\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 89, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/89\hfil Multi-component Allen-Cahn equation] {Multi-component Allen-Cahn equation for elastically stressed solids} \author[T. Blesgen, U. Weikard\hfil EJDE-2005/89\hfilneg] {Thomas Blesgen, Ulrich Weikard} % in alphabetical order \address{Thomas Blesgen \hfill\break Max Planck Institute for Mathematics in the Sciences\\ Inselstra{\ss}e 22-26, 04103 Leipzig, Germany} \email{blesgen@mis.mpg.de} \address{Ulrich Weikard \hfill\break Faculty of Mathematics, University of Duisburg-Essen\\ Lotharstra{\ss}e 65, 47048 Duisburg, Germany} \email{wkd@math.uni-duisburg.de} \date{} \thanks{Submitted November 18, 2004. Published August 15, 2005.} \thanks{Supported by the German Research Community DFG within priority program 1095} \subjclass[2000]{35K15, 74B10, 74N20} \keywords{Linear elasticity; modeling and simulation of multiscale problems} \begin{abstract} The vector-valued Allen-Cahn equations are combined with elasticity where a linear stress-strain relationship is assumed. A short physical derivation of the generalised model is given and global existence and uniqueness of the solution are shown under suitable growth conditions on the nonlinearity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} The Allen-Cahn equation, introduced in \cite{AC}, provides a well-established framework for the mathematical description to free boundary problems for phase transitions. Unlike sharp interface models, it postulates a diffuse interface with a small thickness $\gamma>0$. The equations have been the subject of intense mathematical investigations, see for instance \cite{Amy,Bates,AJM,Bronsard,S1,S2}; and adequate numerical methods have been developed for their solution, see \cite{Krause,Schmidt}, that contain also references to other numerical work. The Allen-Cahn equation has been generalised in many directions, see \cite{Caginalp,Rio} for a generalisation to the phase field equations; \cite{Garcke2}, where also the vector-valued system of Allen-Cahn equations is derived; \cite{Presutti}, where a statistical framework is considered, and finally \cite{Amy} for a mixed Allen-Cahn/Cahn-Hilliard formulation. The physical applications of the Allen-Cahn system are numerous. An overview over the Allen-Cahn and phase field equations is \cite{Chen}, in \cite{Ga3} an overview over the Cahn-Hilliard equation with elasticity is found. Furthermore we mention \cite{KO} and \cite{KM} with applications to dislocations and lattice instabilities, \cite{Ali}, where droplet motion is described, \cite{Bates} for the study of travelling waves, \cite{Steinbach} for applications to crystallisation, and \cite{Blesgen} for diffusion induced segregation phenomena. In this article we consider a generalisation of the vector-valued system of Allen-Cahn equations to linear elasticity. To this end we will first give a short physical derivation of the complete model, then show existence and uniqueness of a solution to the generalised system. The existence proof can be roughly split into two parts. Part~I, presented in sections \ref{secdiffA} to \ref{secdiffZ}, treats the case of polynomial free energy densities that fulfill the mild growth conditions stated in Section~\ref{secass}. The second part, starting in Section~\ref{seclogA}, treats the physically relevant case of logarithmic free energy densities and makes use of the results shown in Part~I. The employed mathematical methods consist in starting from the time-discrete formulation and finding suitable uniform estimates independent of the time step $\tau>0$ which by well-known compactness results allow to pass to the limit $\tau\to0$. A former version of this argument can be found in \cite{AL} and is classical by now. Our approach will follow closely \cite{Gahabil, Ga1,Ga2} where the elastic Cahn-Hilliard model is treated. \subsection{Derivation of the Model} \label{secder} Let $\Omega\subset{\mathbb R}^D$, $1\le D\le 3$ be a bounded domain with Lipschitz boundary. We introduce the vector $u:=(u_1,\ldots,u_n)$ of non-conserved order parameters. Depending on the physical context, $u_i$ can be either the concentration, or the density, or the volume fraction of the $i$-th phase. These quantities fulfill for $1\le i\le n$ \[ u_i\geq0,\quad u_i\in H^{1,2}(\Omega),\quad\sum_{i=1}^n u_i=1. \] By $H^{m,2}(\Omega)$ we denote the Sobolev space of $m$-times weakly differentiable functions in the Hilbert space $L^2(\Omega)$, by $H^{m,2}_0(\Omega)$ the closure of $C_0^\infty(\Omega)$ w.r.t. $\|\cdot\|_{H^{1,2}(\Omega)}$. By $\|\cdot\|_{H^1}$ we always mean $\|\cdot\|_{H^{1,2}(\Omega)}$. $C_0^\infty(\Omega):=\cap_{m=0}^\infty C_0^m(\Omega)$ where $C_0^m(\Omega)$ is the space of $m$-times continuously differentiable functions over $\Omega$ with compact support. In order to describe elastic effects we consider the displacement field $v(x)$ which describes the position of a material point $x$ in the undeformed body after deformation. We assume that the displacement gradient is small, such that the strain tensor can be approximated by \[ \mathcal{E}=\mathcal{E}(v)=\mathcal{E}_{ij}(v):=\frac{1}{2} \big(\partial_i v_j+\partial_j v_i\big). \] We postulate that the system free energy is of the generalised Landau-Ginzburg form \begin{equation} \label{free} F(u(t),\mathcal{E}(v(t)))=F^\mathrm{out}(\mathcal{E}(v(t))) +\int_\Omega\Big(\frac{\gamma^2}{2}\sum_{i=1}^n|\nabla u_i(x,t)|^2 +f(u(x,t),\mathcal{E}(v(x,t)))\Big)dx. \end{equation} In this formulation, the first term represents energy effects due to applied outer forces, \[ F^\mathrm{out}(\mathcal{E}(v)):=\int_\Omega \overline{W}(\mathcal{E}(v)). \] We assume that there are no external body forces and that the tractions applied to $\partial\Omega$ are dead loads and equal $\overline{S}\vec{n}$, where $\vec{n}$ is the unit outer normal to $\partial\Omega$. We assume that the symmetric tensor $\overline{S}$ defined by this property is constant, i.e. independent of time $t$. The work necessary to transform the undeformed body into the state with corresponding displacement vector $v(t)$ is therefore \[ -\int_{\partial\Omega } v\cdot\overline{S}\vec{n}= -\int_\Omega\nabla v:\overline{S}=-\int_\Omega \mathcal{E}(v):\overline{S} \] and we find that $\overline{W}(\mathcal{E}(v)):=-\mathcal{E}(v):\overline{S}$ describes the energy density of the applied outer forces. The term $\frac{\gamma^2}{2}\sum_{i=1}^n|\nabla u_i|^2$ in (\ref{free}) represents the interfacial energy of the transition layers. Here we assume for simplicity that the contributions enter with the same weight for every interface between any two phases. The last term $f(u,\mathcal{E}(v))$ in (\ref{free}) represents the free energy density. The computations in this article are based on the equality \[ f(u,\mathcal{E}(v))=\bar{f}(u)+W^\mathrm{el}(u,\mathcal{E}(v)) \] with suitable structure and growth conditions on $\bar{f}$, see Section~\ref{secass}. $W^\mathrm{el}$ is the contribution of the elastic energy to $f$. It was first studied by Eshelby, \cite{Esh}. By Hooke's law, a possible ansatz for $W^\mathrm{el}$ is \begin{equation} \label{hook} W^\mathrm{el}(u,\mathcal{E}):=\frac{1}{2}(\mathcal{E} -\overline{\varepsilon}(u)):C(u)(\mathcal{E}-\overline{\varepsilon}(u)). \end{equation} We assume the linear relationship (\emph{Vegard's law}) \begin{equation} \label{Vegard} \overline{\varepsilon}(u):=\sum_{i=1}^n u_i \overline{\varepsilon}_i, \end{equation} where $\overline{\varepsilon}_i:=\overline{\varepsilon}(e_i)$ and $e_i$ is the $i$-th basis vector of ${\mathbb R}^n$. This means $\overline{\varepsilon}(e_i)$ is the eigenstrain when the system is equal to the $i$th pure component. $C(u)$ is the elasticity tensor that maps symmetric tensors in ${\mathbb R}^{D\times D}$ onto itself. We assume that $C$ is symmetric and positive definite. Instead of (\ref{hook}) other forms of $W^\mathrm{el}$ are permitted as long as Assumption~(A4) in Section~\ref{secass} remains valid. We define the time evolution of the unconserved order parameter $u$ as gradient flow of the free energy, \[ \int_\Omega\partial_t u=-\frac{\delta}{\delta u} F(u(t),\mathcal{E}(v(t))). \] Thus for large time $t$, $u(t)$ tends to a local minimiser of $F$. The mechanical equilibrium is attained on a much faster time scale than the time scale significant for diffusion. Therefore we will assume a quasi-static elastic equilibrium, i.e. the displacement $v$ is obtained by solving the elliptic equation \[ \mathop{\rm div}(S)=0\quad\mbox{in }\Omega \] with the stress tensor \[ S:=\partial_{\varepsilon}W^\mathrm{el}(u,\mathcal{E}(v)). \] Hence, for a given stop time $T>0$ we end up with the following model: {\it Find for $t\ge0$ a solution pair $(u,v)$ such that in $\Omega_T:=\Omega\times(0,T)$} \begin{gather} \label{reform1} \partial_t u = \gamma^2\triangle u-P(\partial_u f(u,\mathcal{E}(v))),\\ \label{reform2} \mathop{\rm div}(S) = 0,\\ \label{reform3} S = \partial_{\varepsilon}W^\mathrm{el}(u,\mathcal{E}(v)), \end{gather} with the initial data for $t=0$ in $\Omega$ \begin{equation} \label{reform4} u(\cdot,0) = u_{0}(\cdot) \end{equation} and the boundary conditions for $t>0$ in $\partial\Omega$ \begin{equation} \label{reform5} u=u_d,\quad S\cdot\vec{n}=\overline{S}\cdot\vec{n}. \end{equation} The projection operator $P$ in (\ref{reform1}) is due to algebraic constraints on $\partial_u f(u,\mathcal{E}(v))$. This is explained in the subsequent section. The boundary condition $S\cdot\vec{n}=\overline{S}\cdot\vec{n}$ on $\partial\Omega$ determines $v$ only up to infinitesimal rigid displacements (these are translations and infinitesimal rotations). This fact is well-known for formulations that depend on a linearised strain tensor $\mathcal{E}$. The resulting non-uniqueness in $v$ is of no importance as $v$ only enters through the symmetric term $\mathcal{E}(v)$. \section{Preliminaries to existence theory}\label{secdiffA} In this section we discuss the existence theory to the sharp interface model (\ref{reform1})-(\ref{reform5}). We will show that under suitable growth conditions on the free energy density, stated for polynomial $f$ in Section~\ref{secass} and for logarithmic energies in Section~\ref{seclogA}, discrete solutions to the implicit time discretisation exist. A-priori estimates allow to pass to the limit showing the existence of solutions to the model first with polynomial free energy. This result is then used to generalise to logarithmic free energies. We will carry out the proof for classical Dirichlet boundary data, i.e. set w.l.o.g. $u_d=0$ in (\ref{reform5}). Other boundary conditions are shortly discussed in the remark at the end of this section. We begin by collecting general properties of the model and necessary tools that will be needed in the sequel. The vector of order parameters lies inside the simplex $\Sigma$, \begin{equation} \label{sigma} u\in\Sigma:=\big\{u'=(u_1',\ldots,u_n')\in{\mathbb R}^n : \sum_{i=1}^n u_i'=1\big\}. \end{equation} Notice that the condition $0\le u_i\le1$ in $\Omega$ may be violated for polynomial free energies considered in the first part of this section. If we write (\ref{reform1}) as $\partial_t u=w$, as a consequence of (\ref{sigma}), $w$ fulfills $\sum_{i=1}^n w_i=0$. Thus, with $e:=(1,\ldots,1)\in{\mathbb R}^n$, the right hand side $w$ satisfies $w=P(z)$ for some $z\in{\mathbb R}^n$, where \[ P(z):=z-\frac{1}{n}(z\cdot e)e \] is the projection of ${\mathbb R}^n$ to \[ T\Sigma:=\Big\{ u'=(u'_1,\ldots,u'_n)\in{\mathbb R}^n : \sum_{i=1}^n u'_i=0\Big\}, \] the tangent space to $\Sigma$. Let \begin{gather*} X_1 := \{u'\in H^1_0(\Omega;{\mathbb R}^n): u'\in\Sigma \mbox{ almost everywhere in }\Omega\},\\ X_2 := \{v'\in H^1(\Omega,{\mathbb R}^D):(v',w)_{H^1}=0\mbox{ for all } w\in X_{\rm ird}\}, \end{gather*} where \[ X_{\rm ird}=\{v\in H^1(\Omega,{\mathbb R}^D): \mbox{there exist } b\in {\mathbb R}^D,\; A\in{\mathbb R}^{D\times D}\mbox{ such that }v(x)=Ax+b\} \] is the space of all infinitesimal rigid displacements. Since we have (classical) Dirichlet boundary conditions for the equations of conservation of mass, we consider the space of test functions \[ Y:=H_0^{1,2}(\Omega;{\mathbb R}^n) \] and its dual \[ (H^{1,2}_0(\Omega;{\mathbb R}^n))'=H^{-1,2}(\Omega;{\mathbb R}^n). \] \noindent\bf Remark: \rm If we replace the Dirichlet conditions for $u$ by a Neumann boundary condition or periodic boundary conditions, a (generalised) Poincar{\'e} inequality holds in $H^{1,2}(\Omega)$ and all the results found below continue to hold. \subsection{The weak formulation} \label{secweak} A pair $(u,v)\in L^2(0,T;\,H_0^{1,2}(\Omega;{\mathbb R}^n))\times L^2(0,T;\,X_2)$ is called a {\it weak solution of} (\ref{reform1})-(\ref{reform5}) if \begin{equation} \label{weaka} -\int_{\Omega_T}\partial_t\xi\cdot(u-u_0)+\gamma^2\! \int_{\Omega_T}\nabla u:\nabla\xi +\int_{\Omega_T} P(\partial_u f(u,\mathcal{E}(v)))\cdot\xi=0 \end{equation} for all $\xi\in L^2(0,T;\,H_0^1(\Omega;{\mathbb R}^n))\cap L^\infty(\Omega_T;\,{\mathbb R}^n)$ with $\partial_t\xi\in L^2(\Omega_T)$, $\xi(T)=0$, and \begin{equation} \label{weakb} \int_{\Omega_T} W^\mathrm{el}(u,\mathcal{E}(v)):\nabla\zeta=\int_{\Omega_T} \overline{S}:\nabla\zeta. \end{equation} for all $\zeta\in L^2(0,T;\;H^1(\Omega,{\mathbb R}^D))$. \subsection{The implicit time discretisation} We fix an $M\in{\mathbb N}$ and set $h:=\frac{T}{M}$. For $m\ge1$ and given $u^{m-1}\in X_1$ consider \begin{gather} \label{impl1} \frac{u^m-u^{m-1}}{h} =\gamma^2\triangle u-P( \partial_u f(u^m,\mathcal{E}(v^m))),\\ \label{impl2} \mathop{\rm div}(S^m) = 0,\\ \label{impl3} S^m = \partial_{\varepsilon} W^\mathrm{el}(u^m,\mathcal{E}(v^m)). \end{gather} \subsection{Structural Assumptions} \label{secass} To establish the existence of weak solutions in the sense of Section~\ref{secweak}, the following assumptions are made: \begin{itemize} \item[(A1)] $\Omega\subset{\mathbb R}^D$ is a bounded domain with Lipschitz boundary. \item[(A2)] The free energy density $f$ can be written as \[ f(u',\mathcal{E}(v'))=f^1(u')+f^2(u')+W^\mathrm{el}(u',\mathcal{E}(v')) \quad\mbox{for all }u'\in{\mathbb R}^n,\,v'\in{\mathbb R}^D \] with $f^1\in C^1({\mathbb R}^n;{\mathbb R})$ and convex. Additionally we postulate \item[(A2.1)] $f^1\ge0$. \item[(A2.2)] For all $\delta>0$ there exists a constant $C_\delta>0$ such that \[ |\partial_u f^1(u')|\le\delta f^1(u')+C_\delta\quad\mbox{for all }u'\in \Sigma. \] \item[(A2.3)] There exists a constant $C_1>0$ such that \[ |\partial_u f^2(u')|\le C_1(|u'|+1)\quad\mbox{for all }u'\in\Sigma. \] \item[(A3)] The initial datum $u_0$ fulfills $ f(u_0,\mathcal{E}(v_0))<\infty$, where $v_0$ is the solution of (\ref{weakb}). \item[(A4)] The elastic energy density $W^\mathrm{el}\in C^1({\mathbb R}^n\times{\mathbb R}^{D\times D};{\mathbb R})$ satisfies \noindent(A4.1) $W^\mathrm{el}(u',\mathcal{E}')$ only depends on the symmetric part of $\mathcal{E}'\in{\mathbb R}^{D\times D}$, i.e. \[ W^\mathrm{el}(u',\mathcal{E}')=W^\mathrm{el}(u',(\mathcal{E}')^t) \quad\mbox{for all }u'\in{\mathbb R}^n \mbox{ and }\mathcal{E}'\in {\mathbb R}^{D\times D}. \] \item[(A4.2)] $\partial_{\varepsilon}W^\mathrm{el}(u',\cdot)$ is strongly monotone uniformly in $u'$, i.e. there exists a $c_1>0$ such that for all symmetric $\mathcal{E}_1',\,\mathcal{E}_2'\in{\mathbb R}^{D\times D}$, \[ (\partial_{\varepsilon}W^\mathrm{el}(u',\mathcal{E}_2')- \partial_{\varepsilon}W^\mathrm{el}(u',\mathcal{E}_1')):(\mathcal{E}_2' -\mathcal{E}_1')\ge c_1|\mathcal{E}_2'-\mathcal{E}_1'|^2. \] \item[(A4.3)] There exists a constant $C_1>0$ such that for all $u'\in\Sigma$ and all symmetric $\mathcal{E}'\in{\mathbb R}^{D\times D}$, \begin{gather*} |W^\mathrm{el}(u',\mathcal{E}')| \le C_1(|\mathcal{E}'|^2+|u'|^2+1),\\ |\partial_u W^\mathrm{el}(u',\mathcal{E}')| \le C_1(|\mathcal{E}'|^2+|u'|^2+1),\\ |\partial_{\varepsilon}W^\mathrm{el}(u',\mathcal{E}')| \le C_1(|\mathcal{E}'|+|u'|+1). \end{gather*} \item[(A5)] The energy density of the applied outer forces is given by $\overline{W}(\mathcal{E}')=-\mathcal{E}':\overline{S}$ where $\overline{S}$ is a symmetric constant tensor. \end{itemize} For the rest of this article, we assume without further stating that the assumptions (A1)-(A5) hold. \section{Existence of solutions to the time discrete scheme} For each time step $m\ge1$ in the implicit time discretisation (\ref{impl1})-(\ref{impl3}), given time step size $h>0$, and given $u^{m-1}\in X_1$ we define the discrete energy functional \[ F^{m,h}(u',v'):=F(u',\mathcal{E}(v'))+\frac{1}{2h}\|u'-u^{m-1}\|_{L^2}^2. \] \begin{lemma}[Existence of a minimiser] \label{lem1} For given $u^{m-1}\in X_1$ and any $h>0$ the functional $F^{m,h}$ possesses a minimiser $(u^m,v^m)$ in $X_1\times X_2$. \end{lemma} \begin{proof} The proof is an application of the direct method in the calculus of variations. Combined, (A4.2), (A4.3) imply that $W^\mathrm{el}(u',\mathcal{E}')\ge C(|\mathcal{E}'|^2-|u'|^2)-C$ for a constant $C>0$. With Korn's inequality, see for instance \cite{Ciarlet}, this guarantees the coercivity of $F$ with respect to $v\in X_2$. Similarly, the term $\gamma^2\int_\Omega\sum_{i=1}^n|\nabla u_i|^2$ in the definition of $F$ guarantees with the Poincar{\'e} inequality the coercivity of $F$ w.r.t. $u\in X_1$. Using (A2) on $f^1$ and $f^2$ we thus find that the functional $F^{m,h}$ is weakly lower semicontinuous and coercive in $X_1\times X_2$ and hence possesses a minimiser. \end{proof} The following lemma shows that the energy functional $F^{m,h}$ is the correct one and corresponds to the implicit time discretisation (\ref{impl1})-(\ref{impl3}). \begin{lemma}[Euler-Lagrange equations] \label{lem2} The minimiser $(u^m,v^m)\in X_1\times X_2$ of $F^{m,h}$ fulfills \begin{gather} \label{mini1} \begin{aligned} \int_\Omega\frac{u^m-u^{m-1}}{h}\cdot\xi +\int_\Omega\gamma^2\nabla u^m:\nabla\xi+\int_\Omega P(\partial_u f(u^m,\mathcal{E}(v^m)))\cdot\xi=0&\\ \mbox{for all }\xi\in Y\cap L^\infty(\Omega;{\mathbb R}^n),& \end{aligned} \\ \label{mini2} \int_\Omega\partial_{\varepsilon} W^\mathrm{el}(u^m,\mathcal{E}(v^m)):\nabla\zeta=\int_\Omega \overline{S}:\nabla\zeta\quad\mbox{for all }\zeta\in H^1(\Omega,{\mathbb R}^D). \end{gather} \end{lemma} \begin{proof} We choose directions $\xi\in Y\cap L^\infty(\Omega;{\mathbb R}^n)$ with $\sum_{i=1}^n\xi_i=0$, $\zeta\in X_2\cap L^\infty(\Omega;{\mathbb R}^D)$ and determine variations of $F^{m,h}(u,v)$ with respect to $u$ and $v$ for $\xi,\,\zeta$. The variation w.r.t. $u$ is \begin{equation} \label{2zero} \lim_{s\to0}\Big((F^{m,h}(u^m+s\xi,v^m)-F^{m,h}(u^m,v^m))s^{-1}\Big). \end{equation} Since $f^1$ is convex, we have \[ f^1(u^m)\ge f^1(u^m+s\xi)-s\partial_u f^1(u^m+s\zeta)\cdot\xi. \] This implies \begin{align*} f^1(u^m+s\xi) &\le f^1(u^m)+|s\partial_u f^1(u^m+s\xi)|\,\|\xi\|_{L^\infty}\\ &\le f^1(u^m)+|s|\,f^1(u^m+s\xi)\,\|\xi\|_{L^\infty}+C|s|. \end{align*} The last is by Assumption~(A2.2) with $\delta=1$. Hence, for s small enough, we find \[ \Big|\frac{f^1(u^m+s\xi)-f^1(u^m)}{s}\Big|\le C(f^1(u^m)+1). \] Lebesgue's dominated convergence theorem and Assumption~(A2.3) imply \[ \lim_{s\to0}\frac{1}{s}\Big(\int_\Omega(f^1+f^2)(u^m+s\xi) -(f^1+f^2)(u^m)\Big)=\int_\Omega(\partial_u f^1+\partial_u f^2)(u^m) \cdot\xi. \] With the help of (A4.3) we find \begin{align*} & \lim_{s\to0}\int_\Omega s^{-1}\Big(W^\mathrm{el}(u^m+s\xi, \mathcal{E}(v^m+s\zeta))-W^\mathrm{el}(u^m,\mathcal{E}(v^m))\Big)\\ & =\int_\Omega\Big(\partial_u W^\mathrm{el}(u^m, \mathcal{E}(v^m))\xi +\partial_{\varepsilon}W^\mathrm{el}(u^m,\mathcal{E}(v^m)):\nabla\zeta\Big). \end{align*} The variation of the quadratic form $u\mapsto\frac{1}{2h} \|u^m-u^{m-1}\|_{L^2}^2$ yields \[ \lim_{s\to0}\Big(s^{-1}(2h)^{-1}\,\big(\|u^m+s\xi-u^{m-1}\|_{L^2}^2 -\|u^m-u^{m-1}\|_{L^2}^2\big)\Big) =\big(\frac{u^m-u^{m-1}}{h},\xi\big)_{L^2}. \] Taking into account that $u^m-u^{m-1}$ as well as $\nabla u^m:\nabla\xi$ for every $\xi$ lie on $T\Sigma$, this finally yields (\ref{mini1}). To derive (\ref{mini2}) we vary $F^{m,h}$ with respect to $v$. From the symmetry of $\partial_{\varepsilon}W^\mathrm{el}$ and $\overline{S}$ we find (\ref{mini2}). \end{proof} \subsection{Uniform estimates} In the preceding section we proved the existence of a discrete solution $(u^m,v^m)$ for $1\le m\le M$ and arbitrary $M\in{\mathbb N}$. We define the piecewise constant extension $(u_M,v_M)$ of $(u^m,v^m)_{1\le m\le M}$ by \[ (u_M(t),v_M(t)):=(u^m_M,v^m_M):=(u^m,v^m)\quad\mbox{for } t\in((m-1)h,mh] \] with $u_M(0)=u_0$, and $v_M(0)$ given by Equation~(\ref{impl3}). The piecewise linear extension $(\overline{u}_M,\overline{v}_M)$ for $t=(\beta m+(1-\beta)(m-1))h$ with appropriate $\beta\in[0,1]$ is given by the interpolation \[ (\overline{u}_M,\overline{v}_M)(t):=\beta(u^m_M,v^m_M) +(1-\beta)(u^{m-1}_M,v^{m-1}_M). \] \begin{lemma}[A-priori estimates] \label{lem3} The following a-priori estimates are valid. \begin{itemize} \item[(a)] For all $M\in{\mathbb N}$ and all $t\in[0,T]$ we have the dissipation inequality \[ F(u_M,\mathcal{E}(v_M))(t)+\frac{1}{2}\int_{\Omega_t} |\partial_t\overline{u}_M|^2\le F(u_0,\mathcal{E}(v_0)). \] \item[(b)] There exists a constant $C>0$ such that \begin{gather} \label{apri1} \sup_{0\le t\le T}\Big\{\|u_M(t)\|_{H^1}+\|v_M(t)\|_{H^1}\Big\} \le C,\\ \label{apri2} \sup_{0\le t\le T}\int_\Omega f^1(u_M(t)) +\|\partial_t\overline{u}_M\|_{L^2(\Omega_T)}\le C. \end{gather} \end{itemize} \end{lemma} \begin{proof} Since $(u^m,v^m)$ is a minimiser of $F^{m,h}$, it holds for every $m\ge1$ \begin{equation} \label{start1} F(u^m,\mathcal{E}(v^m))+\frac{1}{2h}\|u^m-u^{m-1}\|_{L^2}^2 \le F(u^{m-1},\mathcal{E}(v^{m-1})). \end{equation} After writing $u^m-u^{m-1}$ as a time derivative, iterating (\ref{start1}) yields \[ F(u^m_M,\mathcal{E}(v^m_M))+\frac{1}{2}\int_0^{mh} \|\partial_t\overline{u}_M\|_{L^2}^2\;d\tau\le F(u_0,\mathcal{E}(v_0)). \] Using the assumptions (A2)-(A4) and with the help of the inequalities of Poincar{\'e} and Korn, this proves the lemma. \end{proof} For the linear interpolation $\overline{u}_M$ of $u^m_M$, the Euler-Lagrange equation (\ref{mini1}) can be rewritten as \begin{equation} \label{inter1} \int_\Omega\partial_t\overline{u}_M(t)\cdot\xi+\int_\Omega \gamma^2\nabla u_M(t):\nabla\xi+\int_\Omega P(\partial_u f(u_M(t),\mathcal{E}(v_M(t))))\cdot\xi=0 \end{equation} for all $\xi\in Y\cap L^\infty(\Omega;{\mathbb R}^n)$, which holds for almost all $t\in(0,T)$. Equation~(\ref{inter1}) controls the variation of $\overline{u}_M$ in time and, together with the uniform estimates of Lemma~\ref{lem3}, allows to show compactness in time. The following theorem is the first main result as it can also be used to proof convergence of numerical solution schemes. In the next part we will show that this limit is in fact a solution to (\ref{reform1})-(\ref{reform5}). \begin{theorem}[Compactness of $(u_M,v_M)$] \label{theo1} There exists a constant $C>0$ such that for all $t_1,t_2\in[0,T]$ \[ \|\overline{u}_M(t_2)-\overline{u}_M(t_1)\|_{L^2}\le C|t_2-t_1|^{1/4}. \] Furthermore, there are subsequences $(u_M)_{M\in\mathcal{N}}$ and $(v_M)_{M\in\mathcal{N}}$ with $\mathcal{N}\subset{\mathbb N}$ and there are $u\in L^\infty(0,T;\,H^1_0(\Omega))$ and $v\in L^\infty(0,T;\,H^1(\Omega))$ such that \begin{tabbing} \quad\=(iii)\quad\=$\partial_u f^k(u_M)$ \=$\to\;$ \=$\partial_u f^k$\qquad\=\kill \>(i)\>$\overline{u}_M$ \>$\to$ \>$u$ \>in $C^{0,\alpha}([0,T];\,L^2(\Omega;{\mathbb R}^n))$ for all $\alpha\in(0,\frac{1}{4})$,\\ \>(ii)\>$u_M$ \>$\to$ \>$u$ \>in $L^\infty(0,T;\,L^2(\Omega;{\mathbb R}^n))$, \\ \>(iii)\>$u_M$ \>$\to$ \>$u$ \>almost everywhere in $\Omega_T$,\\ \>(iv)\>$u_M$\>$\stackrel{*}{\rightharpoonup}$ \>$u$ \>in $L^\infty(0,T;\,H^1_0(\Omega;{\mathbb R}^n))$,\\ \>(v) \>$v_M$ \>$\to$ \>$v$ \>in $L^2(0,T;\,H^1(\Omega))$,\\ \>(vi) \>$\partial_u f^k(u_M)$ \>$\to$ \>$\partial_u f^k(u)$ \>in $L^1(\Omega_T)$ for $k=1,2$ \end{tabbing} as $M\in\mathcal{N}$ tends to infinity. \end{theorem} \begin{proof} For chosen constant $L>0$ let \begin{equation} \label{PLdef} P_L(u'):= \begin{cases} u' & \mbox{if }|u'|\le L,\\ \frac{u'}{|u'|}L & \mbox{if }|u'|>L. \end{cases} \end{equation} In (\ref{inter1}) we test with $\xi:=P_L(\overline{u}_M(t_2)-\overline{u}_M(t_1))$, where $t_1,t_2\in[0,T]$ with $t_10$ and all measurable $E\subset\Omega$ \[ \int_E|\partial_u f^1(u_M)|\le\delta\int_E f^1(u_M)+C_\delta|E| \le\delta C+C_\delta|E|. \] Therefore, $\int_E|\partial_u f^1(u_M)|\to 0$ as $|E|\to 0$ uniformly in $M$ and by Vitali's theorem we find $\partial_u f^1(u_M)\to\partial_u f^1(u)$ in $L^1(\Omega_T)$ as $M\in\mathcal{N}$ tends to infinity. Assumption~(A2.3) yields with Lebesgue's dominated convergence theorem accordingly \[ \partial_u f^2(u_M)\to\partial_u f^2(u). \] \end{proof} \section{Global existence of solutions I}\label{secdiffZ} \begin{theorem}[Global existence of solutions for polynomial free energy] \label{theo2} Let the assumptions of Section~\ref{secass} hold. Then there exists a weak solution $(u,v)$ of (\ref{reform1})-(\ref{reform5}) in the sense of Section~\ref{secweak} such that \begin{itemize} \item[(i)] $u\in C^{0,\frac{1}{4}}([0,T];\,L^2(\Omega;{\mathbb R}^n))$, \item[(ii)] $\partial_t u\in L^2(\Omega_T;\;{\mathbb R}^n)$, \item[(iii)] $v\in L^2(0,T;\,H^1(\Omega))$. \end{itemize} \end{theorem} \begin{proof} We are going to prove that $(u,v)$ introduced in Theorem~\ref{theo1} is the desired weak solution in the sense of Section~\ref{secweak}. From Equation~(\ref{inter1}) we learn \[ -\int_{\Omega_T}\partial_t\xi\cdot(\overline{u}_M-u_0) +\int_{\Omega_T}\gamma^2\nabla u_M:\nabla\xi +\int_{\Omega_T} P(\partial_u f(u_M,\mathcal{E}(v_M)))\cdot\xi=0 \] for all $\xi\in L^2(0,T;\,Y)$ with $\partial_t\xi\in L^2(\Omega_T)$ and $\xi(T)=0$. In this equation we pass to the limit $M\to\infty$ and exploit Theorem~\ref{theo1}. The convergence of the linear expressions is clear. The convergence \[ \int_{\Omega_T}\partial_u f(u_M,\mathcal{E}(v_M))\cdot\xi\to \int_{\Omega_T}\partial_u f(u,\mathcal{E}(v))\cdot\xi \] follows similar to the proof of Theorem~\ref{theo1} with Vitali's theorem by using the growth condition (A2.2) on $f^1$, (A2.3), Estimate~(\ref{apri2}), the almost everywhere convergence of $u_M$ and the boundedness of $\xi$. The generalised Lebesgue convergence theorem, the growth condition (A4.3), and the strong convergence of $\nabla v_M$ and $u_M$ in $L^2(\Omega)$ yield that we can pass to the limit in $\int_\Omega\partial_u W^\mathrm{el}(u_M,\mathcal{E}(v_M))\cdot\xi$. This implies (\ref{weaka}). Similarly we can pass to the limit in (\ref{mini2}) and obtain (\ref{weakb}). This is done in the same way as before by using once more growth condition (A4.3) and the strong convergence of $\nabla v_M$ and $u_M$ in $L^2(\Omega)$. \end{proof} \section{Uniqueness of the solution}\label{secuni} We show uniqueness of a solution to (\ref{reform1})-(\ref{reform3}) under the simplifying assumption that \begin{equation} \label{sim} W^\mathrm{el}(u',\mathcal{E}')=\frac{1}{2}(\mathcal{E}' -\overline{\varepsilon}(u')):C(\mathcal{E}'-\overline{\varepsilon}(u')), \end{equation} with a symmetric constant positive definite tensor $C$ and with $\overline{\varepsilon}(u')$ defined by (\ref{Vegard}). The proof of the following theorem is straightforward and uses an integration in time method and a Gronwall argument. \begin{theorem}[Uniqueness of solutions to the elastic Allen-Cahn system] \label{theo3} Let $W^\mathrm{el}$ be given by (\ref{sim}). Then the solution pair $(u,v)$ obtained in Theorem~\ref{theo2} is unique in the spaces stated in this theorem. \end{theorem} \begin{proof} If there are two pairs of solutions $(u^1,v^1)$, $(u^2,v^2)$ to the equations (\ref{reform1})-(\ref{reform3}), it holds for $k=1,2$ \begin{equation} \begin{gathered} \partial_t u^k = \gamma^2\triangle u^k-P(\partial_u f^1(u^k)+\partial_u f^2(u^k))-P((\overline{\varepsilon}_i:C(\overline{\varepsilon}(v^k) -\overline{\varepsilon}(u^k)))_{1\le i\le n}),\\ 0 =\mathop{\rm div}(C(\mathcal{E}(v^k)-\overline{\varepsilon}(u^k))). \end{gathered} \label{fu2} \end{equation} Let $u:=u^2-u^1$ and $v:=v^2-v^1$. Then $(u,v)$ solves the weak equation \begin{equation} \begin{aligned} \int_{\Omega_T}\partial_t u\cdot\xi &= -\int_{\Omega_T} \gamma^2\nabla u:\nabla\xi-\int_{\Omega_T}( \partial_u(f^1+f^2)(u^2)-\partial_u(f^1+f^2)(u^1))\cdot P\xi\\ &\quad -\int_{\Omega_T}(\overline{\varepsilon}_i:C(\mathcal{E}(v) -\overline{\varepsilon}(u)))_{1\le i\le n}\cdot P\xi \end{aligned}\label{uu1} \end{equation} for every $\xi\in L^2(0,T;\;Y)\cap L^\infty(\Omega_T;\,{\mathbb R}^n)$ with $\partial_t\xi\in L^2(\Omega_T)$ and $\xi(T)=0$. Let $t_0\in(0,T)$. We choose $P_L(u^2-u^1)\mathcal{X}_{(0,t_0)}$ as a test function in the difference of the weak formulations of (\ref{fu2}), where $L>0$ and $P_L(u)$ is defined as in (\ref{PLdef}). In the limit $L\to\infty$ the terms with $P_L(u)$ are replaced by $u$ and we find \begin{equation} \label{uu2} \int_{\Omega_{t_0}} C(\mathcal{E}(v) -\overline{\varepsilon}(u)):\mathcal{E}(v)=0. \end{equation} Similarly we choose $\xi:=P_L(u^2-u^1)\mathcal{X}_{(0,t_0)}$ as test function in (\ref{uu1}) and in the limit $L\to\infty$ we obtain with the help of (\ref{uu2}) \begin{align*} \frac{1}{2}\int_{\Omega_{t_0}}\frac{d}{dt}|u|^2 &= -\int_{\Omega_{t_0}}\gamma^2\nabla u:\nabla u -\int_{\Omega_{t_0}}(\mathcal{E}(v)-\overline{\varepsilon}(u)): C(\mathcal{E}(v)-\overline{\varepsilon}(u))\\ &\quad -\int_{\Omega_{t_0}} (\partial_u(f^1+f^2)(u^2)-\partial_u(f^1+f^2)(u^1))\cdot(u^2-u^1). \end{align*} The convexity of $f^1$ yields \[ \partial_u(f^1(u^2)-f^1(u^1))\cdot(u^2-u^1)\ge0 \] and due to $u(t=0)=0$ we end up with \[ \frac{1}{2}\int_{\Omega_{t_0}}\frac{d}{dt}|u|^2=\frac{1}{2} \|u(t_0)\|_{L^2}^2\le\int_\Omega(\partial_u f^2(u^2) -\partial_u f^2(u^1))\cdot u. \] With Gronwall's inequality, as $f^2$ is Lipschitz continuous, and since $t_0$ was arbitrary, we find $u\equiv0$ in $\Omega_T$ which leads to \[ \int_{\Omega_T}\mathcal{E}(v):C\mathcal{E}(v)=0. \] With Korn's inequality this yields $v\equiv0$ in the whole of $\Omega_T$. \end{proof} \section{Logarithmic free energy} \label{seclogA} In the upcoming three sections we are going to extend Theorem~\ref{theo2} to logarithmic free energies. The results will in particular be valid for the free energy functional, \begin{equation} \label{3log} f(u',\mathcal{E}(v'))=k_B\theta\sum_{j=1}^n u_j' \ln u_j'+\frac{1}{2}u'\cdot Au'+W^\mathrm{el}(u',\mathcal{E}(v')) \end{equation} where $\theta$ denotes the (fixed) temperature and $k_B$ the Boltzmann constant. We will exploit this particular structure of $f$ in the sequel. As is well known the mathematical discussion is much more subtle, $f$ becomes singular as one $u_j$ approaches $0$. To show that $00$ by some $f^\delta$ that fulfills the requirements of Section~\ref{secass} and find suitable a-priori estimates that allow to pass to the limit $\delta\to0$. Despite of the mathematical difficulties, the logarithmic free energy guarantees that the vector $u$ of order parameters lies in the transformed Gibbs simplex \[ G:=\Sigma\cap\big\{u'\in{\mathbb R}^n : u_j'\ge0\mbox{ for } 1\le j\le n\mbox{ and }\sum_{i=1}^nu_i'=1\big\} \] and is therefore physically meaningful. The assumptions (A2) and (A3) of Section~\ref{secass} are replaced by the following assumptions: \begin{itemize} \item[(A2')] $f$ is of the form (\ref{3log}), where $A\in{\mathbb R}^{n\times n}$ is a symmetric positive definite matrix and $\theta>0$ the constant temperature. \item[(A3')] The initial value $u_0=(u_{01},\ldots,u_{0n})\in X_1$ fulfills $u_0\in G$ almost everywhere and \[ \int_\Omega u_{0j}>0\quad\mbox{for }1\le j\le n. \] \end{itemize} The other assumptions are unchanged and continue to hold. To proceed, we define for $d\in{\mathbb R}$ and given $\delta>0$ the regularised free energy functional \[ \psi^\delta(d):=\begin{cases} d\ln d & \mbox{for }d\ge\delta,\\ d\ln\delta-\frac{\delta}{2}+\frac{d^2}{2\delta} & \mbox{for }d<\delta. \end{cases} \] The regularised free energy functional is defined in such a way that $\psi^\delta\in C^2$ and the derivative $(\psi^\delta)'$ is monotone. This definition goes back to the work \cite{EL} by Elliott and Luckhaus. Due to Assumption~(A2'), this leads to \begin{gather} \label{logd} f^\delta(u,\mathcal{E}(v')) = f^{1,\delta}(u)+f^2(u) +W^\mathrm{el}(u',\mathcal{E}(v')),\\ \label{logd1} f^{1,\delta}(u') := k_B\theta\sum_{j=1}^n\psi^\delta(u_j'),\\ \label{logd2} f^2(u') := \frac{1}{2}\, u'\cdot Au'. \end{gather} As can be easily checked, $f^{1,\delta},f^2$ fulfill the assumptions of Section~\ref{secass}. \subsection{Uniform estimates} The following lemma was first stated and proved in \cite{EL} for logarithmic free energies typical for the Cahn-Hilliard system. The proof of Elliott and Luckhaus can be directly transferred to the situation considered here with the regularised free energy defined by (\ref{3log}). \begin{lemma}[Uniform bound from below on $f^\delta$]\label{lem4} There exists a $\delta_0>0$ and a $K>0$ such that for all $\delta\in(0,\delta_0)$ \[ f^{1,\delta}(u)+f^2(u)\ge-K\quad\mbox{for all }u\in\Sigma. \] \end{lemma} Now we summarise the results for the regularised problem proved in Lemma~\ref{lem3} and Theorem~\ref{theo1}. Lemma~\ref{lem5} also states the boundedness and convergence of the numerical solutions as $\delta\searrow0$. \begin{lemma}[A-priori and compactness results for regularised problem] \label{lem5} \quad\newline (a) For all $\delta\in(0,\delta_0)$ there exists a weak solution $(u^\delta,v^\delta)$ of (\ref{reform1})-(\ref{reform5}) with a loga\-rithmic free energy that satisfies (A2'), (A3'), (A4)-(A6) in the sense of Section~\ref{secweak}. \noindent(b) There exists a constant $C>0$ independent of $\;\delta$ such that for all $\delta\in(0,\delta_0)$ \begin{gather*} \sup_{t\in[0,T]}\big\{\|u^\delta(t)\|_{H^1}+\|v^\delta(t)\|_{H^1}\big\} \le C,\\ \sup_{t\in[0,T]}\int_\Omega f^{1,\delta}(u^\delta(t))+ \|\partial_t u^\delta\|_{L^2(\Omega_T)} \le C, \\ \|u^\delta(t_2)-u^\delta(t_1)\|_{L^2}\le C|t_2-t_1|^{1/4} \quad \mbox{for all }t_1,t_2\in[0,T]. \end{gather*} \noindent(c) One can extract a subsequence $(u^\delta)_{\delta\in\mathcal{R}}$, where $\mathcal{R}\subset(0,\delta_0)$ is a countable set with zero as the only accumulation point such that \begin{itemize} \item[(i)] $u^\delta \to u$ in $C^{0,\alpha}([0,T];\, L^2(\Omega;{\mathbb R}^n))$ for all $\alpha\in(0,\frac{1}{4})$, \item[(ii)] $u^\delta \to u$ in $L^\infty(0,T;\,L^2(\Omega;{\mathbb R}^n))$, \item[(iii)] $u^\delta \to u$ almost everywhere in $\Omega_T$, \item[(iv)] $u^\delta \stackrel{*}{\rightharpoonup} u$ in $L^\infty(0,T;\,H^1_0(\Omega;{\mathbb R}^n))$, \item[(v)] $v^\delta \to v$ in $L^2(0,T;\,H^1(\Omega))$ \end{itemize} as $\delta\in\mathcal{R}$ tends to zero. \end{lemma} \begin{proof} Using Lemma~\ref{lem4}, the regularised problem satisfies the assumptions of Section~\ref{secass} and by Theorem~\ref{theo2}, a weak solution for fixed $\delta\in(0,\delta_0)$ exists. This proves (a). Lemma~\ref{lem3} and Theorem~\ref{theo1} imply directly (b). From Lemma~\ref{lem3} it follows that $F^\delta(u_0,\mathcal{E}(v_0))$ does not depend on $\delta$, hence the constant on the right hand side does not depend on $\delta$. Theorem~\ref{theo1} leads to Assertion~(c). \end{proof} \section{Higher integrability for the logarithmic free energy} Since $\varphi^\delta:=(\psi^\delta)'$ will be singular as $\delta\to0$ we introduce for $r>0$ \[ \varphi^\delta_r(d):=\begin{cases} \varphi^\delta(d)|\varphi^\delta(d)|^{r-1} & \mbox{if }\varphi^\delta(d) \neq 0,\\ 0 & \mbox{if }\varphi^\delta(d)=0. \end{cases} \] By definition, $\varphi^\delta_r\in C^0({\mathbb R})$. For $00$ we introduce the function $\varphi^{\delta,\varrho}_r$ with $\varphi^{\delta,\varrho}_r=\varphi^\delta_r$ in ${\mathbb R}\setminus[0,1]$ and define $\varphi^{\delta,\varrho}_r$ in $[0,1]$ such that $\varphi^{\delta,\varrho}_r$ is a $C^1$ function, monotone increasing and $\varphi^{\delta,\varrho}_r\to\varphi^\delta_r$ in $C^0({\mathbb R})$ as $u\searrow 0$. First we need a regularity result on the strain tensor. The following Lemma is taken from \cite{Gahabil} where it is also proved. \begin{lemma}[Higher integrability of the strain tensor]\label{lem6} Suppose that $u\in L^\sigma(\Omega,{\mathbb R}^n)$ for a $\sigma>2$. Then there exists a $p\in(2,\sigma]$ independent of $u$ such that for all $v\in H^1(\Omega,{\mathbb R}^n)$ which fulfill for all $\zeta\in H^1(\Omega,{\mathbb R}^n)$ the identity \[ \int_\Omega\partial_u W^\mathrm{el}(u,\mathcal{E}(v)):\nabla\zeta =\int_\Omega\overline{S}:\nabla\zeta \] the integrability property $\nabla u\in L^p(\Omega,{\mathbb R}^{D\times D})$ holds. In particular, \[ \|\nabla v\|_{L^p(\Omega,{\mathbb R}^{D\times D})}\le C\big( \|\nabla v\|_{L^2(\Omega,{\mathbb R}^{D\times D})} +\|u\|_{L^p(\Omega,{\mathbb R}^n)}+1\big) \] independent of $u$. \end{lemma} Even though by construction $01$ and a constant $C>0$ such that for all $\delta\in(0,\delta_0)$ \begin{equation} \label{lemma7} \|\varphi^\delta(u^\delta_j)\|_{L^q(\Omega_T)}\le C\quad \mbox{for all }1\le j\le n. \end{equation} \end{lemma} \begin{proof} Starting point is the weak formulation (\ref{weaka}) \begin{equation} \begin{aligned} &\int_{\Omega_T} k_B\theta P(\varphi^\delta(u^\delta_i))_{1\le i\le n} \cdot\xi \\ &= -\int_{\Omega_T}\partial_t u^\delta\cdot\xi -\int_{\Omega_T}\gamma^2\nabla u^\delta:\nabla\xi -\int_{\Omega_T}PA u^\delta\cdot\xi -\int_{\Omega_T}P\partial_u W^\mathrm{el}(u^\delta,\mathcal{E}(v^\delta)) \cdot\xi \end{aligned} \label{3mud} \end{equation} which holds for all $\xi\in L^2(0,T;\,H_0^1(\Omega;{\mathbb R}^n))$ with $\partial_t\xi\in L^2(\Omega_T)$, $\xi(T)=0$. We want to use Lemma~\ref{lem6} and notice that due to the Sobolev embedding theorem $u^\delta\in L^\infty(0,T;\;L^s(\Omega))$, where $s=\frac{2D}{D-2}$ if $D\ge3$ and $s\in[1,\infty)$ if $D=2$ and $u^\delta\in L^\infty(\Omega_T)$ for $D=1$. So we find $\nabla u^\delta \in L^\infty(0,T;\;L^p(\Omega))$ for some $p>2$. We choose $p$ such that $p\in(2,4]$ and such that $p\in(2,\frac{2D}{D-2})$ if $D\ge3$. This means that also test functions $\xi\in L^2(0,T;\;H^1(\Omega,{\mathbb R}^D))\cap L^\frac{p}{p-2}(\Omega_T,{\mathbb R}^n)$ are allowed. So we can test (\ref{3mud}) with $\xi:=[\varphi^{\delta,\varrho}_r(u^\delta_j)]_{1\le j\le n}$ for $0j}^n\big(\varphi^\delta(u^\delta_i) -\varphi^\delta(u^\delta_j)\big)\varphi^{\delta,\varrho}_r(u^\delta_j)\Big]\\ & =k_B\theta\;\frac{1}{n}\sum_{i0$ the existence of a constant $C_\alpha$ with \begin{equation} \label{3lst} k_B\theta\;\frac{1}{n}\sum_{i1$, with Vitali's theorem we find \[ \varphi^\delta(u^\delta_k)\to\varphi(u_k)\quad\mbox{in }L^1(\Omega_T). \] So we can pass to the limit in (\ref{weakb}). \end{proof} \subsection*{Acknowledgements} The authors thank the German Research Community DFG for the financial support within the priority program 1095 {\it Analysis, Modeling and Simulation of Multiscale Problems}. \begin{thebibliography}{00} \bibitem{AJM} F. Alessio, L. Jeanjean, P. Montecchiari, Existence of infinitely many stationary layered solutions in ${\mathbb R}^2$ for a class of periodic Allen-Cahn equations, {\it Comm. Part. Diff. Equations} {\bf 27} (2002), 1537-1574 \bibitem{AC} S.M. Allen, J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, {\it Acta Metal.} {\bf 27} (1979), 1085--1095 \bibitem{Ali} N. D. Alikakos, X. Chen, G. Fusco, Motion of a droplet by surface tension along the boundary, {\it Calc. Var. Part. Diff. Equations} {\bf 3} (2000), 233-305 \bibitem{AL} H.W. Alt, S. Luckhaus, Quasilinear elliptic parabolic differential equations, {\it Math. Z.\/} {\bf 183} (1983), 311--338 \bibitem{Bates} P.W. Bates, F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, {\it J. Math. Anal. Appl.} {\bf 273} (2002), 45--57 \bibitem{Presutti} L. Bertini, S. Brassesco, P. Butta and E. Presutti, Stochastic phase field equations: existence and uniqueness, {\it Ann. Henri Poincar{\'e}} {\bf 3} (2002), 87-98 \bibitem{Blesgen} T. Blesgen, S. Luckhaus and K. Bente, Diffusion Induced Segregation in the Case of the Ternary System Sphalerite, Chalcopyrite and Cubanite, {\it Crystal Res. Tech.} {\bf 39} (2004), 969--979 \bibitem{Blow} J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis, {\it European~ Journal~Appl.~Math.} {\bf 2} (1991), 147--179 \bibitem{Bronsard} L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, {\it Arch. Rational Mech. Anal.} {\it 124} (1993), 355--379 \bibitem{Caginalp} G. Caginalp, X. Chen, Phase field equations in the singular limit of sharp interface problems, {\it IMA Vol. Math. Appl.} {\bf 43} (1992), 1--27 \bibitem{Chen} L.Q. Chen, Phase-field models for microstructural evolution, Ann. Rev. Mater. Res. 32 (2002), 113-140. \bibitem{Ciarlet} P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three dimensional elasticity, North-Holland, Amsterdam (1993) \bibitem{EL} C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy", {\it Preprint SFB 256}, University Bonn 1991 \bibitem{S1} C.M. Elliott and R. Sch{\"a}tzle, The limit of the anisotropic double-obstacle Allen-Cahn equation, {\it Proc. Roy. Soc. Edinburgh A} {\bf 126} (1996), 1217--1234 \bibitem{S2} C.M. Elliott, M. Paolini and R. Sch{\"a}tzle, Interface estimates for the fully anisotropic Allen-Cahn equation and anisotropic mean-curvature flow, {\it Math. Models Meth. Appl. Sci.} {\bf 6} (1996) 1103-1118 \bibitem{Esh} J.D. Eshelby, Elastic inclusions and inhomogeneities, {\it Prog. Solid Mech.} {\bf 2} (1961), 89--140 \bibitem{Gahabil} H. Garcke, On mathematical models for phase separation in elastically stressed solids, {\it Habilitation Thesis}, Bonn 2001 \bibitem{Ga1} H. Garcke, On Cahn-Hilliard systems with elasticity, Proc. Roy. Soc. Edinburgh 133 A (2003), 307-331. \bibitem{Ga2} H. Garcke, On a Cahn-Hilliard Model for Phase Separation with Elastic Misfit, Ann. de l'IHP - Analyse non lin{\'e}aire 22(2) (2005), 165-185. \bibitem{Ga3} H. Garcke, Mechanical effects in the Cahn-Hilliard model: A review on mathematical results, Preprints of DFG Priority Program Multiscale Problems No. 139 (2004). \bibitem{Garcke2} H. Garcke, B. Nestler and B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits, {\it Physica D} {\bf 115} (1998), 87--108 \bibitem{KM} K. Kassner, C. Misbah, A phase-field approach for stress-induced instabilities, Europhys. Lett. 46 (1999), 217-223. \bibitem{Schmidt} D. Kessler, R. H. Nochetto and A. Schmidt, A posteriori error control for the Allen-Cahn problem: circumvening Gronwall's inequality, {\it ESAIM: Math. Model. Num. Anal.} {\bf 38} (2004), 129--142 \bibitem{Rio} R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, {\it Physica D} 1993, 410--423 \bibitem{Krause} R. Kornhuber, R. Krause, On multigrid methods for vector-valued Allen-Cahn equations with obstacle potential, {\it 14th International Conference on Domain Decomposition Methods}, New Mexico (2003), 307--314 \bibitem{KO} M Koslowski and M Ortiz, A multi-phase field model of planar dislocation networks, Modelling Simul. Mater. Sci. Eng. {\bf 12}(2004), 1087-1097 \bibitem{Amy} R. dal Paso, L. Giacomelli, A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, {\it Interfaces Free Bound.} {\bf 1} (1999), 199--226 \bibitem{Steinbach} I. Steinbach, F. Pezolla, B. Nestler, M. See{\ss}elbach, R. Prieler, G.J. Schmitz, J.L.L. Rezende, A phase-field concept for multiphase systems, {\it Physica D} {\bf 94} (1996), 135--147 \end{thebibliography} \end{document}