\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 115, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/115\hfil Compactness result for periodic structures] {Compactness result for periodic structures and its application to the homogenization of a diffusion-convection equation} \author[A. Meirmanov, R. Zimin \hfil EJDE-2011/115\hfilneg] {Anvarbek Meirmanov, Reshat Zimin} % in alphabetical order \address{Anvarbek M. Meirmanov \newline Department of mahtematics \\ Belgorod State University \\ ul.Pobedi 85, 308015 Belgorod, Russia} \email{meirmanov@bsu.edu.ru} \address{Reshat Zimin \newline Department of mahtematics \\ Belgorod State University \\ ul.Pobedi 85, 308015 Belgorod, Russia} \email{reshat85@mail.ru} \thanks{Submitted April 10, 2011. Published September 6, 2011.} \subjclass[2000]{35B27, 46E35, 76R99} \keywords{Weak, strong and two-scale convergence; homogenization; \hfill\break\indent diffusion-convection} \begin{abstract} We prove the strong compactness of the sequence $\{c^{\varepsilon}(\mathbf{x},t)\}$ in $L_2(\Omega_T)$, $\Omega_T=\{(\mathbf{x},t):\mathbf{x}\in\Omega \subset \mathbb{R}^3, t\in(0,T)\}$, bounded in $W^{1,0}_2(\Omega_T)$ with the sequence of time derivative $\{\partial/\partial t\big(\chi(\mathbf{x}/\varepsilon) c^{\varepsilon}\big)\}$ bounded in the space $L_2\big((0,T); W^{-1}_2(\Omega)\big)$. As an application we consider the homogenization of a diffusion-convection equation with a sequence of divergence-free velocities $\{\mathbf{v}^{\varepsilon}(\mathbf{x},t)\}$ weakly convergent in $L_2(\Omega_T)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{assumption}[theorem]{Assumption} \section{Introduction} \label{Introduction} There are several compactness criteria and among them Tartar's method of compensated compactness \cite{Tartar} and the method suggested by Aubin in \cite{Aubin} (see also \cite{Lions}). These methods intensively used in the theory of nonlinear differential equations. As a rule, the first one has applications in stationary problems, while the second method is used in non-stationary nonlinear equations. In the present publication we discuss the method, closed to the Aubin compactness lemma. In its simplest setting, this result provides the strong compactness in $L_2(\Omega_T)$ (throughout the article, we use the customary notation of function spaces and norms \cite{Lions,LSU}) to the sequence of functions $\{c^{\varepsilon}(\mathbf{x},t)\}$ bounded in $L_{\infty}\big((0,T);L_2(\Omega)\big)\cap W^{1,0}_2(\Omega_T)$ with the sequence of the time derivatives $\{\partial c^{\varepsilon}/\partial t\}$ bounded in $L_2\big((0,T); W^{-1}_2(\Omega)\big)$. But in many applications (especially in homogenization), the second condition on a boundedness of the time derivatives in some dual space is not always satisfied. Sometimes, instead of the last condition, one has the boundedness of time derivatives in a dual space $L_2\big((0,T); W^{-1}_2(\Omega^{\varepsilon}_f)\big)$, defined on some periodic subdomain $\Omega^{\varepsilon}_f\subset \Omega$. Using new ideas of Nguetseng's two-scale convergence method \cite{NGU} we prove that even under this weak condition the sequence $\{c^{\varepsilon}(\mathbf{x},t)\}$ still remains strongly compact in $L_2(\Omega_T)$. The main point here is the fact, that if for some $t_0\in(0,T)$, $\lim_{\varepsilon\to 0}\varepsilon^2 \int_{\Omega}|\nabla c^{\varepsilon}(\mathbf{x},t_0)|^2\,dx=0,$ then the bounded in $L_2(\Omega)$ sequence $\{c^{\varepsilon}(\mathbf{x},t_0)\}$ contains a subsequence, which two-scale converges in $L_2(\Omega)$ to some function $\bar{c}(\mathbf{x},t_0)$. Recall that, in general, any bounded in $L_2(\Omega)$ sequence $\{u^{\varepsilon}\}$ contains a two-scale convergent subsequence $\{u^{\varepsilon_k}\}$, where the limiting function $U(\mathbf{x},\mathbf{y})$ is 1-periodic in variable $\mathbf{y} \in Y=(0,1)^{n}$: $\int_{\Omega}u^{\varepsilon_k}(\mathbf{x}) \varphi(\mathbf{x},\frac{\mathbf{x}}{\varepsilon_k})dx \to\iint _{\Omega Y}U(\mathbf{x},\mathbf{y}) \varphi(\mathbf{x},\mathbf{y})dydx$ for any smooth function $\varphi(\mathbf{x},\mathbf{y})$, 1-periodic in the variable $\mathbf{y}$. In particular, for $\varphi(\mathbf{x},\mathbf{y})=\varphi_0(\mathbf{y}) \cdot h(\mathbf{x})$, where $\varphi_0 \in L_2(Y)$ and $h \in L_{\infty}(\Omega)$. A similar compactness result has been proved in \cite{AAPP} under different assumptions on the sequence $\{c^{\varepsilon}(\mathbf{x},t)\}$. More precisely, the corresponding \cite[Lemma 4.2]{AAPP} states, that if for all $\varepsilon>0$ $0\leqslant c^{\varepsilon}(\mathbf{x},t)\leqslant M_0, \int_{\Omega_T} |c^{\varepsilon}(\mathbf{x} +\triangle \mathbf{x},t)-c^{\varepsilon}(\mathbf{x},t)|^2\,dx\,dt \leqslant M_0 \omega(|\triangle \mathbf{x}|),$ with some $\omega(\xi)$, such that $\omega(\xi)\to 0$ as $\xi\to 0$, and $\|\frac{\partial}{\partial t}(\chi^{\varepsilon} c^{\varepsilon})\| _{L_2\big((0,T); W^{-1}_2(\Omega)\big)}\leqslant M_0,$ where $0<\chi^{-}\leqslant\chi^{\varepsilon}\leqslant\chi^{+}<1$, $\chi^{\pm}=const$, then the family $\{c^{\varepsilon}\}$ is a compact set in $L_2(\Omega_T)$. As an application of our result we consider the homogenization of diffusion-convection equation $$\frac{\partial c^{\varepsilon}}{\partial t}+ \mathbf{v}^{\varepsilon}\cdot\nabla c^{\varepsilon}= \triangle c^{\varepsilon}, \quad \mathbf{x}\in \Omega^{\varepsilon},\; t\in(0,T), \label{0.1}$$ with boundary and initial conditions \begin{gather} \big(\nabla c^{\varepsilon}-\mathbf{v}^{\varepsilon} c^{\varepsilon}\big) \cdot \boldsymbol{\nu}=0, \quad \mathbf{x}\in \partial\Omega^{\varepsilon}\backslash S,\; t\in(0,T), \label{0.2} \\ c^{\varepsilon}(\mathbf{x},t)=0, \quad \mathbf{x}\in S\cap\partial\Omega^{\varepsilon},\; t\in(0,T), \label{0.3} \\ c^{\varepsilon}(\mathbf{x},0)=c_0(\mathbf{x}), \quad \mathbf{x}\in \Omega^{\varepsilon}. \label{0.4} \end{gather} In \eqref{0.2}, $\boldsymbol{\nu}$ is the unit outward normal vector to the boundary $\partial\Omega^{\varepsilon}$ and $S=\partial\Omega$. We assume that velocities $\mathbf{v}^{\varepsilon}$ are uniformly bounded in $L_8\big((0,T);L_4(\Omega)\big)$: $$\int_0^{T}\Big(\int_{\Omega} |\mathbf{v}^{\varepsilon}|^{4}dx\Big)^2dt\leqslant M_0^2, \label{0.5}$$ and $$\nabla\cdot \mathbf{v}^{\varepsilon}=0, \mathbf{x}\in\Omega_T. \label{0.6}$$ As usual, the solution to the problem \eqref{0.1}--\eqref{0.4} is understood in a weak sense as a solution of the integral identity $$\int_{\Omega^{\varepsilon}_T}\Big(c^{\varepsilon} \frac{\partial\phi}{\partial t}-\big(\nabla c^{\varepsilon} -\mathbf{v}^{\varepsilon} c^{\varepsilon}\big)\cdot \nabla \phi\Big)\,dx\,dt =-\int_{\Omega^{\varepsilon}}c_0(\mathbf{x})\phi(\mathbf{x},0)\,dx \label{0.7}$$ for any smooth functions $\phi$, such that $\phi(\mathbf{x},T)=0$. Homogenization means the limiting procedure in \eqref{0.7} as $\varepsilon\to 0$ and the main problem here is how to pass to the limit in the nonlinear term $c^{\varepsilon} \mathbf{v}^{\varepsilon}\cdot \nabla \phi.$ It has been done for velocities with a special structure $\mathbf{v}^{\varepsilon}=\mathbf{v}^{\varepsilon} (\mathbf{x}), \text{or} \mathbf{v}^{\varepsilon}=\mathbf{v}(\mathbf{x},t, \frac{\mathbf{x}}{\varepsilon})$ (see, for example, \cite{AGP, AC,BLP,BJP,Hornung,HJ}). However, in the general case we need the strong compactness in $L_2(\Omega_T)$ of the sequence $\{c^{\varepsilon}\}$. Our compactness result and the energy estimate $\max_{00 an existence of some constant C_{\eta} such that \[ \|\tilde{c} ^{\varepsilon_k}-c\|_{\mathbb{H}^{0}}(t)\leqslant \eta\|\tilde{c} ^{\varepsilon_k}-c\|_{\mathbb{H}^1}(t)+ C_{\eta}\|\tilde{c} ^{\varepsilon_k}-c\|_{\mathbb{H}^{-1}}(t)$ for all $k$ and for all $t\in[0,T]$ (see \cite{Lions}). Therefore, \begin{align*} \int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\| _{\mathbb{H}^{0}}^2(t)dt &\leqslant \eta\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\| _{\mathbb{H}^1}^2(t)dt+ C_{\eta}\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\| _{\mathbb{H}^{-1}}^2(t)dt\\ &\leqslant 2 \eta M_0^2+ C_{\eta}\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\| _{\mathbb{H}^{-1}}^2(t)dt. \end{align*} Due to the compact imbedding $\mathbb{H}^{0}\to\mathbb{H}^{-1}$, the weak convergence in $\mathbb{H}^{0}$ of the sequence $\{\tilde{c} ^{\varepsilon_k}(\mathbf{x},t_0)\}$ to the function $c(\mathbf{x},t_0)$ for all $t_0\in Q$, and the dominated convergence theorem \cite{KF} one has $\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\| _{\mathbb{H}^{-1}}^2(t)dt \to 0 \quad \text{as } k\to\infty.$ This last fact and the arbitrary choice of the constant $\eta$ prove the statement of the lemma. \end{proof} \section{Proof of Theorem \ref{thm1.2}} To simplify the proof we additionally suppose that \begin{assumption} \label{assum2}\rm \begin{itemize} \item[(1)] $Y_{s}\subset Y, \gamma \cap \partial Y=\emptyset$; \item[(2)] the domain $\Omega$ is a unit cube; \item[(3)] $1/\varepsilon$ is an integer. \end{itemize} \end{assumption} As before, we divide the proof by several steps. As a first step we state the well-known existence and uniqueness result for solutions of the problem \eqref{0.1}--\eqref{0.3} (see \cite{LSU}). \begin{lemma}\label{lem3.1} Under conditions of Theorem \ref{thm1.2} for all $\varepsilon >0$ the problem \eqref{0.1}--\eqref{0.4} has a unique solution $c^{\varepsilon}\in L_{\infty}\big((0,T);L_2(\Omega^{\varepsilon})\big)\cap W^{1,0}_2(\Omega_T^{\varepsilon})$ and \max_{0