Homogenization of Trajectory Statistical Solutions for the 3D Incompressible Micropolar Fluids with Rapidly Oscillating Terms

Abstract

This article studies the 3D incompressible micropolar fluids with rapidly oscillating terms. The authors prove that the trajectory statistical solutions of the oscillating fluids converge to that of the homogenized fluids provided that the oscillating external force and angular momentum possess some weak homogenization. The results obtained indicate that the trajectory statistical information of the 3D incompressible micropolar fluids has a certain homogenization effect with respect to spatial variables. In addition, our approach is also valid for a broad class of evolutionary equations displaying the property of global existence of weak solutions without a known result of global uniqueness, including some model hydrodynamic equations, MHD equations and non-Newtonian fluids equations.

1. Introduction

This article studies the following three-dimensional (3D) incompressible micropolar fluids equations with rapidly oscillating external force and angular momentum

$$\begin{array}{cc}\hfill & \frac{\partial u}{\partial t}-(\nu +{\nu }_r)\Delta u+(u·\nabla )u+\nabla p=2{\nu }_r\nabla \times \omega +f(x,\frac{x}{ϵ}),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \nabla ·u=0,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \frac{\partial \omega }{\partial t}-(c_a+c_d)\Delta \omega +(u·\nabla )\omega -(c_0+c_d-c_a)\nabla \mathrm{div}\omega +4{\nu }_r\omega =2{\nu }_r\nabla \times u+g(x,\frac{x}{ϵ}),\hfill \end{array}$$

(3)

supplemented with the initial and boundary conditions

$$\begin{array}{cc}\hfill & u(x,t){|}_{t=0}=u_0,u(x,t){|}_{\partial \Omega \times (0,\infty )}=0,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \omega (x,t){|}_{t=0}={\omega }_0,\omega (x,t){|}_{\partial \Omega \times (0,\infty )}=0,\hfill \end{array}$$

(5)

where the unknown functions $u(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))$, $p(x,t)$ and $\omega (x,t)=({\omega }_1(x,t),{\omega }_2(x,t),{\omega }_3(x,t))$ represent, respectively, the velocity vector, the pressure of the fluid and the angular velocity of rotation of particles at a physical point $x\in \Omega$ and at the moment of time t. The functions ${\displaystyle f(x,\frac{x}{ϵ})}$ and ${\displaystyle g(x,\frac{x}{ϵ})}$ stand for oscillating external force and angular momentum, respectively, and the positive parameter ${ϵ}^{-1}$ characterizes the oscillation frequency of the external force and angular momentum. In addition, the parameters $\nu$, ${\nu }_r$, $c_a$, $c_d$, $c_0$ are positive constants, and we assume $c_0+c_d>c_a$ and that $\Omega \subset {\mathbb{R}}^3$ is a bounded domain with smooth boundary $\partial \Omega$ throughout the article.

Equations (1)–(3) describe the motion of micropolar fluids with a microstructure that belongs to a kind of fluid with a nonsymmetric stress tensor that we call a polar fluid. Physically, micropolar fluids may represent fluids consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluids particles is ignored. The micropolar fluid models can be considered as an essential generalization of the classical Naiver–Stokes models. Obviously, when ${\nu }_r=0$ and $g=\omega =(0,0,0)$, then Equations (1)–(3) reduce to the incompressible Navier–Stokes equations (see [1]).

The theory of micropolar fluids originated in the early 20th century (see e.g., [2]). The mathematical theory of micropolar fluids has been widely studied. For example, ukaszewicz introduces in [3] the framework of the theory for micropolar fluids, and at the same time, in the 2D space, the existence local or global of weak solutions with singular initial data were investigated in [4]; the existence of an attractor was studied in [5,6]; the well-posedness and large-time decay with only angular viscosity dissipation was investigated in [7,8]; in the 3D space, the existence of global weak solution of the generalized micropolar equations was established in [9]; homogenization of past a porous media with nonzero spin boundary condition was studied in [10]; and the existence of global classical solution to the Cauchy problem with vacuum was considered in [11]. However, up until now, as we know, there is no reference investigating the trajectory statistical solutions of incompressible micropolar fluids containing rapidly oscillating external force and angular momentum.

Notice that Equations (1)–(3) can be regarded as some kind of perturbation of the classical equations describing the motions of micropolar fluids [3], with rapidly oscillating external force ${\displaystyle f(x,\frac{x}{ϵ})}$ and angular momentum ${\displaystyle g(x,\frac{x}{ϵ})}$ in place of given external force and angular momentum. The question of whether the trajectory statistical solutions of Equations (1)–(3) converge to those of the classical micropolar fluid equations arises naturally. This is the motivation of this article.

The main result within this article is the homogenization of the trajectory statistical solutions for Equations (1)–(3). The key problem that we solve is the convergent behavior of the trajectory statistical solutions as $ϵ\to 0^{+}$. We first show the existence of a trajectory attractor ${\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ and a trajectory statistical solution ${\rho }_{ϵ,w_{ϵ}}$ for Equations (1)–(3). Simultaneously, we study the following homogenized equations

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}-(\nu +{\nu }_r)\Delta u+(u·\nabla )u+\nabla p=2{\nu }_r\nabla \times \omega +\overline{f}\left(x\right),}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \nabla ·u=0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \frac{\partial \omega }{\partial t}-(c_a+c_d)\Delta \omega +(u·\nabla )\omega -(c_0+c_d-c_a)\nabla \mathrm{div}\omega +4{\nu }_r\omega =2{\nu }_r\nabla \times u+\overline{g}\left(x\right),\hfill \end{array}$$

(8)

where ${\displaystyle (\overline{f}\left(x\right),\overline{g}\left(x\right)):=\overline{F}\left(x\right)\in \widehat{H}}$ (see notations in Section 2) is given. We assume that the oscillating terms ${\displaystyle (f(x,\frac{x}{ϵ}),g(x,\frac{x}{ϵ})):=F(x,\frac{x}{ϵ})\in \widehat{H}}$ for any $ϵ>0$ and that $F(x,\frac{x}{ϵ})\in \widehat{H}$ has the average ${\displaystyle \overline{F}\left(x\right)}$ in ${\widehat{H}}_{\mathrm{w}}$ as $ϵ\to 0^{+}$. Then, we verify the homogenization of the trajectory statistical solutions for Equations (1)–(3), based on the homogenization theory of attractors and the structure of the trajectory statistical solutions.

Precisely, the essential step in our proof is to establish that

$$\begin{array}{c}\hfill ϵ\to 0^{+}⟹\left\{\begin{array}{cc}{w}_{ϵ}(·)\to w(·)\mathrm{in}\mathrm{the}\mathrm{topology}{\Theta }_{\mathrm{loc}}^{+},\hfill & \hfill \\ {\mathcal{A}}_{ϵ}^{\mathrm{tr}}\to {\mathcal{A}}_0^{\mathrm{tr}}\mathrm{in}\mathrm{the}\mathrm{topology}{\Theta }_{\mathrm{loc}}^{+},\hfill & \hfill \end{array}\right.\end{array}$$

(9)

where ${\mathcal{A}}_0^{\mathrm{tr}}$ is the trajectory attractor of the homogenized Equations (6)–(8), $w_{ϵ}(·)$ lies in the trajectory space of Equations (1)–(3), and the limiting function $w(·)$ is a bounded complete trajectory of the homogenized Equations (6)–(8). In terms of the convergent relations in (9), we prove the homogenization of the trajectory statistical solutions by establishing

$$\begin{array}{c}\hfill \underset{ϵ\to 0^{+}}{lim}{\int }_{{\mathcal{A}}_{ϵ}^{\mathrm{tr}}}\psi \left(v\right)\mathrm{d}{\rho }_{ϵ,w_{ϵ}}\left(v\right)={\int }_{{\mathcal{A}}_0^{\mathrm{tr}}}\psi \left(v\right)\mathrm{d}{\rho }_{0,w}\left(v\right),\forall \psi \in C\left({\mathcal{F}}^{+}\right),\end{array}$$

(10)

where ${\rho }_{ϵ,w_{ϵ}}$ and ${\rho }_{0,w}$ are the trajectory statistical solutions of Equations (1)–(3) and (6)–(8) constructed via $w_{ϵ}$ and w, respectively, and $C\left({\mathcal{F}}^{+}\right)$ is the collection of continuous functions defined on ${\mathcal{F}}^{+}$.

The motivation and mathematical analysis introducing above can be described concisely by the following figure:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{Known}:\mathrm{Equations}\left(1\right)–(3)ϵ\to 0^{+}⟹\mathrm{Equations}\left(6\right)–\left(9\right).\hfill & \hfill \\ \mathrm{Question}:{\rho }_{ϵ,w_{ϵ}}⟶{\rho }_{0,w}?\hfill & \hfill \\ \mathrm{Anwser}:\left\{\begin{array}{cc}\hfill w_{ϵ}⟶w& \hfill \\ \hfill {\mathcal{A}}_{ϵ}^{\mathrm{tr}}⟶{\mathcal{A}}_0^{\mathrm{tr}}& \hfill \end{array}\right.⟹{\rho }_{ϵ,w_{ϵ}}⟶{\rho }_{0,w}.\hfill & \hfill \end{array}\right.\end{array}$$

The homogenization theory of partial differential equations is very fruitful; see e.g., [12,13,14,15] and the references therein. The homogenization of attractors for evolution equations with rapidly oscillating terms has been widely studied. For instance, the homogenization of attractors for evolution equations with oscillating with respect to space variables has been investigated in [16,17,18,19], with oscillation with respect to time variables having been studied in [20,21,22,23] and with singular oscillating terms having been investigated in [24,25]. In addition, References [26,27,28,29] have studied the homogenization of attractors for stochastic partial differential equations with randomly oscillating terms. At the same time, homogenization theory has been widely applied to the hydrodynamic and MHD problems, where the asymptotic technique is used to investigate the large-scale behavior; see e.g., [30] for a typical application.

Closely related to some invariant measures, statistical solutions and trajectory statistical solutions characterize the probability distribution of solutions for the evolution equations within the corresponding space. In the theory of fluid mechanics, the invariant measures and statistical solutions have been proven to be very useful in understanding turbulence (see [1]). This is due to the fact that most relevant physical quantities (such as velocity and kinetic energy) of the fluids are actually measurements of time-averaged quantities. In recent years, statistical solutions, trajectory statistical solutions and invariant measures for evolution equations have been paid more and more attention and obtained many research results. For instance, the existence of invariant measures for well-posed dissipative systems has been investigated in [31,32,33,34,35]; the existence of statistical solutions and the trajectory statistical solutions of deterministic evolutionary equations has been researched in [36,37,38,39,40,41,42,43,44,45,46,47,48,49]; the existence of invariant sample measure for the two-dimensional stochastic Navier–Stokes equations has been studied in [50]; and the existence of statistical solutions for impulsive differential equations has been explored in [51,52]. However, there is no references discussing the homogenization of the invariant measures, statistical solutions and trajectory statistical solutions for evolution equations. The issue concerning homogenization of statistical solutions and trajectory statistical solutions should be interesting and of considerable significance.

It is worth mentioning that the approach presented in this article could be extended and give us a tool of much wider applicability. For example, we can use the approach presented in this article to study the homogenization of trajectory statistical solutions for a broad class of evolutionary equations displaying the similar property of global existence of weak solutions without a known result of global uniqueness [16,19], including some model hydrodynamic equations, MHD equations and non-Newtonian fluids equations such as 3D incompressible NS equations [53], 3D NS-like systems [36], 3D NS-$\alpha$ model [37], 2D dissipative Euler equations [48,54], Ginzburg–Landau equations [29], etc. At the same time, our approach can also be applied to above-mentioned equations, 3D Navier–Stokes-Voight equation [21] and damped semilinear wave equations [23,41], investigating the homogenization of trajectory statistical solutions with respect to time variables. For another broad class of non-autonomous evolutionary equations that has global well-posedness, say 2D incompressible NS equations [17,42,53], 2D micropolar fluid flows [5], magneto-micropolar fluids [55,56], two-phase flow model [18], globally modified NS equations [39,45], Klein–Gordon–Schrödinger equations [49], etc., one usually uses the pullback attractors to depict the asymptotic behavior of solutions. In this situation, we can investigate the homogenization (also degenerated regularity) of statistical solutions for these mentioned equations, via the homogenization (also degenerated regularity) of pullback attractors and by exploring the structure of statistical solutions.

The rest of the article is organized as follows. In Section 2, we first introduce the mathematical settings where we settle the problem. Then, we show the existence of trajectory attractors and trajectory statistical solutions for Equations (1)–(3). In Section 3, we first recall some known results concerning the homogenized Equations (6)–(8). Then, we prove the homogenization of the trajectory statistical solutions of Equations (1)–(3).

2. Existence of Trajectory Attractors and Trajectory Statistical Solutions

In this section, we first introduce some mathematical settings where we settle the problem. Then, we recall some known results concerning the existence of solutions, trajectory attractors and trajectory statistical solutions for Equations (1)–(3).

Firstly, we introduce some notations and operators. Usually, we use ${\mathbb{L}}^p(\Omega )={\left(L^p(\Omega )\right)}^3$ and ${\mathbb{W}}^{m,p}(\Omega )={\left(W^{m,p}(\Omega )\right)}^3$ to denote the 3D vector Lebesgue space and Sobolev space with norms ${\parallel ·\parallel }_p$ and ${\parallel ·\parallel }_{m,p}$, respectively. ${\mathbb{W}}_0^{m,p}(\Omega )$ is the closure of $\{\phi :\phi =({\phi }_1,{\phi }_2,{\phi }_3)\in {\left(C_0^{\infty }(\Omega )\right)}^3\}$ in ${\mathbb{W}}^{m,p}(\Omega )$ with norm ${\parallel ·\parallel }_{m,p}$. When $p=2$, we write ${\mathbb{W}}_0^m(\Omega )={\mathbb{H}}_0^m$, ${\mathbb{W}}^{m,p}(\Omega )={\mathbb{H}}^m$ and ${\parallel ·\parallel }_p=\parallel ·\parallel$. Set $\mathcal{V}=\{\phi =({\phi }_1,{\phi }_2,{\phi }_3)\in {\left(C_0^{\infty }(\Omega )\right)}^3:\nabla ·\phi =0\}$, and denote by H the closure of $\mathcal{V}$ in ${\mathbb{L}}^2(\Omega )$ with norm ${\parallel ·\parallel }_H=\parallel ·\parallel$ and inner product $(·,·)$, by V the closure of $\mathcal{V}$ in ${\mathbb{H}}_0^1(\Omega )$ with norm ${\parallel ·\parallel }_V={\parallel ·\parallel }_{1,2}$. Furthermore, we write $\hat{H}=H\times {\mathbb{L}}^2(\Omega )$ and endowed with inner product $(·,·)$ and norm ${\parallel ·\parallel }_{\hat{H}}=\parallel ·\parallel$

$$\begin{array}{cc}\hfill (\Phi ,\Psi )& =(\varphi ,\xi )+(\phi ,\zeta ),\Phi =(\varphi ,\phi ),\Psi =(\xi ,\zeta )\in \hat{H},\hfill \\ \hfill \parallel \Phi \parallel & ={(\parallel \varphi \parallel }^2+{\parallel \phi \parallel }^2{)}^{1/2},\Phi =(\varphi ,\phi )\in \hat{H},\hfill \end{array}$$

and write $\widehat{V}=V\times {\mathbb{H}}_0^1(\Omega )$ endowed with norm ${\parallel ·\parallel }_{\widehat{V}}$

$$\begin{array}{c}\hfill {\parallel \Phi \parallel }_{\widehat{V}}={(\parallel \varphi \parallel }_V^2+{\parallel \phi \parallel }_{1,2}^2{)}^{1/2},\Phi =(\varphi ,\phi )\in \widehat{V}.\end{array}$$

At the same time, we denote by $H^{\ast }$, $V^{\ast }$, ${\hat{H}}^{\ast }=H^{\ast }\times {\mathbb{L}}^2(\Omega )$ and ${\widehat{V}}^{\ast }=V^{\ast }\times {\mathbb{H}}^{-1}(\Omega )$, respectively, the dual spaces of H, V, $\hat{H}$ and $\widehat{V}$, where ${\mathbb{H}}^{-1}(\Omega )$ is the dual space of ${\mathbb{H}}_0^1(\Omega )$. We obviously have $V\hookrightarrow H=H^{\ast }\hookrightarrow V^{\ast },\widehat{V}\hookrightarrow \hat{H}={\hat{H}}^{\ast }\hookrightarrow {\widehat{V}}^{\ast }$ with compact embedding. From now on, we use the same symbol $(·,·)$, for the sake of brevity to denote the inner product in ${\mathbb{L}}^2(\Omega )$, H and $\widehat{H}$ and use the same symbol $\langle ·,·\rangle$ to denote the dual pairing between the spaces V and $V^{\ast }$, $\widehat{V}$ and ${\widehat{V}}^{\ast }$, and ${\mathbb{H}}_0^m(\Omega )$ and ${\mathbb{H}}^{-1}(\Omega )$ whenever it is clear from the context.

Next, we define

$${\mathrm{d}}_{\mathrm{s}}(w,v)=\parallel w-v\parallel ,{\mathrm{d}}_{\mathrm{w}}(w,v)=\sum _{\mathbf{k}\in {\mathbb{Z}}^3}\frac{1}{2^{\left|\mathbf{k}\right|}}\frac{|w_{\mathbf{k}}-v_{\mathbf{k}}|}{1+|w_{\mathbf{k}}-v_{\mathbf{k}}|},w,v\in \hat{H},$$

where $w_{\mathbf{k}}$ and $v_{\mathbf{k}}$ are Fourier coefficients of w and v, respectively. In fact, ${\mathrm{d}}_{\mathrm{s}}$ and ${\mathrm{d}}_{\mathrm{w}}$ stand for the strong metric and weak metric in $\hat{H}$, respectively. The symbol $(\hat{H},{\mathrm{d}}_{•})$ means the space $\hat{H}$ endowed with the metric ${\mathrm{d}}_{•}$ ($•=$ s or w). For compact interval $[a,b]\subset \mathbb{R}$, We denote by $C([a,b];{\hat{H}}_{•})$ the space of ${\mathrm{d}}_{•}$ continuous $\hat{H}$-valued functions on $[a,b]$ endowed with the metric

$${\mathrm{d}}_{C([a,b];{\hat{H}}_{•})}(w,v)=\underset{t\in [a,b]}{sup}{\mathrm{d}}_{•}(w\left(t\right),v\left(t\right)),$$

and denote by $C([a,\infty );{\hat{H}}_{•})$ the space of ${\mathrm{d}}_{•}$ continuous $\widehat{H}$-valued functions on $[a,\infty )$ endowed with the metric

$$\begin{array}{c}\hfill {\mathrm{d}}_{C([a,\infty );{\hat{H}}_{•})}(w,v)=\sum _{i\in \mathbb{N}}\frac{1}{2^i}\frac{sup\{{\mathrm{d}}_{•}(w\left(t\right),v\left(t\right)):a⩽t⩽a+i\}}{1+sup\{{\mathrm{d}}_{•}(w\left(t\right),v\left(t\right)):a⩽t⩽a+i\}}.\end{array}$$

(11)

Now, we define some operators. First, the linear operators $A_1:V\to V^{\ast }$ and $A_2:{\mathbb{H}}_0^1(\Omega )\to {\mathbb{H}}^{-1}(\Omega )$ are defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\langle A_{1}u,\varphi \rangle =(\nu +{\nu }_r)(\nabla u,\nabla \varphi ),\forall u,\varphi \in V,\hfill & \hfill \\ \langle A_2\omega ,\phi \rangle =(c_a+c_d)(\nabla \omega ,\nabla \phi ),\forall \omega ,\phi \in {\mathbb{H}}_0^1(\Omega ).\hfill & \hfill \end{array}\right.\end{array}$$

(12)

Obviously, we have $D\left(A_1\right)=V\cap {\mathbb{H}}_0^2(\Omega )$ and $D\left(A_2\right)={\mathbb{H}}_0^1(\Omega )\cap {\mathbb{H}}_0^2(\Omega )$. We also use the following classical trilinear form $b(·,·,·)$

$$\begin{array}{c}\hfill {\displaystyle b(u,\varphi ,\phi )=\sum _{j,k=1}^3{\int }_{\Omega }{u}_j\frac{\partial {\varphi }_k}{\partial x_j}{\phi }_k\mathrm{d}x,\forall u,\varphi ,\phi \in {\mathbb{H}}_0^1(\Omega ),}\end{array}$$

(13)

which is continuous on $V\times {\mathbb{H}}_0^1(\Omega )\times V$ and (see e.g., [57])

$$\begin{array}{c}\hfill b(u,\varphi ,\phi )=-b(u,\phi ,\varphi ),b(u,\varphi ,\varphi )=0,\forall u\in V,\varphi ,\phi \in {\mathbb{H}}_0^1(\Omega ).\end{array}$$

(14)

For every $u\in V$ and $\omega \in {\mathbb{H}}_0^1(\Omega )$, we define $B_1(u,\omega ):V\times {\mathbb{H}}_0^1(\Omega )\mapsto {\mathbb{H}}^{-1}(\Omega )$ as

$$\langle B_1(u,\omega ),\phi \rangle =b(u,\omega ,\phi ),\forall \phi \in {\mathbb{H}}_0^1(\Omega ).$$

Furthermore, we denote for $w=(u,\omega )\in \widehat{V}$, $f(x,\frac{x}{ϵ})\in H$ and $g(x,\frac{x}{ϵ})\in {\mathbb{L}}^2(\Omega )$ that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}Aw=\left(A_{1}u,A_2\omega \right),\hfill & \hfill \\ B\left(w\right)=\left(B_1(u,u),B_1(u,\omega )\right),\hfill & \hfill \\ Nw=\left(-2{\nu }_r\nabla \times \omega ,4{\nu }_r\omega +(c_a-c_0-c_d)\nabla \mathrm{div}\omega -2{\nu }_r\nabla \times u\right),\hfill & \hfill \\ {\displaystyle F_{ϵ}=\left(f(x,\frac{x}{ϵ}),g(x,\frac{x}{ϵ})\right)=(f_{ϵ},g_{ϵ}).}\hfill & \hfill \end{array}\right.\end{array}$$

Using above notations and operators, we write problems (1)–(5) as

$$\begin{array}{cc}\hfill & \frac{\mathrm{d}w}{\mathrm{d}t}+Aw+B\left(w\right)+Nw=F_{ϵ}\mathrm{in}{\mathcal{D}}^{\prime }((0,\infty ),{\widehat{V}}^{\ast }),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & w\left(0\right)=w_0=(u_0,{\omega }_0).\hfill \end{array}$$

(16)

We next specify the definition of weak solutions to problems (15) and (16).

Definition 1.

A weak solution of problems (15) and (16) on $[0,\infty )$ is an $\hat{H}$-valued function $w\left(t\right)=\left(u\right(t),\omega (t\left)\right)$ defined for $t\in [0,\infty )$ with $w_0=(u_0,{\omega }_0)$, which satisfies

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}t}\in L_{\mathrm{loc}}^{4/3}([0,\infty );{\widehat{V}}^{\ast }),w(·)\in C([0,\infty );{\hat{H}}_{\mathrm{w}})\cap L_{\mathrm{loc}}^2([0,\infty );\widehat{V}),}\hfill & \hfill \\ {\displaystyle (\frac{\mathrm{d}w}{\mathrm{d}t},v)+\langle Aw,v\rangle +\langle B\left(w\right),v\rangle +\langle Nw,v\rangle =\langle F_{ϵ},v\rangle ,\forall v\in \widehat{V},}\hfill & \hfill \end{array}\right.\end{array}$$

in the distribution sense ${\mathcal{D}}^{\prime }(0,\infty )$, and satisfies the following energy inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \frac{1}{2}{\parallel w\left(t\right)\parallel }^2+{\int }_0^t\langle Aw\left(s\right),w\left(s\right)\rangle \mathrm{d}s+{\int }_0^t\langle Nw\left(s\right),w\left(s\right)\rangle \mathrm{d}s\hfill \\ \hfill ⩽& \frac{1}{2}{\parallel w_0\parallel }^2+{\int }_0^t(F_{ϵ},w\left(s\right))\mathrm{d}s,\forall t⩾0,\hfill \end{array}\end{array}$$

(17)

in the sense that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -\frac{1}{2}{\int }_0^t{\parallel w\left(s\right)\parallel }^2{\psi }^{\prime }\left(s\right)\mathrm{d}s+{\int }_0^t\langle Aw\left(s\right),w\left(s\right)\rangle \psi \left(s\right)\mathrm{d}s+{\int }_0^t\langle Nw\left(s\right),w\left(s\right)\rangle \psi \left(s\right)\mathrm{d}s\hfill \\ \hfill ⩽& {\int }_0^t(F_{ϵ},w\left(s\right))\psi \left(s\right)\mathrm{d}s,\forall \psi (·)\in C_0^{\infty }\left([0,t]\right)with\psi (·)⩾0,\forall t⩾0.\hfill \end{array}\end{array}$$

(18)

The following assumption concerning the weak homogenization on the rapidly oscillating external force and angular momentum plays a prominent role in our investigation.

(H) Suppose that $F_{ϵ}\in \widehat{H}$ for every $ϵ>0$, and that $F_{ϵ}$ has the average $\overline{F}=(\overline{f}\left(x\right),\overline{g}\left(x\right))$$\in \widehat{H}$ as $ϵ\to 0^{+}$ in the space ${\widehat{H}}_{\mathrm{w}}$:

$$\begin{array}{c}\hfill (F_{ϵ},\Phi )⟶(\overline{F},\Phi )asϵ\to 0^{+},\forall \Phi \in \widehat{H}.\end{array}$$

(19)

For simplicity, we consider $ϵ\in (0,1]$. Then, (19) implies that

$$\begin{array}{c}\hfill \underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel <+\infty .\end{array}$$

(20)

Proceeding as the proofs of [58] (Lemma 2.1), we can obtain the following lemma about the existence and estimate of weak solutions to problem (15) and (16). The difference is that we replace the the positive constant $R={\left(\frac{2\lambda }{\delta }\left[\frac{\lambda }{\nu }{\parallel f\parallel }^2+\frac{\lambda }{c_a+c_d}{\parallel g\parallel }^2\right]\right)}^{1/2}$ in [58] (Lemma 2.1) by $R_{ϵ}$.

Lemma 1

(cf. [58] Lemma 2.1). Suppose that (H) holds. Then, to every $w_0=(u_0,{\omega }_0)\in \widehat{H}$ and each $ϵ\in (0,1]$, there corresponds at least one weak solution $w_{ϵ}\left(t\right)$ of problems (15) and (16). Moreover, there exists a time $t_1=t_1(ϵ,\Omega ,c_a,c_d,\nu ,f_{ϵ},g_{ϵ})>0$ such that

$$\begin{array}{c}\hfill w_{ϵ}\left(t\right)\in X_{ϵ}:=\{v\in \widehat{H}:\parallel v\parallel ⩽R_{ϵ}=(\frac{2\lambda }{\delta }[\frac{\lambda }{\nu }\parallel f_{ϵ}{\parallel }^2+\frac{\lambda }{c_a+c_d}{\parallel g_{ϵ}\parallel }^2]{)}^{1/2}\},\forall t⩾t_1,\end{array}$$

(21)

where $\delta =min\{c_a+c_d,\nu \}$ and λ is a constant depending only on Ω.

In terms of Lemma 1, we give the definition of the trajectory space and kernel of Equation (15), respectively, as

$$\begin{array}{cc}\hfill & {\mathcal{T}}_{ϵ}^{+}=\{w\left(t\right):w\left(t\right)\mathrm{is}\mathrm{a}\mathrm{weak}\mathrm{solution}\mathrm{of}\left(15\right)\mathrm{and}w\left(t\right)\in X_{ϵ}\mathrm{for}\mathrm{all}t\in [0,\infty )\},\hfill \\ \hfill & {\mathcal{K}}_{ϵ}=\{w\left(t\right):w\left(t\right)\mathrm{is}\mathrm{a}\mathrm{weak}\mathrm{solution}\mathrm{of}\left(15\right)\mathrm{and}w\left(t\right)\in X_{ϵ}\mathrm{for}\mathrm{all}t\in (-\infty ,+\infty )\}.\hfill \end{array}$$

Next, we put

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{X}_0:=\{v\in \hat{H}:{\parallel v\parallel }^2⩽R_0=(\frac{2\lambda }{\delta }[\frac{\lambda }{\nu }\parallel \overline{f}{\parallel }^2+\frac{\lambda }{c_a+c_d}{\parallel \overline{g}\parallel }^2]{)}^{1/2}\},\hfill & \hfill \\ X:=\{v\in \hat{H}:{\parallel v\parallel }^2⩽c_1(\underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel ,R_0\}),\hfill & \hfill \end{array}\right.\end{array}$$

where $c_1(\underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel ,R_0)$ is a positive constant depending only on $\underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel$ and $R_0$ such that

$$\left({\cup }_{0<ϵ⩽1}{X}_{ϵ}\right)\cup X_0\subset X.$$

In order to consider the same topology ${\Theta }_{\mathrm{loc}}^{+}$ in trajectory spaces ${\mathcal{T}}_{ϵ}^{+}$ ($0<ϵ⩽1$) for Equations (1)–(3) and in trajectory space ${\mathcal{T}}_0^{+}$ for Equations (6)–(8), in the unified space ${\mathcal{F}}^{+}$, we denote by

$${\mathcal{F}}^{+}=C_{\mathrm{loc}}([0,\infty );X_{\mathrm{w}})$$

the space of continuous functions from $[0,\infty )$ to $X_{\mathrm{w}}$, where $X_{\mathrm{w}}=(X,{\mathrm{d}}_{\mathrm{w}})$ is the space X endowed with the weak distance. Define the weak distance in ${\mathcal{F}}^{+}$ as

$$\begin{array}{c}\hfill {\mathrm{d}}_{{\mathcal{F}}^{+}}(w,v)={\mathrm{d}}_{C([0,\infty );X_{\mathrm{w}})}(w,v)=\sum _{i\in \mathbb{N}}\frac{1}{2^i}\frac{sup\{{\mathrm{d}}_{X_{\mathrm{w}}}(w\left(t\right),v\left(t\right)):0⩽t⩽i\}}{1+sup\{{\mathrm{d}}_{X_{\mathrm{w}}}(w\left(t\right),v\left(t\right)):0⩽t⩽i\}},\end{array}$$

(22)

which is compatible with the compact-open topology (denoted by ${\Theta }_{\mathrm{loc}}^{+}$) of ${\mathcal{F}}^{+}$ (cf. [59]). For a set $\mathcal{Q}\subset {\mathcal{F}}^{+}$ and some $r>0$, we set

$$\mathcal{B}(\mathcal{Q},r)=\{v\in {\mathcal{F}}^{+}|{\mathrm{d}}_{{\mathcal{F}}^{+}}(v,\mathcal{Q})=\underset{q\in \mathcal{Q}}{inf}{\mathrm{d}}_{{\mathcal{F}}^{+}}(v,q)<r\}.$$

In what follows, we recall some concepts and known results relative to trajectory attractors and trajectory statistical solutions for Equation (15). We begin with the natural translation operator ${\left\{T\left(t\right)\right\}}_{t⩾0}$, which is defined on ${\mathcal{F}}^{+}$ as

$$\begin{array}{c}\hfill T\left(t\right)w\left(s\right)=w(s+t),w\in {\mathcal{F}}^{+},t⩾0.\end{array}$$

(23)

It is obvious that $T\left(s\right){\mathcal{T}}_{ϵ}^{+}\subset {\mathcal{T}}_{ϵ}^{+}$ for any $s⩾0$.

Definition 2.

(1)

A set $\mathcal{Q}\subset {\mathcal{F}}^{+}$ is said to uniformly attract a set $D\subset {\mathcal{T}}_{ϵ}^{+}$ if, to every $\epsilon >0$, there corresponds a $t_{\epsilon }>0$ such that $T\left(t\right)D\subset \mathcal{B}(\mathcal{Q},\epsilon )$, $\forall t⩾t_{\epsilon }$. A set $\mathcal{Q}\subset {\mathcal{F}}^{+}$ is called to be a trajectory attracting set in ${\mathcal{T}}_{ϵ}^{+}$ if it uniformly attracts ${\mathcal{T}}_{ϵ}^{+}$. We call a set $\mathcal{U}\subset {\mathcal{F}}^{+}$ a trajectory attractor in ${\mathcal{T}}_{ϵ}^{+}$ if $\mathcal{U}$ is the minimal compact trajectory attracting set in ${\mathcal{T}}_{ϵ}^{+}$ and $T\left(t\right)\mathcal{U}=\mathcal{U}$ for all $t⩾0$.

(2)

A generalized Banach limit, which we denote by ${\mathrm{LIM}}_{t\to +\infty }$, is any linear continuous functional defined on the space of all bounded real-valued functions on $[0,\infty )$ that satisfies the following two properties:

(i)

${\mathrm{LIM}}_{t\to +\infty }\zeta \left(t\right)⩾0$ for nonnegative functions $\zeta (·)$ on $[0,\infty )$;

(ii)

${\mathrm{LIM}}_{t\to +\infty }\zeta \left(t\right)=\underset{t\to +\infty }{lim}\zeta \left(t\right)$ if the usual limit $\underset{t\to +\infty }{lim}\zeta \left(t\right)$ exists.

(3)

A Borel probability measure ρ on ${\mathcal{F}}^{+}$ is a ${\mathcal{T}}_{ϵ}^{+}$-trajectory statistical solution over $[0,\infty )$ (or simply a trajectory statistical solution) for Equation (15) if the following two properties hold:

(a)

For any $\mathcal{B}\in \mathcal{B}\left({\mathcal{F}}^{+}\right)$ (the collection of Borel sets of ${\mathcal{F}}^{+}$), there holds

$$\rho \left(\mathcal{B}\right)=sup\left\{\rho \left(E\right)|E\in \mathcal{B}\left({\mathcal{F}}^{+}\right)\mathrm{and}E\subset \mathcal{B}\right\};$$

(b)

ρ is supported by a Borel subset of ${\mathcal{F}}^{+}$ contained in ${\mathcal{T}}_{ϵ}^{+}$.

For every $ϵ\in (0,1]$, the existence of the trajectory attractors and trajectory statistical solutions for Equation (15) has been proved in [58]. To be quite explicit, we have the following:

Theorem 1

(cf. [58] Theorems 2.1 and 3.1). Suppose that assumption(H )holds.

(1)

For every $ϵ\in (0,1]$, the natural translation semigroup ${\left\{T\left(t\right)\right\}}_{t⩾0}$ possesses a trajectory attractor ${\mathcal{A}}_{ϵ}^{\mathrm{tr}}$, which satisfies

$$\begin{array}{c}\hfill {\mathcal{A}}_{ϵ}^{\mathrm{tr}}={\Pi }_{+}{\mathcal{K}}_{ϵ}=\left\{w(·){|}_{[0,\infty )}\right|w\in {\mathcal{K}}_{ϵ}\}\subset {\mathcal{T}}_{ϵ}^{+},\end{array}$$

(24)

where ${\Pi }_{+}$ denotes the restriction operator to $[0,\infty )$.

(2)

Let ${\mathrm{LIM}}_{t\to +\infty }$ be a given generalized Banach limit. Then, to each $w_{ϵ}\in {\mathcal{T}}_{ϵ}^{+}$, there corresponds a unique Borel probability measure ${\rho }_{ϵ,w_{ϵ}}$ on ${\mathcal{F}}^{+}$ such that

$$\begin{array}{c}\hfill {\int }_{{\mathcal{F}}^{+}}\psi \left(v\right)\mathrm{d}{\rho }_{ϵ,w_{ϵ}}\left(v\right)={\mathrm{LIM}}_{t\to +\infty }\frac{1}{t}{\int }_0^t\psi \left(T\left(s\right)w_{ϵ}\right)\mathrm{d}s,\forall \psi \in C\left({\mathcal{F}}^{+}\right).\end{array}$$

(25)

Furthermore, ${\rho }_{ϵ,w_{ϵ}}$ is supported by ${\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ and is a trajectory statistical solution for Equation (15).

3. Homogenization of the Trajectory Statistical Solutions

The goal of this section is to prove the homogenization of the trajectory statistical solutions ${\rho }_{ϵ,w_{ϵ}}$ obtained by Theorem 1 as $ϵ\to 0^{+}$. The essential steps are to prove the convergence of the sequence $w_{{ϵ}_n}(·)\in {\mathcal{T}}_{{ϵ}_n}^{+}$ to some solution $w(·)\in {\mathcal{T}}_0^{+}$, and ${\mathcal{A}}_{{ϵ}_n}^{\mathrm{tr}}$ to ${\mathcal{A}}_0^{\mathrm{tr}}$ for any sequence ${ϵ}_n\to 0^{+}$.

We begin with some known results for the homogenized Equations (6)–(8). First, we use the notations introduced in §2, for $w=(u,\omega )$, to write the weak form of problem (6)–(8) as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}w}{\mathrm{d}t}+Aw+B\left(w\right)+Nw=\overline{F}\mathrm{in}{\mathcal{D}}^{\prime }((0,\infty );{\widehat{V}}^{\ast }),}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & w\left(0\right)=w_0=(u_0,{\omega }_0).\hfill \end{array}$$

(27)

The definition of solutions to problems (26)–(27) is almost the same as Definition 1 with the average function $\overline{F}$ in place of the oscillating function $F_{ϵ}$.

Lemma 2

([58] Lemma 2.1). Let $\overline{F}\in \widehat{H}$. Then, to every $w_0=(u_0,{\omega }_0)\in \widehat{H}$, problems (26) and (27) correspond at least one weak solution $w\left(t\right)$. Furthermore, there exists a time $t_2=t_2(\Omega ,c_a,c_d,\nu ,\overline{f},\overline{g})>0$ yielding

$$\begin{array}{c}\hfill w\left(t\right)\in X_0,\forall t⩾t_2.\end{array}$$

(28)

Correspondingly, the trajectory space ${\mathcal{T}}_0^{+}$ and kernel ${\mathcal{K}}_0$ of Equation (26) are defined, respectively, as

$$\begin{array}{cc}\hfill & {\mathcal{T}}_0^{+}=\{w\left(t\right):w\left(t\right)\mathrm{is}\mathrm{a}\mathrm{weak}\mathrm{solution}\mathrm{of}\left(26\right)\mathrm{and}w\left(t\right)\in X_0\mathrm{for}\mathrm{all}t\in [0,\infty )\},\hfill \\ \hfill & {\mathcal{K}}_0=\{w\left(t\right):w\left(t\right)\mathrm{is}\mathrm{a}\mathrm{weak}\mathrm{solution}\mathrm{of}\left(26\right)\mathrm{and}w\left(t\right)\in X_0\mathrm{for}\mathrm{all}t\in (-\infty ,+\infty )\}.\hfill \end{array}$$

The existence of a trajectory attractor and a trajectory statistical solution to Equation (26) has been proved in [58] (Theorems 2.1 and 3.1).

Lemma 3

([58] Theorems 2.1 and 3.1). Let $\overline{F}\in \widehat{H}$.

(1)

The natural translation semigroup ${\left\{T\left(t\right)\right\}}_{t⩾0}$ possesses a trajectory attractor ${\mathcal{A}}_0^{\mathrm{tr}}$, which satisfies

$$\begin{array}{c}\hfill {\mathcal{A}}_0^{\mathrm{tr}}={\Pi }_{+}{\mathcal{K}}_0=\left\{w(·){|}_{[0,\infty )}\right|w\in {\mathcal{K}}_0\}\subset {\mathcal{T}}_0^{+}.\end{array}$$

(29)

(2)

Let ${\mathrm{LIM}}_{t\to +\infty }$ be a given generalized Banach limit. Then, to every $w\in {\mathcal{T}}_0^{+}$, there corresponds a unique Borel probability measure ${\rho }_{0,w}$ on ${\mathcal{F}}^{+}$ such that

$$\begin{array}{c}\hfill {\int }_{{\mathcal{F}}^{+}}\gamma \left(v\right)\mathrm{d}{\rho }_{0,w}\left(v\right)={\mathrm{LIM}}_{t\to +\infty }\frac{1}{t}{\int }_0^t\gamma \left(T\left(s\right)w\right)\mathrm{d}s,\forall \gamma \in C\left({\mathcal{F}}^{+}\right),\end{array}$$

(30)

and ${\rho }_{0,w}$, which is carried by ${\mathcal{A}}_0^{\mathrm{tr}}$, is a trajectory statistical solution of Equation (26).

The convergence of the sequence from the trajectory spaces ${\mathcal{T}}_{{ϵ}_n}^{+}$ to some element within the trajectory space ${\mathcal{T}}_0^{+}$ plays a key role in our investigation. This convergent relation is stated in the following lemma.

Lemma 4.

Suppose that assumption (H) holds. Let the sequence ${\left\{{ϵ}_n\right\}}_{n⩾1}\subset (0,1]$, which tends to zero as $n\to \infty$, and the sequence ${\left\{w_{{ϵ}_n}\left(t\right)\right\}}_{n⩾1}$ with $w_{{ϵ}_n}\left(t\right)\in {\mathcal{T}}_{{ϵ}_n}^{+}$ for each n satisfy that

$$\begin{array}{c}\hfill w_{{ϵ}_n}(·)⟶w(·)\mathrm{in}\mathrm{the}\mathrm{topology}{\Theta }_{\mathrm{loc}}^{+}\mathrm{as}n\to \infty .\end{array}$$

(31)

Then $w\in {\mathcal{T}}_0^{+}$.

Proof. 

We prove under the conditions of Lemma 4 that $w(·)$ is a solution of problems (26) and (27) that satisfies (28). We begin by proving that w satisfies Equation (26). In fact, we have that

$$\begin{array}{c}\hfill \frac{\mathrm{d}{w}_{{ϵ}_n}\left(t\right)}{\mathrm{d}t}+A{w}_{{ϵ}_n}\left(t\right)+B\left(w_{{ϵ}_n}\left(t\right)\right)+N{w}_{{ϵ}_n}\left(t\right)=F_{ϵ}\end{array}$$

(32)

in the sense that ${\mathcal{D}}^{\prime }((0,T);{\widehat{V}}^{\ast })$ as $w_{{ϵ}_n}(·)\in {\mathcal{T}}_{{ϵ}_n}^{+}$. From [58] ((2.33), (2.34)), we see that

$$\begin{array}{c}\hfill {\left\{w_{{ϵ}_n}\left(t\right)\right\}}_{n⩾1}isboundedin{L}^2([0,T];\widehat{V})\cap L^{\infty }([0,T];\widehat{H}),\end{array}$$

(33)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{w}_{{ϵ}_n}\left(t\right)}{\mathrm{d}t}{\parallel }_{L^{4/3}([0,T];{\widehat{V}}^{\ast })}⩽& \parallel A{w}_{{ϵ}_n}{\left(t\right)\parallel }_{L^{4/3}([0,T];{\widehat{V}}^{\ast })}+{\parallel B\left(w_{{ϵ}_n}\left(t\right)\right)\parallel }_{L^{4/3}([0,T];{\widehat{V}}^{\ast })}\hfill \\ \hfill & +\parallel N{w}_{{ϵ}_n}{\left(t\right)\parallel }_{L^{4/3}([0,T];{\widehat{V}}^{\ast })}+{\parallel F_{{ϵ}_n}\parallel }_{L^{4/3}([0,T];{\widehat{V}}^{\ast })}\hfill \\ \hfill \lesssim & \parallel w_{{ϵ}_n}{\left(t\right)\parallel }_{L^2([0,T];\widehat{V})}+\parallel w_{{ϵ}_n}{\left(t\right)\parallel }_{L^{\infty }([0,T];\widehat{H})}^{1/2}{\parallel w_{{ϵ}_n}\left(t\right)\parallel }_{L^2([0,T];\widehat{V})}^{3/2}\hfill \\ \hfill & +\parallel w_{{ϵ}_n}{\left(t\right)\parallel }_{L^2([0,T];\widehat{V})}+T\parallel F_{{ϵ}_n}\parallel ,\hfill \end{array}\end{array}$$

(34)

where the symbol $a\lesssim b$ means that $a⩽cb$ for a universal constant $c>0$ that only depends on the parameters coming from the problem. We then conclude from (20), (33) and (34) that ${\displaystyle \frac{\mathrm{d}{w}_{{ϵ}_n}\left(t\right)}{\mathrm{d}t}}$ and $B\left(w_{{ϵ}_n}\left(t\right)\right)$ are uniform (with respect to $n\in \mathbb{N}$) bounded in $L^{4/3}([0,T];{\widehat{V}}^{\ast })$. Consequently, we deduce from (31), (33) and (34) that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{w}_{{ϵ}_n}\left(t\right)\rightharpoonup w\left(t\right)\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }([0,T];\widehat{H})\mathrm{as}{ϵ}_n\to 0^{+},\hfill & \hfill \\ w_{{ϵ}_n}\left(t\right)\rightharpoonup w\left(t\right)\mathrm{weakly}\mathrm{in}{L}^2([0,T];\widehat{V})\mathrm{as}{ϵ}_n\to 0^{+},\hfill & \hfill \\ {\displaystyle \frac{\mathrm{d}{w}_{{ϵ}_n}\left(t\right)}{\mathrm{d}t}\rightharpoonup \frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}\mathrm{weakly}\mathrm{in}{L}^{4/3}([0,T];{\widehat{V}}^{\ast })\mathrm{as}{ϵ}_n\to 0^{+}.}\hfill & \hfill \end{array}\right.\end{array}$$

(35)

At the same time, by the definitions of the operators A and N, we have as ${ϵ}_n\to 0^{+}$ that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}A{w}_{{ϵ}_n}\left(t\right)\rightharpoonup Aw\left(t\right)\mathrm{weakly}\mathrm{in}{L}^2([0,T];{\widehat{V}}^{\ast }),\hfill & \hfill \\ N{w}_{{ϵ}_n}\left(t\right)\rightharpoonup Nw\left(t\right)\mathrm{weakly}\mathrm{in}{L}^2([0,T];{\widehat{V}}^{\ast }).\hfill & \hfill \end{array}\right.\end{array}$$

(36)

For the nonlinear term $B\left(w_{{ϵ}_n}\left(t\right)\right)$, when ${ϵ}_n\to 0^{+}$, we can use the same derivations as that in [57] to arrive at

$$\begin{array}{c}\hfill B\left(w_{{ϵ}_n}\left(t\right)\right)\rightharpoonup B\left(w\left(t\right)\right)\mathrm{weakly}\mathrm{in}{L}^{4/3}([0,T];{\widehat{V}}^{\ast }).\end{array}$$

(37)

Then, (32) and (35)–(37) yield

$$\begin{array}{c}\hfill \frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}+Aw\left(t\right)+B\left(w\left(t\right)\right)+Nw\left(t\right)=\overline{F}\mathrm{in}{\mathcal{D}}^{\prime }((0,T);{\widehat{V}}^{\ast }).\end{array}$$

(38)

The fact that $w(·)$ satisfies the energy inequality can be proven by passing to limit of the following inequality:

$$\begin{array}{cc}\hfill & -\frac{1}{2}{\int }_0^t{\parallel w_{{ϵ}_n}\left(s\right)\parallel }^2{\psi }^{\prime }\left(s\right)\mathrm{d}s+{\int }_0^t\langle A{w}_{{ϵ}_n}\left(s\right),w_{{ϵ}_n}\left(s\right)\rangle \psi \left(s\right)\mathrm{d}s+{\int }_0^t\langle N{w}_{{ϵ}_n}\left(s\right),w_{{ϵ}_n}\left(s\right)\rangle \psi \left(s\right)\mathrm{d}s\hfill \\ \hfill ⩽& {\int }_0^t(F_{ϵ},w_{{ϵ}_n}\left(s\right))\psi \left(s\right)\mathrm{d}s,\forall \psi (·)\in C_0^{\infty }\left([0,t]\right)\mathrm{with}\psi \left(t\right)⩾0,\forall t⩾0.\hfill \end{array}$$

Finally, from [58] (2.5), we see that $w(·)$ satisfies (28). This completes the proof. □

We next introduce a family of sets ${\left\{{\mathfrak{B}}_{ϵ}\right\}}_{0<ϵ⩽1}$, of which the main purpose is to investigate the the convergence of the trajectory attractors ${\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ to ${\mathcal{A}}_0^{\mathrm{tr}}$ as $ϵ\to 0^{+}$. Denote

$${\mathfrak{B}}_{ϵ}=\left\{w_{ϵ}\mid w_{ϵ}\in {\mathcal{T}}_{ϵ}^{+},{\parallel w_{ϵ}\parallel }_{{\mathfrak{B}}_{ϵ}}⩽c_2(\underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel ,R_0),\mathrm{for}\mathrm{all}{w}_{ϵ}\in {\mathcal{B}}_{ϵ}\right\},$$

where

$$\parallel w_{ϵ}{\parallel }_{{\mathfrak{B}}_{ϵ}}=\underset{t⩾0}{sup}\parallel w_{ϵ}\left(t\right)\parallel +\underset{t⩾0}{sup}({\int }_t^{t+1}\parallel w_{ϵ}{\left(s\right)\parallel }_{\widehat{V}}^2{\mathrm{d}s)}^{1/2}+\underset{t⩾0}{sup}({\int }_t^{t+1}\parallel {\partial }_s{w}_{ϵ}\left(s\right){{\parallel }_{{\widehat{V}}^{\ast }}^{4/3}\mathrm{d}s)}^{3/4},$$

$c_2(\underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel ,R_0)$ is positive constant that is large enough and depends only on $\underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel$ and $R_0$, and can be chosen for our purpose. Obviously, we have

$$\begin{array}{c}\hfill T\left(t\right){\mathfrak{B}}_{ϵ}\subset {\mathfrak{B}}_{ϵ},\forall ϵ\in (0,1].\end{array}$$

(39)

Similar to that, as pointed out in Section 2, we can consider the same topology ${\Theta }_{\mathrm{loc}}^{+}$ in ${\mathfrak{B}}_{ϵ}$ ($0<ϵ⩽1$) in the unified space ${\mathcal{F}}^{+}$.

Lemma 5.

Let assumption(H)hold and $ϵ\in (0,1]$. Then,

$$\begin{array}{c}\hfill T\left(t\right){\mathfrak{B}}_{ϵ}\to {\mathcal{A}}_0^{\mathrm{tr}}inthetopology{\Theta }_{\mathrm{loc}}^{+}ast\to +\infty andϵ\to 0^{+},\end{array}$$

(40)

where (40) is interpreted as follows: for ${\left\{w_{ϵ}(·)\right\}}_{0<ϵ⩽1}$ with $w_{ϵ}(·)\in {\mathfrak{B}}_{ϵ}$, there exists some $w\in {\mathcal{A}}_0^{\mathrm{tr}}$ such that $T\left(t\right)w_{ϵ}\to w$ in the topology ${\Theta }_{\mathrm{loc}}^{+}$ as $t\to +\infty$ and $ϵ\to 0^{+}$.

Proof. 

We establish (40) via the argument of contradiction. Suppose the converse: there is a neighborhood $\mathcal{O}\left({\mathcal{A}}_0^{\mathrm{tr}}\right)$ of ${\mathcal{A}}_0^{\mathrm{tr}}$ in the topology ${\Theta }_{\mathrm{loc}}^{+}$ and two sequences ${ϵ}_n\to 0^{+}$, $t_n\to +\infty$ as $n\to \infty$, yielding

$$\begin{array}{c}\hfill T\left(t_n\right){\mathfrak{B}}_{{ϵ}_n}\neg \subset \mathcal{O}\left({\mathcal{A}}_0^{\mathrm{tr}}\right).\end{array}$$

(41)

Then, there are solutions $w_{{ϵ}_n}\in {\mathfrak{B}}_{{ϵ}_n}$ that satisfy

$$\begin{array}{c}\hfill w_{{ϵ}_n}\left(t\right):=T\left(t_n\right)w_{{ϵ}_n}\left(t\right)\notin \mathcal{O}\left({\mathcal{A}}_0^{\mathrm{tr}}\right).\end{array}$$

(42)

Now, for each n, $w_{{ϵ}_n}\left(t\right)$ is a solution of Equation (15) with $ϵ={ϵ}_n$ on the interval $[-t_n,+\infty )$ and $w_{{ϵ}_n}\left(t\right)$ is the backward time shift in the solution $w_{{ϵ}_n}\left(t\right)$ by $t_n$. The definition of ${\mathfrak{B}}_{ϵ}$ says that

$$\begin{array}{cc}\hfill & \underset{t⩾-t_n}{sup}\parallel w_{{ϵ}_n}\left(t\right)\parallel +\underset{t⩾-t_n}{sup}({\int }_t^{t+1}\parallel w_{{ϵ}_n}{\left(s\right)\parallel }_{\widehat{V}}^2{\mathrm{d}s)}^{1/2}+\underset{t⩾-t_n}{sup}({\int }_t^{t+1}\parallel {\partial }_s{w}_{{ϵ}_n}\left(s\right){{\parallel }_{{\widehat{V}}^{\ast }}^{4/3}\mathrm{d}s)}^{3/4}\hfill \\ \hfill ⩽& c_2(\underset{0<ϵ⩽1}{sup}\parallel F_{ϵ}\parallel ,R_0).\hfill \end{array}$$

(43)

Now, for any given $T>0$, we consider ${ϵ}_n$ with the picked index n such that $t_n⩾T$. Hence, (43) indicates that we can extract a subsequence (still denote by) $\left\{w_{{ϵ}_n}\left(s\right)\right\}$ and that there is a function $w\left(t\right)$ defined on $(-T,T)$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{w}_{{ϵ}_n}\left(t\right)\rightharpoonup w\left(t\right)weaklystarin{L}^{\infty }(-T,T;\widehat{H})as{ϵ}_n\to 0^{+},\hfill & \hfill \\ w_{{ϵ}_n}\left(t\right)\rightharpoonup w\left(t\right)weaklyin{L}^2(-T,T;\widehat{V})as{ϵ}_n\to 0^{+},\hfill & \hfill \\ {\displaystyle \frac{\mathrm{d}{w}_{{ϵ}_n}\left(t\right)}{\mathrm{d}t}\rightharpoonup \frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}weaklyin{L}^{4/3}(-T,T;{\widehat{V}}^{\ast })as{ϵ}_n\to 0^{+}.}\hfill & \hfill \end{array}\right.\end{array}$$

Proceeding the diagonal procedure, we construct a function $w\left(t\right)$ that is defined on $\mathbb{R}$ and satisfies

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{w}_{{ϵ}_n}\left(t\right)\rightharpoonup w\left(t\right)weaklystarin{L}^{\infty }(-T,T;\widehat{H})as{ϵ}_n\to 0^{+},\forall T>0,\hfill & \hfill \\ w_{{ϵ}_n}\left(t\right)\rightharpoonup w\left(t\right)weaklyin{L}^2(-T,T;\widehat{V})as{ϵ}_n\to 0^{+},\forall T>0,\hfill & \hfill \\ {\displaystyle \frac{\mathrm{d}{w}_{{ϵ}_n}\left(t\right)}{\mathrm{d}t}\rightharpoonup \frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}weaklyin{L}^{4/3}(-T,T;{\widehat{V}}^{\ast })as{ϵ}_n\to 0^{+},\forall T>0.}\hfill & \hfill \end{array}\right.\end{array}$$

(44)

We now denote by

$$\mathcal{Y}=\left\{v\left(t\right):v\left(t\right)\in L^2([-T,T];\widehat{V})\cap L^{\infty }([-T,T];\widehat{H}),\frac{\mathrm{d}v\left(t\right)}{\mathrm{d}t}\in L^{4/3}([-T,T];{\widehat{V}}^{\ast })\right\}$$

and endow it with the norm

$${\parallel v\parallel }_{\mathcal{Y}}={\parallel v\parallel }_{L^2([-T,T];\widehat{V})}+{\parallel v\parallel }_{L^{\infty }([-T,T];\widehat{H})}+{\parallel \frac{\mathrm{d}v\left(t\right)}{\mathrm{d}t}\parallel }_{L^{4/3}([-T,T];{\widehat{V}}^{\ast })}.$$

Then, [3] (Lemma 2.3.1, $P_{55}$) proved that

$$\begin{array}{c}\hfill \mathcal{Y}\subset C([-T,T];{\widehat{H}}_{\mathrm{w}}).\end{array}$$

(45)

Notice that the topology ${\Theta }_{\mathrm{loc}}^{+}$ is compatible with the weak distance defined by (22). Thus, (44) and (45) imply that the topology given by (44) is stronger than ${\Theta }_{\mathrm{loc}}^{+}$ in ${\mathcal{F}}^{+}$. Consequently, (44) gives

$$\begin{array}{c}\hfill w_{{ϵ}_n}(·)\to w(·)\mathrm{in}\mathrm{the}\mathrm{topology}{\Theta }_{\mathrm{loc}}^{+}\mathrm{as}{ϵ}_n\to 0^{+}.\end{array}$$

(46)

Now, Lemma 4 indicates that $w(·)\in {\mathcal{A}}_0^{\mathrm{tr}}$. This fact, combined with (43) and (44), says that $w\left(t\right)\in {\mathcal{K}}_0$. Therefore, (29) gives ${\Pi }_{+}w\in {\Pi }_{+}{\mathcal{K}}_0={\mathcal{A}}_0^{\mathrm{tr}}$, and (46) implies that

$${\Pi }_{+}{w}_{{ϵ}_n}\to {\Pi }_{+}w\mathrm{in}\mathrm{the}\mathrm{topology}{\Theta }_{\mathrm{loc}}^{+}\mathrm{as}{ϵ}_n\to 0^{+},$$

which means that, for large enough n,

$$\begin{array}{c}\hfill {\Pi }_{+}{w}_{{ϵ}_n}\in \mathcal{O}\left({\Pi }_{+}w\right)\subset \mathcal{O}\left({\mathcal{A}}_0^{tr}\right).\end{array}$$

(47)

(47) contradicts (41). This completes the proof. □

As seen below, Lemma 5 implies that the trajectory attractors ${\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ converge as $ϵ\to 0^{+}$ to ${\mathcal{A}}_0^{\mathrm{tr}}$. The convergence of this type was researched in [60] for the convective Brinkman–Forchheimer equations.

The main results of this article read as follows.

Theorem 2.

Suppose that assumption (H) holds. Let ${\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ and ${\mathcal{A}}_0^{\mathrm{tr}}$ be the trajectory attractors obtained by Theorem 1(1) and Lemma 3(1), respectively. Let $w_{ϵ}\in {\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ and $w_{ϵ}\to w$ in the topology ${\Theta }_{\mathrm{loc}}^{+}$ as $ϵ\to 0^{+}$, and ${\rho }_{ϵ,w_{ϵ}}$ and ${\rho }_{0,w}$ be the trajectory statistical solutions obtained by Theorem 1(2) and Lemma 3(2), respectively. Then,

$$\begin{array}{cc}\hfill & {\mathcal{A}}_{ϵ}^{\mathrm{tr}}\to {\mathcal{A}}_0^{\mathrm{tr}}inthetopology{\Theta }_{\mathrm{loc}}^{+}asϵ\to 0^{+},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & \underset{ϵ\to 0^{+}}{lim}{\int }_{{\mathcal{A}}_{ϵ}^{\mathrm{tr}}}\gamma \left(v\right)\mathrm{d}{\rho }_{ϵ,w_{ϵ}}\left(v\right)={\int }_{{\mathcal{A}}_0^{\mathrm{tr}}}\gamma \left(v\right)\mathrm{d}{\rho }_{0,w}\left(v\right),\forall \gamma \in C\left({\mathcal{F}}^{+}\right).\hfill \end{array}$$

(49)

Proof. 

Employing Lemma 5 with ${\mathfrak{B}}_{ϵ}={\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ and the invariant property of ${\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ and letting $t\to +\infty$, we obtain that

$${\mathcal{A}}_{ϵ}^{\mathrm{tr}}=T\left(t\right){\mathcal{A}}_{ϵ}^{\mathrm{tr}}\to {\mathcal{A}}_0^{\mathrm{tr}}\mathrm{in}\mathrm{the}\mathrm{topology}{\Theta }_{\mathrm{loc}}^{+}\mathrm{as}ϵ\to 0^{+}.$$

This proves (48).

We next establish (49). Let ${\left\{w_{ϵ}(·)\right\}}_{0<ϵ⩽1}$ with $w_{ϵ}(·)\in {\mathcal{A}}_{ϵ}^{\mathrm{tr}}$ for each $ϵ\in (0,1]$, and $w_{ϵ}(·)\to w(·)$ in the topology ${\Theta }_{\mathrm{loc}}^{+}$ as $ϵ\to 0^{+}$. Lemma 4 gives $w(·)\in {\mathcal{A}}_0^{\mathrm{tr}}$. At the same time, Theorem 1(2) and Lemma 3(2) say that, to $w_{ϵ}$ and w, there corresponds, respectively, trajectory statistical solutions ${\rho }_{ϵ,w_{ϵ}}$ and ${\rho }_{0,w}$ such that

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{T}}_{ϵ}^{\mathrm{tr}}}\gamma \left(v\right)\mathrm{d}{\rho }_{ϵ,w_{ϵ}}\left(v\right)={\mathrm{LIM}}_{t\to +\infty }\frac{1}{t}{\int }_0^t\gamma \left(T\left(s\right)w_{ϵ}\right)\mathrm{d}s,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{T}}_0^{\mathrm{tr}}}\gamma \left(v\right)\mathrm{d}{\rho }_{0,w}\left(v\right)={\mathrm{LIM}}_{t\to +\infty }\frac{1}{t}{\int }_0^t\gamma \left(T\left(s\right)w\right)\mathrm{d}s\hfill \end{array}$$

(51)

for a given generalized Banach limit ${\mathrm{LIM}}_{t\to +\infty }$ and given $\gamma \in C\left({\mathcal{F}}^{+}\right)$. Notice that ${\mathrm{LIM}}_{t\to +\infty }$ is a given linear continuous functional, the translation semigroup $T(·)$ is continuous on ${\mathcal{F}}^{+}$ with respect to the topology ${\Theta }_{\mathrm{loc}}^{+}$ (cf. [53]), $\gamma \in C\left({\mathcal{F}}^{+}\right)$ is a given continuous function, and $w_{ϵ}\to w$ in the topology ${\Theta }_{\mathrm{loc}}^{+}$ as $ϵ\to 0^{+}$. We have from (50) and (51) that

$$\begin{array}{cc}\hfill \underset{ϵ\to 0^{+}}{lim}{\int }_{{\mathcal{T}}_{ϵ}^{\mathrm{tr}}}\gamma \left(v\right)\mathrm{d}{\rho }_{ϵ,w_{ϵ}}\left(v\right)=& \underset{ϵ\to 0^{+}}{lim}{\mathrm{LIM}}_{t\to +\infty }\frac{1}{t}{\int }_0^t\gamma \left(T\left(s\right)w_{ϵ}\right)\mathrm{d}s\hfill \\ \hfill =& {\mathrm{LIM}}_{t\to +\infty }\frac{1}{t}{\int }_0^t\gamma \left(T\left(s\right)\underset{ϵ\to 0^{+}}{lim}{w}_{ϵ}\right)\mathrm{d}s\hfill \\ \hfill =& {\mathrm{LIM}}_{t\to +\infty }\frac{1}{t}{\int }_0^t\gamma \left(T\left(s\right)w\right)\mathrm{d}s\hfill \\ \hfill =& {\int }_{{\mathcal{T}}_0^{\mathrm{tr}}}\gamma \left(v\right)\mathrm{d}{\rho }_{0,w}\left(v\right).\hfill \end{array}$$

This ends the proof. □

4. Conclusions and Remarks

In this article, we prove the homogenization of the trajectory statistical solutions for the 3D incompressible micropolar fluids with rapidly oscillating external force and angular momentum, based on the homogenization theory of attractors and the structure of the trajectory statistical solutions. The essential steps are to verify that the solutions within the trajectory space and the trajectory attractors of the oscillating equations converge to those of the homogenized ones, as the oscillating external force and angular momentum tend toward their averages. Our results reveal that the trajectory statistical information of the 3D incompressible micropolar fluids has a certain homogenization effect with respect to spatial variables.

To our best knowledge, this article is the first one investigating the homogenization issue of trajectory statistical solutions for evolution equations. The main novelty is that we find that the subtle bones of the homogenization theory of attractors and the structure of the trajectory statistical solutions could be used to study the homogenization of trajectory statistical solutions for evolution equations. We want to point out that the approach presented in this article could be extended and give us a tool of much wider applicability.

(1)

We can use the approach presented in this article to study the homogenization of trajectory statistical solutions for a broad class of evolutionary equations displaying the property of global existence of weak solutions without a known result of global uniqueness [16,19], including some model hydrodynamic equations; MHD equations; and non-Newtonian fluids equations such as 3D incompressible NS equations [53,58], 3D NS-like systems [36], 3D NS-$\alpha$ model [37], 2D dissipative Euler equations [48,54], 3D MHD equations [35,56,61,62,63,64], 3D non-Newtonian and micropolar fluids equations [7,65,66,67,68,69,70], Ginzburg–Landau equations [29], etc. At the same time, our approach can also be applied to the abovementioned equations, 3D Navier–Stokes–Voight equation [21] and damped semilinear wave equations [23,41], investigating the homogenization of trajectory statistical solutions with respect to time variables.

(2)

For another broad class of non-autonomous evolutionary equations that has global well-posedness, say 2D incompressible NS equations [17,42,53], 2D micropolar fluid flows [5], magneto-micropolar fluids [56,71], two-phase flow model [18], globally modified NS equations [39,45], Klein–Gordon–Schrödinger equations [49], etc., one usually use the pullback attractors to depict the asymptotic behavior of solutions. In this situation, we can investigate the homogenization (also degenerated regularity) of statistical solutions for these mentioned equations, via the homogenization (also degenerated regularity) of pullback attractors and by exploring the structure of statistical solutions.

We will investigate these issues in some other articles.