2002 The mesa-limit of the porous-medium equation and the Hele-Shaw problem
Noureddine Igbida
Differential Integral Equations 15(2): 129-146 (2002). DOI: 10.57262/die/1356060869


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 $\frac{1}{\vert\Omega\vert}(\int_\Omega u_0+ t\int_{\partial\Omega}g).$


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Noureddine Igbida. "The mesa-limit of the porous-medium equation and the Hele-Shaw problem." Differential Integral Equations 15 (2) 129 - 146, 2002. https://doi.org/10.57262/die/1356060869


Published: 2002
First available in Project Euclid: 21 December 2012

zbMATH: 1011.35080
MathSciNet: MR1870466
Digital Object Identifier: 10.57262/die/1356060869

Primary: 35K57
Secondary: 35K35 , 35R35 , 76D27 , 76S05

Rights: Copyright © 2002 Khayyam Publishing, Inc.


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Vol.15 • No. 2 • 2002
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