Differential and Integral Equations

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

Noureddine Igbida

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

Article information

Source
Differential Integral Equations, Volume 15, Number 2 (2002), 129-146.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060869

Mathematical Reviews number (MathSciNet)
MR1870466

Zentralblatt MATH identifier
1011.35080

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35K35: Initial-boundary value problems for higher-order parabolic equations 35R35: Free boundary problems 76D27: Other free-boundary flows; Hele-Shaw flows 76S05: Flows in porous media; filtration; seepage

Citation

Igbida, Noureddine. The mesa-limit of the porous-medium equation and the Hele-Shaw problem. Differential Integral Equations 15 (2002), no. 2, 129--146. https://projecteuclid.org/euclid.die/1356060869


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