\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conf. 17 (2009), pp. 255--265.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{255} \title[\hfilneg EJDE-2009/Conf/17 \hfil Existence of solutions] {Existence of solutions for thermoelastic semiconductor equations} \author[X. Wu\hfil EJDE/Conf/17 \hfilneg] {Xiaoqin Wu} \address{Xiaoqin Wu \newline Department of Mathematics, Computer and Information Sciences\\ Mississippi Valley State University\\ Itta Bena, MS 38941, USA} \email{xpaul\_wu@yahoo.com} \thanks{Published April 15, 2009.} \subjclass[2000]{35M10, 35Q99} \keywords{Thermoelastic semiconductor equations; existence} \begin{abstract} We study a model for the semiconductor problem that consists of a system of dynamic thermoelasticity equations of displacement and semiconductor equations. This problem arises from the observation that semiconductor devices are too often cracked and broken because of the thermal stresses. Since the heat source generated by Joule heating is quadratic in the gradient of the electrical potential, this causes some problem even in analysis. We establish the existence theorem of a weak solution. The proof is based on time retarding. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}{Definition}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}{Lemma}[section] \section{Introduction} Semiconductors are electrical devices which are extensively used in industrial world and in our daily life. The so-called ``Semiconductor Problem'' is a system of a stationary charge conservation equation of electrical current and two parabolic equations of electrons and holes. That is, \begin{equation} \label{e1.1} \begin{gathered} -\mathop{\rm div}(\sigma\nabla\varphi)=p-n+f\\ n_t-\mathop{\rm div}(d_n \nabla n-\mu_n \sigma n\nabla\varphi)=g(n,p)(1-np)\\ p_t-\mathop{\rm div}(d_p\nabla p+ \mu_p \sigma p\nabla\varphi)=g(n,p)(1-np) \end{gathered} \end{equation} where $\varphi$ represents the electrostatic potential, $n$ and $p$ the densities of electrons and holes, $\sigma$ the electrical conductivity, $d_n$ and $d_p$ the diffusion coefficients, $\mu_n$ and $\mu_p$ the mobilities of electrons and holes, respectively, $f$ the net impurity, $g(n,p)(1-np)$ the general rate of recombination-generation. This model, which appeared in the 1950's, has, since then, received much attention (see \cite{A_A_R:1, B_F, R:1, YIN:1}). It is a common phenomenon that electrical devices produce heat or temperature. One of the main reasons for considering the thermoelastic semiconductor problem is the observation that these devices are too often cracked and broken because of the thermal stresses. However, the literature on this problem has not dealt with the thermoelastic aspects of the process, only with the thermal and electrical conductions. In this paper we fill in this gap and consider a new model for the thermoelastic semiconductor. The thermoelastic behavior is taken into account, resulting in a fully coupled system of equations for the temperature, electrical potential and elastic displacements. This model consists of an elliptic equation of charge conservation of electrical current, three hyperbolic equations of displacement of the device, a parabolic equation of the energy caused by Joule's heating, and two parabolic equations of the densities of electrons and holes. For the sake of simplicity, we assume that the material constitutive behavior can be adequately described by linear thermoelasticity. The nonlinearity resides in the electrical conductivity. Moreover, Joule's heating introduces a source term in the heat equation that is quadratic in the gradient of the electrical potential. This feature makes the problem interesting also from the mathematical point of view. In this paper we establish the existence of a weak solution for the problem. This paper is organized as follows. We present the classical model of thermoelastic semiconductor in Section \ref{sec2} where we explain in the combined process. The formulation of a weak solution for this model is presented in Section \ref{sec3} together with the assumptions on the data. The statement of our main result is given in Theorem \ref{sec3:thm1}. The proof of the existence of a weak solution is presented in Section \ref{sec4}. It is based on a sequence of time retarded problems. The basic idea is the following. First we give a prior estimates, and then construct a sequence of solutions of approximate problems. By passing to the limits of the sequence, we will obtain a weak solution. \section{A Model of Thermoelastic Semiconductor Problem} \label{sec2} Let $\Omega\subset\mathbb{R}^N$ ($N\ge 2$) be a bounded domain with smooth boundary $\Gamma=\partial\Omega$, representing the isothermal reference configuration of the thermoelastic body, the semiconductor in our case. We assume that $\Gamma$ is divided into two relatively open parts $\Gamma_D$ and $\Gamma_N$ such that $\Gamma_D\cap\Gamma_N=\emptyset$ and $\overline{\Gamma}_D\cup \overline{\Gamma}_N=\Gamma$. We denote by $\nu=(\nu_1, \dots, \nu_N)$ the outward unit normal to $\Gamma$. We assume that the body is held fixed on $\Gamma_D$, and the temperature and potential are prescribed there. On $\Gamma_N$ the body is free, electrically insulted and exchanging heat with the environment. We choose the Dirichlet conditions on $\Gamma_D$ for the three fields for the sake of simplicity. We use $\theta$ to represent the temperature field, $\varphi$ the electrical potential and $u=(u_1, \dots, u_N)$ the displacement field. Let $T>0$ and set $\Omega_T=\Omega\times(0, T)$. The behavior of the system is governed by the energy equation, the equation of charge conservation and the equations of linear elasticity. In a non-dimensional form we may write the system (see, e.g., \cite{A_C:1, C_C:1, D_L:1, H_R_S:1, YIN:1}) as \begin{equation} \label{SemiEls} \begin{gathered} -\mathop{\rm div}(\sigma(\theta)\nabla\varphi)=p-n+f,\\ \frac{\partial n}{\partial t}-\mathop{\rm div}(d_n\nabla n-\mu_n \sigma(\theta)n\nabla\varphi)=g(n,p)(1-np),\\ \frac{\partial p}{\partial t}-\mathop{\rm div}(d_p\nabla p+\mu_p \sigma(\theta)p\nabla\varphi)=g(n,p)(1-np),\\ \frac{\partial\theta}{\partial t}-\frac{\partial}{\partial x_j}\Big(k_{ij}(\theta)\frac{\partial \theta}{\partial x_i}\Big) =\sigma(\theta) | \nabla \varphi |^2-m_{ij}\frac{\partial^2 u_i}{\partial t\partial x_j},\\ \rho \frac{\partial^2 u_i}{\partial t^2}-\frac{\partial}{\partial x_j}\Big(a_{ijkl}\frac{\partial u_k}{\partial x_l}-m_{ij}\theta\Big)=f_i. \end{gathered} \end{equation} Here and below, $i, j, k, l=1, \dots, N$ and summation over repeated indices is employed. The density $\rho>0$ is a constant; $K=\{k_{ij}\}$ and $M=\{m_{ij}\}$ are the heat conduction and thermal expansion tensors, respectively, and $A=\{a_{ijkl}\}$ is the elasticity tensor; $\mathbf{f}=(f_1, \dots, f_N)$ represents the density of body force. Finally, $\sigma=\sigma(\theta)$ is the temperature-dependent electrical conductivity. The first equation of \eqref{SemiEls} represents the charge conservation of the electrical potential. The second and third ones represent the exchange of the densities of electrons and holes of the semiconductor. The fourth one represents the heat transfer equation in terms of the deformation of the semiconductor and the energy, produced by the so-called Joule heating \[ J=\sigma(\theta) | \nabla \varphi |^2 \] generated by the electrical current for the temperature $\theta$. The last one of \eqref{SemiEls} is the dynamic thermoelasticity equations of displacement of the semiconductor. To complete the classical formulation of the problem we have to specify the initial and boundary conditions. We set \begin{equation} \label{IniCon} \begin{gathered} \varphi=\varphi_b, \ n=n_b, \ p=p_b, \ u=0, \ \theta=\theta_b, \ \text{on }\Gamma_D\times(0,T), \\ \frac{\partial\varphi}{\partial \nu}=0, \ \frac{\partial n}{\partial \nu}=0, \ \frac{\partial p}{\partial \nu}=0, \ \frac{\partial u_i}{\partial \nu}=0, \ i=1, \dots, N, \ \text{on }\Gamma_N\times(0,T), \\ -k_{ij}\frac{\partial \theta}{\partial x_i}\nu_j=h(\theta-\theta_a),\ \text{on }\Gamma_N\times(0,T), \\ n=n_0, \ p=p_0, \ u=u_0, \ u_t=u_1, \ \theta=\theta_0, \ \text{in }\Omega\times\{t=0\}, \end{gathered} \end{equation} where $n_0, p_0$ are the initial densities of electrons and holes, respectively, $u_0, u_1$ the initial displacements and velocities, respectively, $\theta_0$ the initial temperature, $n_b$, $p_b$ are the density on $\Gamma_D$ respectively, $\theta_b$ the temperature and $\varphi_b$ the potential there, $\theta_a$ the ambient temperature near $\Gamma_N$, and $h$ the heat exchange coefficient. Our goal for this ``thermoelastic semiconductor'' problem is: \begin{center} Find $\{\varphi, n, p, \theta, u\}$ such that \eqref{SemiEls}-\eqref{IniCon} are satisfied. \end{center} The precise assumptions on the data and the weak formulation are given in the next section. We note that, if (the electrical conductivity) $\sigma=0$ at large temperature, the first equation of \eqref{SemiEls} for $\varphi$ degenerates and the heating term in the fourth equation of \eqref{SemiEls} vanishes and This degeneracy makes it necessary to consider a capacity solution (\cite{XU:1,XU:2}) to this model. Recently, Wu and Xu (\cite{W_X}) consider a model of thermoelastic thermistor problem with the degenerate electrical conductivity and obtain the existence of its capacity solutions. The authors use time retardation to construct an approximate model whose solution is easily to be obtained and whose limit is the solution of the original model. The method used in this paper is similar to the one in that paper. We must note that when $\rho\partial^2u_i/\partial t^2$ may be omitted; i.e., small $\rho$ or small accelerations, the resulting problem is quasi-static and we will consider in future. \section{Weak Formulation and Main Result} \label{sec3} We present a weak formation for our model, the assumptions on the problem data and the statement of our existence results. For simplicity we extend $n_b, p_b, \theta_b$ to functions defined on $\overline{\Omega}_T$, and denote it by $n_b, p_b, \theta_b$. This means that they have to satisfy a compatibility condition \begin{gather*} n_b, p_b, \theta_b\in H^1(\Omega_T), \\ \mathop{\rm ess\,sup}_{(\Gamma_D\times(0,T))\cup(\Omega\times{0})}{|n_b|,\ |p_b|, \ |\theta_b|}<\infty,\\ \varphi_b\in L^2(0,T; H^1(\Omega))\cap L^\infty(\Omega_T). \end{gather*} First, we introduce the following function spaces: \begin{gather*} V_0=H^1_D(\Omega)=\{w\in H^1(\Omega): w=0\text{ on }\Gamma_D\}, \ U_0=H^1_D(\Omega)^N \end{gather*} and denote by $V_0'$ and $U_0'$ their dual spaces, respectively. Finally, we assume that $\rho>0$ is constant and that \begin{gather*} u_0(x)=(u_{01}, \dots, u_{0N})\in U_0, \\ u_1(x)=(u_{11}, \dots, u_{1N})\in (L^2(\Omega))^N, \\ f=(f_1, \dots, f_N)\in (L^2(\Omega_T))^N, \ h\in (0,\infty), \\ a_{ijkl}, \frac{\partial}{\partial t}a_{ijkl}\in L^\infty(\Omega_T), a_{ijkl}=a_{jikl}=a_{klij}, \\ a_{ijkl}\eta_{ij}\eta_{kl}\ge \lambda|\eta|^2=\lambda\sum_{i,j=1}^N\eta_{ij}^2, \ \forall \eta=(\eta_{ij}),\\ \eta_{ij}=\eta_{ji}, \lambda>0, \\ m_{ij}, \frac{\partial}{\partial t}m_{ij}\in W^{1, \infty}(\Omega_T), \ m_{ij}=m_{ji}, \\ k_{ij}\in L^\infty(\Omega_T), \ k_{ij}=k_{ji}, \\ k_{ij}\xi_i\xi_j\ge\lambda_1|\xi|^2, \ \forall \xi=(\xi_{ij}), \lambda_1>0,\\ 00$ , let $\varepsilon=\frac{T}{m}$ and define the time-retarded function $g_\varepsilon$ of $g$ by \[ g_\varepsilon=\begin{cases} g(t-\varepsilon) &\text{if } t>\varepsilon,\\ g_0 &\text{if } t\in[0,\varepsilon], \end{cases} \] where $g_0$ will be given. Consider the following approximation problem: for all $\psi\in V_0, \gamma\in V_0\cap L^\infty(\Omega), w\in U_0$ \begin{equation} \label{ApprProblem} \begin{gathered} \int_{\Omega}\sigma(\theta^\varepsilon_\varepsilon)\nabla\varphi^\varepsilon\nabla\psi dx =\int_{\Omega}(p^\varepsilon-n^\varepsilon+f)\psi dx, \\ \langle n^\varepsilon_t,\psi\rangle +\int_{\Omega}(d_n\nabla n^\varepsilon -\mu_n\sigma(\theta^\varepsilon_\varepsilon)n^\varepsilon\nabla\varphi^\varepsilon)\nabla\psi dx =\int_{\Omega}g(n^\varepsilon,p^\varepsilon)(1-n^\varepsilon p^\varepsilon)\psi dx, \\ \langle p^\varepsilon,\psi\rangle +\int_{\Omega}(d_p\nabla p^\varepsilon +\mu_p\sigma(\theta^\varepsilon_\varepsilon)p^\varepsilon\nabla\varphi^\varepsilon)\nabla \psi dx =\int_{\Omega}g(n^\varepsilon,p^\varepsilon)(1-n^\varepsilon p^\varepsilon)\psi dx, \\ \begin{aligned} &\langle \theta^\varepsilon,\gamma\rangle +\int_{\Omega}k_{ij} (\theta_\varepsilon^\varepsilon)\frac{\partial \theta^\varepsilon}{\partial x_i} \frac{\partial\gamma}{\partial x_j}dx+\int_{\Gamma_N} h(\theta^\varepsilon-\theta_b)\gamma\,dS\\ &=-\int_{\Omega}\sigma(\theta^\varepsilon_\varepsilon)\varphi^\varepsilon\nabla \varphi^\varepsilon\nabla\gamma dx -\int_{\Omega}\varphi^\varepsilon(n^\varepsilon-p^\varepsilon +f)\gamma dx-\int_{\Omega}m_{ij} \frac{\partial v_i^\varepsilon}{\partial x_j}\gamma dx, \end{aligned} \\ \rho \langle v_t^\varepsilon,w\rangle +\int_{\Omega}a_{ijkl} \frac{\partial u_k^\varepsilon}{\partial x_l}\frac{\partial w_i}{\partial x_j}dx-\int_{\Omega}m_{ij}\theta^\varepsilon \frac{\partial w_i}{\partial x_j}dx=\int_{\Omega}f_iw_i dx, \\ v^\varepsilon=u_t^\varepsilon \end{gathered} \end{equation} with the following boundary conditions: \begin{gather*} \varphi^\varepsilon=\varphi_b, \ n^\varepsilon=n_b, \ p^\varepsilon=p_b, \ u^\varepsilon=0, \ \theta=\theta_b, \ \text{on }\Gamma_D\times(0,T), \\ \frac{\partial\varphi^\varepsilon}{\partial \nu}=0, \ \frac{\partial n^\varepsilon}{\partial \nu}=0, \ \frac{\partial p^\varepsilon}{\partial \nu}=0, \ \frac{\partial u_i^\varepsilon}{\partial \nu}=0, \ i=1, \dots, N, \text{ on }\Gamma_N\times(0,T), \\ -k_{ij}\frac{\partial \theta^\varepsilon}{\partial x_i}\nu_j=h(\theta^\varepsilon-\theta_a), \ \text{on }\Gamma_N\times(0,T), \\ n^\varepsilon=n_0, \ p^\varepsilon=p_0, \ u^\varepsilon=u_0, \ v^\varepsilon=u_1, \ \theta^\varepsilon=\theta_0, \ \text{in }\Omega\times\{t=0\}, \end{gather*} Here, according to the definition above, $$ \theta^\varepsilon_\varepsilon=\theta^\varepsilon(t-\varepsilon)=\theta_0, \ \text{for }t\in[0,\varepsilon] $$ is given. With $\sigma(\theta^\varepsilon_\varepsilon)$ given and the fact that $\sigma$ is bounded between positive constants, the first three equations of \eqref{ApprProblem} are the system of standard elliptic and parabolic equations of semiconductor devices. The existence and uniqueness of the solutions of these three equations in $[0,\varepsilon]$ are well known (e.g., see \cite{YIN:1}). Once we obtain $(\varphi^\varepsilon, n^\varepsilon, p^\varepsilon)$, the rest three equations of \eqref{ApprProblem} are standard linear thermoelasticity equations whose existence and uniqueness of the solution $(\theta^\varepsilon,u^\varepsilon,v^\varepsilon)$ in $[0,\varepsilon]$ are also well known (e.g., see \cite{R_R:1}). We proceed inductively on each time interval $[k\varepsilon,(k+1)\varepsilon]$ for $0\le k\le m$ to obtain the unique solution of \eqref{ApprProblem} in $[0,T]$. To take limit as $\varepsilon\to 0$ in \eqref{ApprProblem} we need some a priori estimates. The following result is cited from (\cite{YIN:1}). \begin{lemma} \label{lemma1} There is a constant $C$ independent of $\varepsilon$ such that \begin{gather*} 0\le n^\varepsilon, \ p^\varepsilon\le C, \ \|\varphi^{\varepsilon}\|_{L^\infty(\Omega_T)}\le C, \ \|\varphi^\varepsilon\|_{H^1(\Omega)}\le C, \\ \|n^\varepsilon\|_{L^2(0,T;H^1(\Omega))}+\|p^\varepsilon\|_{L^2(0,T;H^1(\Omega))}\le C. \end{gather*} \end{lemma} With the estimates of this lemma and using the same method in \cite{R_R:1}, we have the following a priori estimate results for the last two equations of \eqref{ApprProblem}. \begin{lemma}\label{lemma2} The following estimate holds \begin{align*} &\sup_{0\le t\le T}\|\theta^{\varepsilon}(t)\|_{L^2}^2+\int_0^T \|\nabla \theta^{\varepsilon}(\tau)\|_{L^2}^2d\tau +\int_0^T\int_{\Gamma_N}h|\theta^{\varepsilon}-\theta_b|^2\,ds\\ &+\sup_{0\le t\le T}\|u^{\varepsilon}(x,t)\|^2_{U_0}+\sup_{0\le t\le T} \|v^{\varepsilon}(x,t)\|_{L^2}^2\,dx \le C, \end{align*} where $C$ is a positive constant that depends on the data but is independent of $m$. \end{lemma} \begin{proof} Using $\theta^\varepsilon-\theta_b$ as a test function in the forth equation of \eqref{ApprProblem}, we have \begin{align*} &\int_{\Omega}\frac{\partial\theta^\varepsilon}{\partial t}(\theta^\varepsilon-\theta_b)dx +\int_{\Omega}k_{ij}(\theta_\varepsilon^\varepsilon)\frac{\partial\theta^\varepsilon}{\partial x_i} \frac{\partial(\theta^\varepsilon-\theta_b)}{\partial x_j}dx +\int_{\Gamma_N}h|\theta^{\varepsilon}-\theta_b|^2\,ds\\ &=-\int_{\Omega}\sigma(\theta_\varepsilon^\varepsilon)\varphi^\varepsilon\nabla\varphi^\varepsilon\nabla (\theta^\varepsilon-\theta_b)dx-\int_{\Omega}\varphi^\varepsilon(n^\varepsilon-p^\varepsilon+f)(\theta^\varepsilon -\theta_b)dx\\ &\quad -\int_{\Omega}m_{ij}\frac{\partial v_i^\varepsilon}{\partial x_j} (\theta^\varepsilon-\theta_b)dx, \end{align*} from which we derive \begin{align*} &\frac{1}{2}\frac{d}{dt}\|\theta^\varepsilon-\theta_b\|_{L^2}^2dx +\int_{\Omega}k_{ij}(\theta_\varepsilon^\varepsilon)\frac{\partial (\theta^\varepsilon-\theta_b)} {\partial x_i}\frac{\partial(\theta^\varepsilon-\theta_b)}{\partial x_j}dx +\int_{\Gamma_N}h|\theta^{\varepsilon}-\theta_b|^2\,ds\\ &=- \int_{\Omega}\frac{\partial \theta_b}{\partial t}(\theta^\varepsilon -\theta_b)dx- \int_{\Omega}k_{ij}(\theta_\varepsilon^\varepsilon) \frac{\partial \theta_b}{\partial x_i}\frac{\partial(\theta^\varepsilon-\theta_b)} {\partial x_j}dx\\ &\quad -\int_{\Omega}\sigma(\theta_\varepsilon^\varepsilon)\varphi^\varepsilon\nabla\varphi^\varepsilon \nabla(\theta^\varepsilon-\theta_b)dx-\int_{\Omega}\varphi^\varepsilon(n^\varepsilon-p^\varepsilon +f)(\theta^\varepsilon-\theta_b)dx\\ &\quad -\int_{\Omega}m_{ij}\frac{\partial v_i^\varepsilon}{\partial x_j}(\theta^\varepsilon -\theta_b)dx\\ &\le C\int_{\Omega}|\theta^\varepsilon-\theta_b|dx+\frac{1}{2}\int_{\Omega}k_{ij} (\theta_\varepsilon^\varepsilon)\frac{\partial \theta_b}{\partial x_i} \frac{\partial \theta_b}{\partial x_j}dx +\frac{1}{2}\int_{\Omega}k_{ij}(\theta_\varepsilon^\varepsilon)\frac{\partial(\theta^\varepsilon -\theta_b)}{\partial x_i}\frac{\partial(\theta^\varepsilon-\theta_b)} {\partial x_j}dx\\ &\quad -\int_{\Omega}\sigma(\theta_\varepsilon^\varepsilon)\varphi^\varepsilon\nabla\varphi^\varepsilon \nabla(\theta-\theta_b)dx-\int_{\Omega}\varphi^\varepsilon(n^\varepsilon-p^\varepsilon+f) (\theta^\varepsilon-\theta_b)dx\\ &\quad -\int_{\Omega}m_{ij}\frac{\partial v_i^\varepsilon}{\partial x_j}(\theta^\varepsilon -\theta_b)dx. \end{align*} With the estimates in Lemma \ref{lemma1} and by the Poincar\'e inequality, we have \begin{align*} &\frac{d}{dt}\|\theta^\varepsilon-\theta_b\|_{L^2}^2+C_1\|\nabla(\theta^\varepsilon -\theta_b)\|^2_{L^2}+\int_{\Gamma_N}h|\theta^{\varepsilon}-\theta_b|^2\,ds\\ &\le C_2-\int_{\Omega}m_{ij}\frac{\partial v_i^\varepsilon}{\partial x_j}(\theta^\varepsilon -\theta_b)dx, \end{align*} Using $v_i$ as a test function in the fifth equation of \eqref{ApprProblem}, we have \[ \frac{1}{2}\rho\frac{d}{dt}\|v\|_{L^2}^2+\int_{\Omega}a_{ijkl} \frac{\partial u_k^\varepsilon}{\partial x_l}\frac{\partial v_i^\varepsilon}{\partial x_j}dx -\int_{\Omega}m_{ij}\theta^\varepsilon \frac{\partial v_i^\varepsilon}{\partial x_j}dx =\int_{\Omega}f_iv_idx \] Noting that $v_i^\varepsilon=\frac{\partial u^\varepsilon_i}{\partial t}$, we have \begin{align*} &\frac{1}{2}\rho\frac{d}{dt}\|v\|_{L^2}^2+\frac{1}{2}\frac{d}{dt} \int_{\Omega}a_{ijkl}\frac{\partial u_k^\varepsilon}{\partial x_l} \frac{\partial u_i^\varepsilon}{\partial x_j}dx\\ &=\frac{1}{2}\int_{\Omega}\frac{\partial a_{ijkl}}{\partial t} \frac{\partial u_k^\varepsilon}{\partial x_l}\frac{\partial u_i^\varepsilon}{\partial x_j}dx +\int_{\Omega}m_{ij}\theta^\varepsilon \frac{\partial v_i^\varepsilon}{\partial x_j}dx +\int_{\Omega}f_iv_idx\\ &\le C+C\|\nabla u^\varepsilon\|_{L^2}^2+\int_{\Omega}m_{ij}\theta^\varepsilon \frac{\partial v_i^\varepsilon}{\partial x_j}dx+C\|v\|^2_{L^2}. \end{align*} Integrating by parts and adding the above two estimates we obtain \begin{align*} &\frac{d}{dt}\Big(\|\theta^\varepsilon-\theta_b\|_{L^2}^2+\rho\|v\|_{L^2}^2 +\int_{\Omega}a_{ijkl}\frac{\partial u_k^\varepsilon}{\partial x_l} \frac{\partial u_i^\varepsilon}{\partial x_j}dx\Big)\\ &+C_1\|\nabla(\theta^\varepsilon-\theta_b)\|^2_{L^2}+\int_{\Gamma_N}h|\theta^{\varepsilon}-\theta_b|^2\,ds\\ &\le C+C\|v^\varepsilon\|_{L^2}^2+C\|\nabla u^\varepsilon\|^2_{L^2} \end{align*} The assumption on $a_{ijkl}$ and the Gronwall inequality derive the desired estimates. \end{proof} To use the energy estimates in Lemmas \ref{lemma1}-\ref{lemma2} to extract convergence subsequences of $\{\varphi^\varepsilon\}, \{n^\varepsilon\}, \{p^\varepsilon\}, \{\theta^{\varepsilon}\}, \{u^\varepsilon\}$ and $\{v^{\varepsilon}\}$, we need the following lemma. \begin{lemma} \label{lemma3} Let $B_0, B, B_1$ be Banach spaces with $B_0\subset B\subset B_1$; assume $B_0\hookrightarrow B$ is compact and $B\hookrightarrow B_1$ is continuous. Let $1