\documentclass[twoside]{article} \usepackage{amssymb} \usepackage{amstex} \usepackage{amsthm} \pagestyle{myheadings} \markboth{\hfil Analysis of the Mushy Region \hfil EJDE--1997/04}% {EJDE--1997/04\hfil Mike O'Leary \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1997}(1997), No.\ 04, pp. 1--14. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Analysis of the mushy region in conduction- convection problems with change of phase \thanks{ {\em 1991 Mathematics Subject Classifications:} 35R35, 76D99, 76R99, 80A22. .\newline\indent {\em Key words and phrases:} conduction, convection, free boundary, Stefan problem, \newline\indent mushy region \newline\indent \copyright 1997 Southwest Texas State University and University of North Texas.\newline\indent Submitted August 12, 1996. Published January 30, 1997.} } \date{} \author{Mike O'Leary} \maketitle \begin{abstract} A conduction-convection problem with change of phase is studied, where convective motion of the liquid affects the change of phase. The mushy region is the portion of the system to which temperature and enthalpy do not assign a phase, solid or liquid. In this paper we show that the enthalpy density remains constant in time almost everywhere in the mushy region. \end{abstract} \newtheorem{Theorem}{Theorem} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Lemma}[Theorem]{Lemma} \newcommand\cl{\mathcal} \newcommand\bb{\mathbb} \newenvironment{pf}{\textsc{Proof: }}{\qed\endtrivlist\vspace{1 em}} \def\al{\alpha} \def\bt{\beta} \def\ep{\epsilon} \def\Om{\Omega} \def\Omt{{\Omega_T}} \def\Omb{\overline\Omega} \def\Ombt{{\overline\Omega}_T} \def\om{\omega} \def\gm{\gamma} \def\Chi{\chi} \def\kp{\kappa} \def\lm{\lambda} \def\dl{\delta} \def\sg{\sigma} \def\Sg{\Sigma} \def\th{\theta} \def\zt{\zeta} \def\v{\bold v} \def\ol{\overline} \def\pl{\partial} \def\gd{\nabla} \def\dt{\!\cdot\!} \def\lp{\Delta} \newcommand\sgn{\operatorname{sgn}} \def\meas{\operatorname{meas}} \def\essup{\operatornamewithlimits{ess\,sup}} \def\div{\operatorname{div}} \def\myline{-\!\!\!\!-\!\!\!\!-\!\!\!\!-\!\!\!\!} \section{Introduction} The conduction--convection problem with change of phase is the problem of determining the temperature and motion of a system with liquid and solid components where the evolution and change of phase depend on both conduction and the convective motion of the liquid phase. The phase of the material is determined by the temperature $u(x,t)$ and enthalpy density, or thermal energy density, $w(x,t)$ of the material, normalized so that $w=0$ for solid at the phase change temperature, which for convenience we select to be zero. Then if $u(x,t)<0$, or more generally if $w(x,t)\leq 0$, the material is solid, while if $u(x,t)>0$, or more generally if $w(x,t)\geq L$, where $L$ is the latent heat of fusion per unit mass, the material is liquid. The mushy region is that portion of the material for which $u(x,t)=0$ and $00$, and let $\Omt=\Om\times(0,T)$. Let $\v\in L_2(\Omt)$ be weakly solenoidal, let $u\in L_\infty(\Omt)\cap L_2(0,T;W^1_2(\Om))$ and let $w\subseteq\bt(u)$ be any selection from the graph. Suppose that $$\label{mushy-7} w_t-\lp K(u)+\v\dt\gd u=0$$ weakly in $\Omt$. If M(t)=\{x\in\Om:00$. Let$\v\in L_2(0,T;W^1_2(\Om))\cap L_\infty(0,T;L_2(\Om))$be weakly solenoidal, let$g\in L_\infty(\pl\Om\times(0,T))\cap L_2(0,T;W^1_2(\pl\Om))$and suppose that$w_o\in C^2(\Omb)$and$u_o=\bt^{-1}(w_o)$. Then there exists a function$u\in L_\infty(\Omt)\cap L_2(0,T;W^1_2(\Om))$, and a function$w\subseteq\bt(u)so that \begin{alignat}{2} w_t-\lp K(u)&+\v\dt\gd u=0&\hspace{0.5 in}&\text{weakly in }\Omt, \label{mushy-7/96-8}\\ u\big|_{\pl\Om}&=g&&\text{as traces,} \\ w(\cdot,t)&\overset{t\downarrow 0}{\myline\rightharpoonup}w_o&&\text{weakly in }L_2(\Om). \label{mushy-7/96-10} \end{alignat} Further, if M(t)=\{x\in\Om:00 be a regularizing parameter, and let $\bt_\ep$ and $K_\ep$ be smooth approximations of $\bt$ and $K$ so that \begin{xalignat}{2} \bt_\ep(0)&=0, & 0<\bt_o&\leq\bt'_\ep(s)\leq\tfrac 1\ep, \\ K_\ep(0)&=0, & 0ds ,\\ u^i_\ep\big|_{\pl\Om}&=\frac 1h\int_{ih}^{(i+1)h}g_\ep(x,s)\>ds, \\ w^i_{\ep}&=\bt_\ep(u^i_\ep),\hspace{0.2 in}i=1,2,\dots,n-1 \\ w^o_\ep(x)&=w_o\subseteq\bt(u_o). \end{align} For $\ep$, $n$ and $i$ fixed, this is a quasi-linear elliptic equation for $w^i_{\ep}=\bt_\ep(u^i_\ep)$. Monotonicity of $\bt_\ep$ and $K_\ep$ imply the classical maximum principle (c.f. \cite[Chapter 3, \S 1]{LaUr68}) $$\label{mushy-7/96-23} \sup_{\ol\Om}|w^i_{\ep}(x)|\leq\max\{\sup_{\pl\Om\times(0,T)}|\bt_\ep\circ g_\ep|,\sup_{\Omb}|w^{i-1}_\ep|\}$$ for $1\leq i\leq N-1$. Standard theory \cite[Theorem 15.11, Theorem 6.19]{GiTr84} implies we have a classical solution $w^i_{\ep}\in C^{3}(\ol\Om)$ for each $\ep$ and $i$; boundedness of the data and the maximum principle imply that $\| w^i_\ep\|_{L_\infty(\Om)}$ and $\| u^i_\ep\|_{L_\infty(\Om)}$ are bounded uniformly in $\ep$ and $i$. For each $h=T/n$, define the functions \begin{alignat}{2} u_{\ep,h}(x,t)&=u^i_\ep(x)&&\text{ if }ih\leq t<(i+1)h, \\ w_{\ep,h}(x,t)&=w^i_{\ep}(x)&&\text{ if }ih\leq t<(i+1)h, \\ \v_{\ep,h}(x,t)&=\v_\ep^i(x)&&\text{ if }ih\leq t<(i+1)h. \end{alignat} Since \eqref{mushy-7/96-19} holds classically, multiply by $h u^i_\ep$, integrate over $\Om$ and sum to obtain \begin{multline} \label{mushy-28} \sum_{i=1}^{n-1} K_o h\int_\Om|\gd u^i_\ep|^2\>dx \leq \sum_{i=1}^{n-1}h\int_{\pl\Om}u^i_\ep \gd K_\ep(u^i_\ep)\dt\nu\>d\sg +\int_\Om w^o u^1_\ep\>dx \\ -\int_\Om w^{n-1}_\ep u^{n-1}_\ep\>dx +\sum_{i=2}^{n-1}\int_\Om w^{i-1}_{\ep}(u^i_\ep-u^{i-1}_\ep)\>dx\\ +\sum_{i=1}^{n-1}h\int_\Om u^i_\ep\v^i_\ep\dt\gd u^i_\ep\>dx, \end{multline} where $\nu$ is the outward unit normal to $\Om$. For each $2\leq i\leq n-1$, the continuity of $\bt_\ep$ and the mean value theorem for integrals implies the existence of a number $a_i(x)$ between $u^{i-1}_\ep(x)$ and $u^i_\ep(x)$ so that $$\bt_\ep(a_i)(u^i_\ep-u^{i-1}_\ep)=\int_{u^{i-1}_\ep}^{u^i_\ep}\bt_\ep(s)\>ds;$$ thus \begin{multline} \label{mushy-30} \sum_{i=2}^{n-1}\int_\Om w_\ep^{i-1}(u^i_{\ep}-u^{i-1}_\ep) \>dx=\\ \int_\Om\int_{u^1_\ep}^{u^{n-1}_\ep}\bt_\ep(s)\>ds\>dx+\sum_{i=2}^{n-1}\int_\Om\big(\bt_\ep(u^{i-1}_\ep)-\bt_\ep(a_i)\big)(u^i_\ep-u^{i-1}_\ep)\>dx. \end{multline} Monotonicity of $\bt_\ep$ implies that the last term above is nonpositive. Young's inequality applied to the first term on the right side of \eqref{mushy-28} together with \eqref{mushy-30} imply \begin{multline} \label{mushy-7/96-31} \int_h^T\!\!\!\int_{\Om}|\gd u_{\ep,h}|^2\>dx\>dt\leq C\int_0^T\!\!\!\int_{\pl\Om}\{|g_\ep|^2+|\gd g_\ep|^2\}d\sg(x)\>dt +C\int_\Om w^o u^1_\ep\>dx\\ +C\int_\Om\int_{u^1_\ep}^{u^{n-1}_\ep}\bt_\ep(s)\>ds\>dx+C\int_h^T\!\!\!\int_{\Om} u_{\ep,h}\v_{\ep,h}\dt\gd u_{\ep,h}\>dx\>dt\leq C, \end{multline} as boundedness of the data, the maximum principle, and Young's inequality applied to the last term above imply $\|\gd u_{\ep,h}\|_{L_2(\Om\times(h,T))}$ is bounded uniformly in $\ep$ and $h$. Next, we wish to show the pre-compactness of $\{w_{\ep,h}\}$ in $L_{1}(\Omt)$; to that end we begin with an estimate of $\gd w_{\ep,h}$. Let $\Om'\Subset\Om$ and let $\zt\in C^\infty_0(\Om)$ be a cutoff function so that $\zt(x)=1$ if $x\in\Om'$. Let $\dl>0$ and define $$\phi^i(x)=-\frac{\pl}{\pl x_k} \left\{\zt(x)\frac{w^i_{\ep,x_k}}{\sqrt{(w^i_{\ep,x_k})^2+\dl}}\right\} =-\frac{\pl}{\pl x_k}(\zt\sgn^\dl w^i_{\ep,x_k})$$ for some $k=1,2,\dots,N$. Multiply \eqref{mushy-7/96-19} by $h \phi_i$, integrate, and sum from $i=1$ to $m$, for some arbitrary $1\leq m\leq n-1$ to obtain \begin{multline} \label{mushy-7/96-34} \sum_{i=1}^m\int_{\Om}(w^i_{\ep,x_k}-w^{i-1}_{\ep,x_k})\zt\sgn^\dl w^i_{\ep,x_k}\>dx +h\sum_{i=1}^m\int_\Om\lp K_\ep(u^i_\ep)\frac{\pl}{\pl x_k}(\zt\sgn^\dl w^i_{\ep,x_k})\>dx \\ +h\sum_{i=1}^m\int_\Om\frac{\pl}{\pl x_k}(\v^i_\ep\dt\gd u^i_\ep)\zt\sgn^\dl w^i_{\ep,x_k}\>dx=0. \end{multline} To estimate the first term, note that $$\begin{split} \lim_{\dl\downarrow 0}\sum_{i=1}^m&\int_\Om (w^i_{\ep,x_k}-w^{i-1}_{\ep,x_k})\zt\sgn^\dl w^i_{\ep,x_k}\>dx \\ =&\int_{\Om} |w^m_{x_k}|\zt\>dx-\int_{\Om}|w^o_{x_k}|\zt\>dx-\sum_{i=1}^m\int_{\Om} |w^{i-1}_{\ep,x_k}|\zt(\sgn w^i_{\ep,x_k}\sgn w^{i-1}_{\ep,x_k}-1)\>dx\\ \geq &\int_{\Om} |w^m_{x_k}|\zt\>dx-\int_{\Om}|w^o_{x_k}|\zt\>dx. \end{split}$$ To handle the second term in \eqref{mushy-7/96-34}, begin with a pair of integrations by parts yielding \begin{multline} h\sum_{i=1}^m\int_\Om\lp K_\ep(u^i_\ep)\frac{\pl}{\pl x_k}(\zt\sgn^\dl w^i_{\ep,x_k})\>dx \\ =h\sum_{i=1}^m\int_\Om \gd\big\{(K_\ep\circ\bt_\ep^{-1})'(w^i_\ep)w^i_{\ep,x_k}\big\} (\zt \gd\sgn^\dl w^i_{\ep,x_k}+\gd\zt\sgn^\dl w^i_{\ep,x_k})\>dx, \end{multline} and integrating by parts again in the last term we obtain \begin{multline} h\sum_{i=1}^m\int_\Om\lp K_\ep(u^i_\ep)\frac{\pl}{\pl x_k}(\zt\sgn^\dl w^i_{\ep,x_k})\>dx \\ =h\sum_{i=1}^m\int_\Om [\gd(K_\ep\circ\bt_\ep^{-1})' (w^i_{\ep})]\zt w^i_{\ep,x_k}\gd\sgn^\dl w^i_{\ep,x_k}\>dx \\ +h\sum_{i=1}^m\int_\Om (K_\ep\circ\bt_\ep^{-1})'(w^i_{\ep})\zt\gd w^i_{\ep,x_k}\gd\sgn^\dl w^i_{\ep,x_k}\>dx \\ -h\sum_{i=1}^m\int_\Om (K_\ep\circ\bt_\ep^{-1})'(w^i_{\ep}) w^i_{\ep,x_k}\div(\gd\zt\sgn^\dl w^i_{\ep,x_k})\>dx\\ =I_1+I_2+I_3. \end{multline} By computation, we see that $$I_1=h\sum_{i=1}^m\int_\Om [\gd(K_\ep\circ\bt_\ep^{-1})'(w^i_{\ep})]\zt w^i_{\ep,x_k} \frac{\dl\gd w^i_{\ep,x_k}}{[(w^i_{\ep,x_k})^2+\dl]^{3/2}}\>dx.$$ Now for almost every $x\in\Om$ $$\frac{\dl w^i_{\ep,x_k}(x)}{[(w^i_{\ep,x_k}(x))^2+\dl]^{3/2}}\leq 1 \hspace{0.5 in}\text{and}\hspace{0.5 in}\lim_{\dl\downarrow 0}\frac{\dl w^i_{\ep,x_k}(x)}{[(w^i_{\ep,x_k}(x))^2+\dl]^{3/2}}=0.$$ Since $w^i_\ep\in C^3(\Omb)$, we can apply Lebesgue's dominated convergence theorem to conclude $$\lim_{\dl\downarrow 0} I_1=\lim_{\dl\downarrow 0}h\sum_{i=1}^m\int_\Om[\gd(K_\ep\circ\bt_\ep^{-1})'(w^i_{\ep})]\zt w^i_{\ep,x_k}\frac{\dl \gd w^i_{\ep,x_k}}{[(w^i_{\ep,x_k})^2+\dl]^{3/2}}dx=0.$$ To estimate $I_2$, note that monotonicity of $K_\ep$ and $\bt_\ep^{-1}$ implies \begin{multline} I_2=h\sum_{i=1}^m\int_\Om(K_\ep\circ\bt_\ep^{-1})'(w^i_{\ep})\zt\gd w^i_{\ep,x_k}\gd\sgn^\dl w^i_{\ep,x_k}\>dx\\ =h\sum_{i=1}^m\int_\Om(K_\ep\circ\bt_\ep^{-1})'(w^i_{\ep})\zt\frac{\dl |\gd w^i_{\ep,x_k}|^2}{[(w^i_{\ep,x_k})^2+\dl]^{3/2}}\geq 0. \end{multline} Finally, decompose $I_3$ as \begin{multline} I_3\> = h\sum_{i=1}^m\int_\Om (K_\ep\circ\bt^{-1}_\ep)'(w^i_{\ep}) w^i_{\ep,x_k}\div(\gd\zt\sgn^{\dl} w^i_{\ep,x_k})\,dx\\ \>=h\sum_{i=1}^m\int_\Om (K_\ep\circ\bt_\ep^{-1})' (w^i_{\ep})\gd\zt w^i_{\ep,x_k}\gd\sgn^\dl w^i_{\ep,x_k}\,dx \\ \>+h\sum_{i=1}^m\int_\Om(K\circ\bt^{-1}_{\ep})'(w^i_{\ep}) w^i_{\ep,x_k}\sgn^\dl w^i_{\ep,x_k}\lp\zt\,dx. \end{multline} The first of these terms tends to zero as $\dl\downarrow 0$ for the same reasons that $I_1$ does, while the second can be written as \begin{multline} \lim_{\dl\downarrow 0}h\sum_{i=1}^m\int_\Om(K_\ep\circ\bt_\ep^{-1})' (w^i_{\ep}) w^i_{\ep,x_k}\sgn^\dl w^i_{\ep,x_k}\lp\zt\,dx \\ =h\sum_{i=1}^m\int_\Om (K_\ep\circ\bt_\ep^{-1})'(w^i_{\ep}) \lp\zt|w^i_{\ep,x_k}|\,dx \\ =h\sum_{i=1}^m\int_\Om\left|\frac{\pl}{\pl x_k} K_\ep(u^i_\ep) \right|\lp\zt\,dx \end{multline} where we have again used the monotonicity of $K_\ep$ and $\bt_\ep^{-1}$. To estimate the last term of \eqref{mushy-7/96-34}, integrate by parts and use the fact that $\v_\ep$ is solenoidal to obtain \begin{multline} h\sum_{i=1}^m\int_\Om\frac{\pl}{\pl x_k}(\v^i_\ep\dt\gd u^i_\ep)\zt\sgn^\dl w^i_{\ep,x_k}\>dx =h\sum_{i=1}^m\int_\Om\frac{\pl\v^i_\ep}{\pl x_k}\dt\gd u^i_\ep\zt\sgn^\dl w^i_{\ep,x_k}\>dx \\ -h\sum_{i=1}^m\int_\Om u^i_{\ep,x_k}\v^i_\ep\dt\gd\zt\sgn^\dl w^i_{\ep,x_k}\>dx \\ -h\sum_{i=1}^m\int_\Om(\bt^{-1}_\ep)'(w^i_{\ep})w^i_{\ep,x_k}\zt\v^i_\ep\dt\gd\sgn^\dl w^i_{\ep,x_k}\>dx. \end{multline} The last term above vanishes as $\dl\downarrow 0$ for the same reasons that $I_1$ did, thus \begin{multline} \lim_{\dl\downarrow 0}h\sum_{i=1}^m\int_\Om\frac{\pl}{\pl x_k}(\v^i_\ep\dt\gd u^i_\ep)\zt\sgn^\dl w^i_{\ep,x_k}\>dx\\ \geq-h\sum_{i=1}^m\int_\Om\left|\frac{\pl \v^i_\ep}{\pl x_k}\right| |\gd u^i_\ep| \> \zt\>dx -h\sum_{i=1}^m\int_\Om\left|\v^i_\ep\right|\> |u^i_{\ep,x_k}|\> |\gd\zt| \>dx . \end{multline} Combining these results, we obtain for any $\Om'\Subset\Om$ and for any $0\leq t\leq T$ \begin{multline} \label{mushy-7/96-48} \int_{\Om'} |\gd {w_{\ep,h}}(x,t)|\>dx \leq\int_\Om |\gd w_o|\>dx\\ +C\int_h^T\!\!\!\int_{\Om}\big\{ |\gd K_\ep(u_{\ep,h})|+ |\gd\v_{\ep,h}|\>|\gd u_{\ep,h}| + |\v_{\ep,h}|\>|\gd u_{\ep,h}|\big\}dx\leq C \end{multline} for a constant $C$ depending only upon the data, including $\| w_o\|_{C^1(\ol\Om)}$, $\Om$, and $\Om'$, but independent of $\ep$ and $h$. To obtain the pre-compactness of $\{w_{\ep,h}\}$, we require an additional estimate. Let $0dx\\ =h\sum_{i=m_1+1}^{m_2}\int_{\Om}K(u^i_{\ep,h})\lp\psi\>dx +h\sum_{i=m_1+1}^{m_2}\int_\Om u^i_{\ep,h}\v^i_{\ep,h}\dt\gd\psi\>dx. \end{multline} Thus \begin{multline} \label{mushy-7/96-45} \left|\int_{\Om} \left\{w_{\ep,h}(x,t+\tau)-w_{\ep,h}(x,t)\right\}\psi \>dx\right| \\ \leq \int_{t}^{t+\tau}\!\!\int_\Om\left|K(u_{\ep,h})\lp\psi+u_{\ep,h} \v_{\ep,h}\dt\gd\psi\right|dx\>ds\\ \leq C\tau\sup_{\Om}\{|\lp\psi|+|\gd\psi|\} \end{multline} where$C$depends only upon$K_1$,$|\Om|$,$\| u_{\ep,h}\|_{L_\infty(\Omt)}$and$\sup_{00$. Let$rdx\leq M. If there is a constant $\gm$ so that for any $\psi\in C^2_0(B_r)$ and almost every $0dx\right|\leq\gm \tau\>\sup_{B_r}\big\{|\psi|+|\gd\psi|+|\lp\psi|\big\}, then for almost every$0dx\leq C\min_{0<\sg0\}$is measurable. If$\meas\Sg>0$, then $$\meas S=\iint_{\!\!\Omt}\chi_S\>dx\>dt\geq\int_\Sg\meas\{x:(x,t)\in S\}\>dt>0$$ so that$\meas \Sg=0$. Let$0\leq t_1\leq t_2\leq T$with$t_1, t_2\in[0,T]\setminus\Sg$. Recall the definition M(t_2)=\{x\in\Om:00$ and define M_\dl(t_2)=\{x\in\Om:\dl0$; by Egoroff's theorem there is a set$E_\eta\subseteq\Om$so that$w_{h_j}(x,t_2)\rightarrow w(x,t_2)$uniformly in$E_\eta$and so that$\meas(\Om\setminus E_\eta)\leq\eta$. Thus, there exists an integer$J$so that if$j>J$and$x\in M_\dl(t_2)\cap E_\eta$, then$00$. Suppose that$\v_1,\v_2\in L_2(\Omt)$are weakly solenoidal. If$w_i\subseteq\bt (u_i)$and$u_i\in L_\infty(\Omt)\cap L_2(0,T,W^1_2(\Om))$satisfy \begin{gather} \frac{\pl}{\pl t}w_i-\lp K(u_i)+\v_i\dt\gd u_i=0 \\ u_i\big|_{\pl\Om}=g \\ w_i\big|_{t=0}=w_{o,i} \end{gather} for$i=1,2$, then \essup_{0<\taudx -\int_0^\tau\!\!\int_\Om\big\{w_i\phi_t+K(u_i)\lp\phi\big\}dx\>dt\\ +\int_0^\tau\!\!\int_{\pl\Om} K(g)\frac{\pl\phi}{\pl\nu}\>d\sg(x)\>dt +\int_0^\tau\!\!\int_\Om u_i\v_i\dt\gd\phi\>dx\>dt=\int_\Om w_{o,i}\phi(x,0)\>dx \end{multline} where$\frac{\pl\phi}{\pl\nu}$is the derivative in the direction of the outward normal. Subtract these identities for$i=1,2$to obtain \begin{multline} \label{mushy-7/96-90} \int_{\Om}w(x,\tau)\phi(x,\tau)\>dx-\int_0^\tau\!\!\int_\Om w\big\{\phi_t+\kp\mu\lp\phi+\mu\v_2\dt\gd\phi\big\}dx\>dt\\ =\int_0^\tau\!\!\int_\Om u_1\v\dt\gd\phi\>dx\>dt+\int_0^\tau\!\!\int_\Om w_o\phi(x,0)\>dx \end{multline} where $$\kp=\frac{K(u_2)-K(u_1)}{u_2-u_1}\hspace{0.5 in}\text{and}\hspace{0.5 in}\mu=\frac{u_2-u_1}{w_2-w_1}.$$ Note that$00$,$\dl>0$, and$\sg>0$, and let$\v_{2,\dl}$,$\mu_\sg$, and$\kp_\sg$be smooth approximations of$\v_2$,$\mu$and$\kp$, with$\v_{2,\dl}$solenoidal. Let$\phi_o\in C_0^3(\ol\Om)and consider \begin{align} \frac{\pl \phi_{\dl,\ep,\sg}}{\pl t}+(\kp_\sg\mu_\sg+\ep)\lp\phi_{\dl,\ep,\sg} &+\mu_\sg\v_{2,\dl}\gd\phi_{\dl,\ep,\sg}=0, \label{mushy-7/96-92}\\ \phi_{\dl,\ep,\sg}\big|_{\pl\Om\times(0,\tau)}&=0, \\ \phi_{\dl,\ep,\sg}(x,\tau)&=\phi_o(x). \end{align} The usual parabolic theory \cite[Chapter 4, Theorem 5.2]{LSU68} implies that this problem has a solution\phi_{\dl,\ep,\sg}\in C^{2,1}_{x,t}(\Omb_\tau)$. To obtain uniform estimates, multiply \eqref{mushy-7/96-92} by$\lp\phi_{\dl,\ep,\sg}$and integrate over$\Om$to obtain \begin{multline} -\frac 12\frac d{dt}\int_{\Om}|\gd\phi_{\dl,\ep,\sg}|^2\>dx\\ +\int_{\Om}(\kp_\sg\mu_\sg+\ep)|\lp\phi_{\dl,\ep,\sg}|^2\>dx +\int_\Om \mu_\sg\v_{2,\dl}\dt\gd\phi_{\dl,\ep,\sg}\lp\phi_{\dl,\ep,\sg}\>dx=0. \end{multline} Then Young's inequality implies the existence of a constant depending on$K_1$and$\bt_o$so that $$-\frac d{dt}\int_\Om |\gd\phi_{\dl,\ep,\sg}|^2\>dx +\int_\Om (\kp_\sg\mu_\sg+\ep)|\lp\phi_{\dl,\ep,\sg}|^2\>dx \leq C\int_\Om |\v_{2,\dl}|^2|\gd\phi_{\dl,\ep,\sg}|^2\>dx.$$ Thus if we allow$C$to depend upon$\dl$through$\|\v_{2,\dl}\|_{L_\infty(\Omt)}$, we can apply Gronwall's inequality to obtain the estimate $$\sup_{0<\tau<\tau}\int_\Om |\gd\phi_{\dl,\ep,\sg}|^2\>dx +\int_0^\tau\!\!\int_\Om(\kp_\sg\mu_\sg+\ep)|\lp\phi_{\dl,\ep,\sg}|^2\>dx\>dt \leq C_\dl\int_\Om |\gd\phi_o|^2\>dx.$$ This, together with the maximum principle, allows us to send$\sg\downarrow 0$and implies the existence of a function$\phi_{\dl,\ep}\in W^{2,1}_2(\Om_\tau)$so that \begin{gather} \frac{\pl \phi_{\dl,\ep}}{\pl t}+(\kp\mu+\ep)\lp\phi_{\dl,\ep}+\mu\v_{2,\dl}\dt\gd\phi_{\dl,\ep}=0, \\ \phi_{\dl,\ep}\big|_{\pl\Om\times(0,\tau)}=0, \\ \phi_{\dl,\ep}(x,\tau)=\phi_o(x), \end{gather} and so that \begin{gather} \sup_{\Omb\times[0,\tau]}|\phi_{\dl,\ep}|\leq\sup_{\Omb}|\phi_o|,\label{mushy-7/96-101} \\ \int_0^\tau\!\!\int_\Om\left\{\ep |\lp\phi_{\dl,\ep}|^2+|\gd\phi_{\dl,\ep}|^2+\left|\frac{\pl\phi_{\dl,\ep}}{\pl t}\right|^2\right\}dx\>dt\leq C_\dl\int_\Om|\gd\phi_o|^2\>dx. \label{mushy-7/96-102} \end{gather} Substitute this function$\phi_{\dl,\ep}$into \eqref{mushy-7/96-90} to obtain \begin{multline} \int_{\Om} w(x,\tau)\phi_o(x)\>dx=-\int_0^\tau\!\!\int_\Om\left\{\ep w\lp\phi_{\dl,\ep}+\phi_{\dl,\ep}(\v_2-\v_{2,\dl})\dt\gd u\right\}dx\>dt\\ -\int_0^\tau\!\!\int_\Om \phi_{\dl,\ep}\v\dt\gd u_1\>dx\>dt+\int_\Om w_o\phi_{\dl,\ep}(x,0)\>dx \end{multline} where we have integrated by parts. The estimate \eqref{mushy-7/96-102} implies \begin{multline} \left|\int_0^\tau\!\!\int_\Om \ep w\lp\phi_{\dl,\ep}\>dx\>dt\right|\\ \leq\| w\|_{L_\infty(\Omt)}(\ep|\Om|\tau)^{1/2}\left(\int_0^\tau\!\!\int_\Om \ep|\lp\phi_{\dl,\ep}|^2\>dx\>dt\right)^{\frac 12}\leq C_\dl\sqrt\ep \end{multline} so that we may pass to the limit as$\ep\downarrow 0$to obtain \begin{multline} \left|\int_\Om w(x,\tau)\phi_o(x)\>dx\right| \leq\| \phi_o\|_{L_\infty(\Omt)}\int_0^\tau\int_\Om \{|\v-\v_\dl|\>|\gd u| +|\v|\>|\gd u_2|\}dx\>dt\\ +\| \phi_o\|_{L_\infty(\Om)}\int_\Om |w_o|dx \end{multline} which, by completeness, holds for every$\phi_o\in L_\infty(\Om)$. Let$\dl\downarrow 0$and set$\phi_o=w(x,\tau)$to obtain the result. \end{pf} \noindent\textsc{Proof of Theorem 1.} Let$Q_R(x_o,t_o)=B_R(x_o)\times(t_o-R,t_o)\subseteq\Omt$. Then for almost every$R/2