing set, uniform attracting set, uniform attractor, uniform compact, and uniform asymptotically compact of the family of processed $\{U_\sigma(t, \tau)\}_{t\geq\tau}, \sigma\in\Sigma$, one can refer to ^[9,16].

Let $T(h)$ be a family of operators acting on $\Sigma$, satisfying: $T(h)\sigma(s) = \sigma(s+h), \forall s\in\mathbb{R}$. In this paper, we assume that $\Sigma$ satisfies

$(C1)$ $T(h)\Sigma = \Sigma$, $\forall h\in\mathbb{R}^{+}$;

$(C2)$ translation identity:

Theorem 2.2. ^[3] If the family of processes $\{U_\sigma(t, \tau)\}_{t\geq\tau}, \sigma\in\Sigma$ is $(X, Y)$-uniformly (w.r.t. $\sigma\in\Sigma$) asymptotically compact, then it has a $(X, Y)$-uniform (w.r.t. $\sigma\in\Sigma$) attractor $\mathcal{A}_\Sigma$, $\mathcal{A}_\Sigma$ is compact in $Y$, and it attracts all bounded subsets of $X$ in the topology of $Y$.

In this paper, the letter C represents a positive constant. It may represent different values in different lines, or even in the same line.

3.   Uniform estimations of solutions

In this paper, we chose $\mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$ as the symbol space. Obviously, $T(t)(\mathcal{H}((f_{1}^{0})\times \mathcal{H}(f_{2}^{0})) = \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$, for all $t\geq 0$. $\{T(t)\}_{t\geq0}$ is defined by

which is a translation semigroup and is continuous on $\mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$.

Thanks to Theorem 2.1, when $(u_\tau, \omega_\tau)\in V_1\times V_2$, $f_1, f_2\in L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$, and $\beta > 3$, we can define a process $\{U_{(f_1, f_2)}(t, \tau)\}_{t\geq\tau}$ in $V_1\times V_2$ by

where $(u(t), \omega(t))$ is the solution of Eq (1.1) with external forces $f_1, f_2$ and initial data $(u_\tau, \omega_\tau)$.

Next, let us assume that the external forces $f_1^0(x, t), f_2^0(x, t)$ are tr.c. in $L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$. Then, $f_1^0, f_2^0$ are tr.b. in $L_{\mathrm{loc}}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$, and

Furthermore, we assume $f_1^0, f_2^0$ are uniformly bounded in $\mathbf{L}^2(\Omega)$, i.e., there exists a positive constant $K$, which satisfies

Meanwhile, we suppose the derivatives $\frac{\mathrm{d}f_1^0}{\mathrm{d}t}, \ \frac{\mathrm{d}f_2^0}{\mathrm{d}t}$, labeled as $h_1, h_2$, also belong to $L_{c}^2(\mathbb{R}; \mathbf{L}^2(\Omega))$.

Lemma 3.1. Suppose $(u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2}$ and $(f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$. If $\beta > 3$ then there exists a time $t_{0}$ and constants $\rho_1, I_1$ such that, for any $t\geq t_{0}$,

Proof. Multiplying $(1.1)_{1}$ and $(1.1)_{2}$ with external forces $f_{1}\in\mathcal{H}(f^{0}_{1})$, $f_{2}\in\mathcal{H}(f^{0}_{2})$ by $u$ and $\omega$, respectively, and integrating the results equations on $\Omega$, using H$\ddot{o}$lder's inequality, Young's inequality, and Poincar$\acute{e}$'s inequality, it yields

So, we can obtain that

and by Poincar$\acute{e}$'s inequality, it yields

where $\alpha = \min\{\nu, \gamma\}$. So, by Gronwall's inequality, we get

Therefore, there must exists a time $t_{0}\geq\tau+\frac{1}{\lambda\alpha}\ln\frac{\lambda^{2}\alpha^{2}(\|u_{\tau}\|^{2}+\|\omega_{\tau}\|^{2})}{(1+\lambda\alpha)(\|f_{1}\|^{2}_{L^{2}_{b}}+\|f_{2}\|^{2}_{L^{2}_{b}})},$ such that, $\forall t\geq t_{0}$,

Taking $t\geq t_{0}$, integrating (3.4) from $t$ to $t+1$, and noticing (3.6), we get

Letting $\delta_{1} = \mathrm{min}\{\nu, \gamma, 2\sigma, 2\mu\}$, we have

Letting $I_{1} = \frac{1}{\delta_{1}}(\rho_{1}+\frac{1}{\lambda\alpha}(\|f_{1}\|^{2}_{L^{2}_{b}}+\|f_{2}\|^{2}_{L^{2}_{b}}))$, we have

This completes the proof of Lemma 3.1. □

Lemma 3.2. Assume $\beta > 3$, $(u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2}$ and $(f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$. Then, there exists a time $t_{2}$ and a constant $\rho_2$ such that

for any $t\geq t_{2}$.

Proof. Taking the inner product of $-\Delta u$ in $H_1$ with the first equation of (1.1), we obtain

In ^[18], we find that, when $\beta > 3$,

where $C_1 = \frac{N^2}{\nu+\kappa}+\frac{N^2}{(\nu+\kappa)(N^{\beta-1}+1)}$, and $N$ is sufficiently large such that

And, because

so combining (3.10)–(3.12) with (3.9), we have

where $C_2 = \max\{2C_1, \frac{8\kappa^2}{\nu+\kappa}, \frac{2}{\nu+\kappa}\}$.

Applying uniform Gronwall's inequality to (3.13), we obtaint, $\forall t\geq t_{0}+1\equiv t_{1}$,

where $C_3$ is a positive constant dependent on $C_2$, $I_1$, and $\parallel f_1^0\parallel_{L_b^2}^2$.

Taking the inner product of $-\Delta \omega$ in $H_2$ with the second equation of (1.1), we get

In the last inequality of (3.15), we used Agmon's inequality and the trilinear inequality. Then,

where $C_4 = \max\{\frac{2d_1^2}{\gamma}, \frac{8\kappa^2}{\gamma}, \frac{2}{\gamma}\}$.

By the uniform Gronwall's inequality, we easily obtain that, for $t\geq t_{1}+1\equiv t_{2}$,

where $C_5$ is a positive constant dependent on $C_3, C_4$, and $\parallel f_2^0\parallel_{L_b^2}^2$.

Adding (3.14) with (3.17) yields

for $t\geq t_2$. Hence, Lemma 3.2 is proved. □

Lemma 3.3. Suppose that $(u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2}$ and $(f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$. Then, for $\beta > 3$, there exists a time $t_{3}$ and a constant $\rho_3$ such that

for any $t\geq t_{3}$.

Proof. Multiplying $(1.1)_{1}$ by $u_{t}$, then integrating the equation over $\Omega$, we have

The trilinear inequality (2.4), Agmon's inequality, and Poincar$\acute{e}$'s inequality are used in the last inequality of (3.19).

Hence,

where $C_6 = \max\{\frac{3d_1^2}{\sqrt{\lambda_1}}, 12\kappa^2, 3\}$.

By (3.20), using Lemmas 3.1 and 3.2 and the uniform Gronwall's inequality, we have

Similar to (3.19), multiplying $(1.1)_{2}$ by $\omega_{t}$ and integrating it over $\Omega$, we get

Hence,

By (3.23), using Lemma 3.2 and the uniform Gronwall's inequality, we infer that

The proof of Lemma 3.3 is finished. □

Lemma 3.4. Suppose $(u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2}$ and $(f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$. If $\beta > 3$, then there exists a time $t_{4}$ and a constant $\rho_5$, such that

for any $s\geq t_{4}$.

Proof. Taking the inner products of $u_{t}$ and $\omega_{t}$ with the first and second equations of (1.1), respectively, and using (3.19) and (3.22), we find

where $C_7 = \max\{\frac{3d_1^2}{2\sqrt{\lambda_1}}, 6\kappa^2, \frac{3}{2}\}$. The trilinear inequality (2.4), Agmon's inequality, and Poincar$\acute{e}$'s inequality are used in the last inequality of (3.26).

Integrating (3.26) over $[t, t+1]$ and using Lemmas 3.1–3.3, we get

where $\rho_4$ is a positive constant dependent on $C_7, \rho_2, \rho_3$, $\parallel f_1^0\parallel_{L_b^2}^2$, and $\parallel f_2^0\parallel_{L_b^2}^2$.

We now differentiate $(2.1)_1$ with respect to $t$, then take the inner product of $u_t$ with the resulting equation to obtain

Then, we differentiate $(2.1)_2$ with respect to $t$ and take the inner product with $\omega_t$ to obtain

Adding (3.28) with (3.29), we have

From Lemma 2.4 in ^[15], we know that $G'(u)$ is positive definite, so

For $L_{1}$, using the trilinear inequality (2.5) and Lemma 3.2, we have

For $L_2$, by H$\ddot{o}$lder's inequality, Gagliardo-Nirenberg's inequality, and Young's inequality, we have

By (3.30)–(3.34), we get

Thanks to

and applying uniform Gronwall's inequality to (3.35), we have for any $s\geq t_{3}+1\equiv t_{4}$,

Thus, Lemma 3.4 is proved. □

Lemma 3.5. Suppose $(u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2}$ and $(f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$. Then, for $\beta > 3$, there exists a constant $\rho_6$ such that

for any $t\geq t_{4}$.

Proof. Taking the inner product of $-\Delta u$ in $H_1$ with the first equation of (1.1), we have

Because

combining (3.39) with (3.38), we obtain

From the assumption of $f_1^0(t)$, we can easily get

By Lemmas 3.2 and 3.4, we obtain

Taking the inner product of $A\omega$ with $(2.1)_{2}$, we get

And, by Agmon's inequality,

From the assumption on $f_2^0(t)$, we can easily obtain

By Lemma 3.2, Lemma 3.4, (3.42), (3.43), (3.44), and (3.45), we get

By (3.42) and (3.46), Lemma 3.5 is proved for all $t\geq t_{4}$. □

Lemma 3.6. Suppose $(u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2}$ and $(f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0})$. Then, for $\beta > 3$, there exists a time $t_5$ and a constant $\rho_{7}$ satisfying

Proof. In the proof of Lemma 3.4, from (3.30)–(3.34) we can also get

Integrating (3.48) from $t$ to $t+1$, and according to Lemma 3.4, we have

By Lemma 3.5, we get

So, by Lemma 3.2, applying Agmon's inequality, we get

Taking the derivative of $(2.1)_{1}$ and $(2.1)_{2}$ with respect to $t$, then multiplying by $Au_{t}$ and $A\omega_{t}$, respectively, and integrating the resulting equations over $\Omega$, we then have

For $J_{1}$, $J_2$, using (2.6) and Lemmas 3.2 and 3.5, we have

and

For $J_{3}$ and $J_{4}$, similar to (3.53) and (3.54), we get

For $J_{5}$, $J_6$, and $J_{7}$, applying Hölder's inequality and Young's inequality, we have

and thanks to (3.51),

For $J_8$ and $J_9$, we have

By (3.52)–(3.60), we obtain

Then, by (3.27), (3.49), and using the uniform Gronwall's lemma, we get

Thus, Lemma 3.6 is proved. □

4.   Existence of uniform attractors

In this section, we consider the existence of the $(V_1\times V_2, V_1\times V_2)$-uniform (w.r.t. $(f_1, f_2)\in \mathcal{H}(f_1^0)\times\mathcal{H}(f_2^0)$) attractor and the $(V_1\times V_2, \mathbf{H}^2(\Omega)\times \mathbf{H}^2(\Omega))$-uniform attractor for $\{U_{(f_1, f_2)}(t, \tau)\}_{t\geq\tau}, f_1\times f_2\in \mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$.

Lemma 4.1. Suppose $\beta > 3$. The family of processes $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$, corresponding to (2.1) is $((V_{1}\times V_{2})\times(\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})), V_{1}\times V_{2})$-continuous for $\tau\geq t_5.$

Proof. Let $\tau_n\subset [\tau, +\infty)$ be a time sequence, $U_{(f_{1}^{(n)}, f_{2}^{(n)})}(t, \tau)(u_{\tau_n}, \omega_{\tau_n}) = (u^{(n)}(t), \omega^{(n)}(t))$, $U_{(f_{1}, f_{2})}(t, \tau)(u_{\tau}, \omega_{\tau}) = (u(t), \omega(t))$ and

It is evident that $\bar{u}^{(n)}(t)$ is the solution of

and $\bar{\omega}^{(n)}(t)$ is the solution of the following system

for each $n$.

Taking the inner product of (4.1) with $A\bar{u}^{(n)}$ in $H_1$, we get

Taking the inner product of (4.2) with $A\bar{\omega}^{(n)}$ in $H_2$, we have

Combining (4.3) with (4.4), we get

Due to

and

where $\bar{u}^{(n)}(t) = u(t)-u^{n}(t)$. In the above inequality, we used the fact that

for any $x, y\geq 0$, where $c$ is an absolute constant.

Therefore,

By (4.5)–(4.15) and (4.17), we obtain

Using Gronwall's inequality in (4.18) yields

From Lemmas 3.2 and 3.5, and using Agmon's inequality, we know that

So, from Lemmas 3.2–3.5, we have

for any given $t$ and $\tau$, $t\geq\tau$, $\tau\geq t_5$.

Thus, from (4.19), we have that $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ is $((V_{1}\times V_{2})\times(\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})), V_{1}\times V_{2})$-continuous, for $\tau\geq t_5$. □

By Lemma 3.5, the fact of compact imbedding $\mathbf{H}^2\times \mathbf{H}^2\hookrightarrow V_{1}\times V_{2}$, and Theorem 3.1 in ^[16], we have the following theorems.

Theorem 4.1. Suppose $\beta > 3$. The family of processes $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ with respect to problem (1.1) has a $(V_1\times V_2, V_1\times V_2)$ uniform attractor $\mathcal{A}_{1}$. Moreover,

where $\mathcal{K}_{(f_{1}, f_{2})}(0)$ is the section at $t = 0$ of kernel $\mathcal{K}_{(f_{1}, f_{2})}$ of the processes $\{U_{(f_1, f_2)}(t, \tau)\}_{t\geq\tau}$.

Theorem 4.2. Suppose $\beta > 3$. The family of processes $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ with respect to problem (1.1) has a $(V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega))$-uniform attractor $\mathcal{A}_{2}$. $\mathcal{A}_{2}$ is compact in $\mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)$, and it attracts every bounded subset of $V_{1}\times V_{2}$ in the topology of $\mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)$.

Proof. By Theorem 2.2, we only need to prove that $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ acting on $V_{1}\times V_{2}$ is $(V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega))$-uniform $(w.r.t.\ \ f_{1}\times f_{2}\in \mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2}))$ asymptotically compact.

Thanks to Lemma 3.5, we know that $B = \{(u\times\omega)\in \mathbf{H}^2\times \mathbf{H}^2: \|Au\|^{2}+\|A\omega\|^{2}\leq C\}$ is a bounded $(V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega))$-uniform absorbing set of $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$. Then, we just need to prove that, for any $\tau_{n}\in\mathbb{R}$, any $t\rightarrow +\infty$, and $(u_{\tau_{n}}, \omega_{\tau_n})\in B$, $\{(u_{n}(t), \omega_{n}(t))\}_{n = 0}^{\infty}$ is precompact in $\textbf{H}^{2}(\Omega)\times\textbf{H}^{2}(\Omega)$, where$(u_{n}(t), \omega_{n}(t)) = U_{(f_{1}, f_{2})}(t, \tau_{n})(u_{\tau_{n}}, \omega_{\tau_{n}})$.

Because $V_1\hookrightarrow H_1, V_2\hookrightarrow H_2$ are compact, from Lemma 3.6 we obtain that $\{\frac{d}{dt}u_{n}(t)\}_{n = 0}^{\infty}$, $\{\frac{d}{dt}\omega_{n}(t)\}_{n = 0}^{\infty}$ are precompact in $H_1$ and $H_2$, respectively.

Next, we will prove $\{u_{n}(t)\}_{n = 0}^{\infty}$, $\{\omega_{n}(t)\}_{n = 0}^{\infty}$ are Cauchy sequences in $\mathbf{H}^{2}(\Omega)$. From (2.1), we have

Multiplying (4.21) by $Au_{n_k}(t)-Au_{n_j}(t)$, we obtain

so we have

Multiplying (4.22) by $A\omega_{n_k}(t)-A\omega_{n_j}(t)$ we obtain

so we get

Combining (4.23) with (4.24), we have

Because $V_2\hookrightarrow H_2$ is compact, from Lemma 3.2 we know that $\{\omega_n(t)\}_{n = 0}^\infty$ is precompact in $H_2$. And, using the compactness of embedding $\mathbf{H}^2(\Omega)\hookrightarrow V_1, \mathbf{H}^2(\Omega)\hookrightarrow V_2$ and Lemma 3.5, we have that $\{u_{n}(t)\}_{n = 0}^\infty, \{\omega_n(t)\}_{n = 0}^\infty$ are precompact in $V_1$ and $V_2$, respectively. Considering $V_1\hookrightarrow H_1, V_2\hookrightarrow H_2$ are compact, from Lemma 3.6 we know that $\{\frac{d}{dt}u_n(t)\}_{n = 0}^\infty$, $\{\frac{d}{dt}\omega_n(t)\}_{n = 0}^\infty$ are precompact in $H_1$ and $H_2$, respectively.

Using (2.6), we have

and

From the proof of Lemma 4.2 in ^[15], we have

Taking into account (4.25)–(4.28), we have

(4.29) indicates that the processes $\{U_{(f_1, f_2)}(t, \tau)\}_{t\geq\tau}$ are uniformly asymptotically compact in $\mathbf{H}^2(\Omega)\times\mathbf{H}^2(\Omega)$. So, by Theorem 2.2, it has a $(V_1\times V_2, \mathbf{H}^2(\Omega)\times \mathbf{H}^2(\Omega))$-uniform attractor $\mathcal{A}_2$. □

Theorem 4.3. Suppose $\beta > 3$. The $(V_{1}\times V_{2}, V_{1}\times V_{2})$-uniform attractor $\mathcal{A}_{1}$ of the family of processes $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ is actually the $(V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega))$-uniform attractor $\mathcal{A}_{2}$, i.e., $\mathcal{A}_{1} = \mathcal{A}_{2}$.

Proof. First, we will prove $\mathcal{A}_{1}\subset\mathcal{A}_{2}$. Because $\mathcal{A}_{2}$ is bounded in $\mathbf{H}^2(\Omega)\times \mathbf{\mathbf{H}}^2(\Omega)$, and the embedding $\mathbf{H}^2(\Omega)\times \mathbf{\mathbf{H}}^2(\Omega)\hookrightarrow V_{1}\times V_{2}$ is continuous, $\mathcal{A}_{2}$ is bounded in $V_{1}\times V_{2}$. From Theorem 4.2, we know that $\mathcal{A}_{2}$ attracts uniformly all bounded subsets of $V_{1}\times V_{2}$, so $\mathcal{A}_{2}$ is a bounded uniform attracting set of $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ in $V_{1}\times V_{2}$. By the minimality of $\mathcal{A}_{1}$, we have $\mathcal{A}_{1}\subset\mathcal{A}_{2}$.

Now, we will prove $\mathcal{A}_{2}\subset\mathcal{A}_{1}$. First, we will prove $\mathcal{A}_{1}$ is a $(V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega))$-uniformly attracting set of $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$. That is to say, we will prove

for any $\tau\in\mathbb{R}$ and $B\in \mathcal{B}(V_{1}\times V_{2})$.

If we suppose (4.30) is not valid, then there must exist some $\tau\in\mathbb{R}$, $B\in \mathcal{B}(V_{1}\times V_{2})$, $\varepsilon_{0} > 0$, $(f_{1}^{(n)}, f_{2}^{(n)})\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$, and $t_{n}\rightarrow +\infty$, when $n\rightarrow +\infty$, such that, for all $n\geq 1$,

This shows that there exists $(u_{n}, \omega_{n})\in B$ such that

In the light of Theorem 4.2, $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ has a $(V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega))$-uniform attractor $\mathcal{A}_{2}$ which attracts any bounded subset of $V_{1}\times V_{2}$ in the topology of $\mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)$. Therefore, there exists $(\zeta, \eta)\in\mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)$ and a subsequence of $U_{(f_{1}^{(n)}, f_{2}^{(n)})}(t_{n}, \tau)(u_{n}, \omega_{n})$ such that

On the other side, the processes $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$ have a $(V_{1}\times V_{2}, V_1\times V_2)$-uniform attractor $\mathcal{A}_{1}$, which attracts uniformly any bounded subsets of $V_{1}\times V_{2}$ in the topology of $V_{1}\times V_{2}$. So, there exists $(u, \omega)\in V_{1}\times V_{2}$ and a subsequence of $U_{(f_{1}^{(n)}, f_{2}^{(n)})}(t_{n}, \tau)(u_n, \omega_n)$ such that

From (4.33) and (4.34), we have $(u, \omega) = (\zeta, \eta)$, so (4.33) can also be written as

And, from Theorem 4.1, we know that $\mathcal{A}_{1}$ attracts $B$, so

By (4.34), (4.36), and the compactness of $\mathcal{A}_{1}$ in $V_{1}\times V_{2}$, we have $(u, \omega)\in\mathcal{A}_{1}$. Considering (4.35), we have

which contradicts (4.32). Therefore, $\mathcal{A}_{1}$ is a $(V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega))$-uniform attractor of $\{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau}$, $f_{1}\times f_{2}\in\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})$, and by the minimality of $\mathcal{A}_{2}$, we have $\mathcal{A}_{2}\subset\mathcal{A}_{1}$. □

5.   Conclusions

In this paper, we discussed the existence of uniform attractors of strong solutions for 3D incompressible micropolar equations with nonlinear damping. Based on some translation-compactness assumption on the external forces, and when $\beta > 3$, we made a series of uniform estimates on the solutions in various functional spaces. According to these uniform estimates, we proved the existence of uniform attractors for the process operators corresponding to the solution of the equation in $V_1\times V_2$ and $\mathbf{H}^2\times\mathbf{H}^2$, and verified that the uniform attractors in $V_1\times V_2$ and $\mathbf{H}^2\times\mathbf{H}^2$ are actually the same.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are thankful to the editors and the anonymous reviewers for their valuable suggestions and comments on the manuscript. This work is supported by National Natural Science Foundation of China (Nos. 11601417, 12001420).

Conflict of interest

The authors declare no conflict of interest in this paper.