The mesa-limit of the porous-medium equation and the Hele-Shaw problem

Abstract

We are interested in the limit, as $m\to\infty,$ of the solution $u_m$ of the porous-medium equation $u_t = \Delta u^m$ in a bounded domain $\Omega$ with Neumann boundary condition, $\frac{\partial u^m}{\partial n}= g$ on $\partial\Omega,$ and initial datum $u(0)=u_0\geq 0.$ It is well known by now that this kind of limit turns out to be singular. In the case $g\equiv 0,$ it was proved that there exists an initial boundary layer ${\underline u}_0,$ the so-called mesa, and $u_m(t)\to {\underline u}_0$ in $L^1(\Omega),$ for any $t>0,$ as $m\to\infty.$ In this work, we generalize this result to the case of arbitrary $g\in L^2(\partial \Omega),$ we prove that the initial boundary layer is still ${\underline u}_0$ and in general (even in the regular case) the limit function is not a solution of a Hele--Shaw problem. There exists a time interval $I$ where the limit of $u_m,$ as $m\to\infty,$ is the unique solution of a Hele--Shaw problem and elsewhere, $u_m$ conveges to the constant function $\frac{1}{\vert\Omega\vert}(\int_\Omega u_0+ t\int_{\partial\Omega}g).$