Interfaces and Free Boundaries

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Volume 13, Issue 3, 2011, pp. 373–395
DOI: 10.4171/IFB/263

Published online: 2011-10-10

Long-time behavior of a Hele-Shaw type problem in random media

Norbert Požár[1]

(1) UCLA, Los Angeles, USA

We study the long-time behavior of an exterior Hele-Shaw problem in random media with a free boundary velocity that depends on the position in dimensions n ≥ 2. A natural rescaling of solutions that is compatible with the evolution of the free boundary leads to the homogenization of the free boundary velocity. By studying a limit obstacle problem for a Hele-Shaw problem with a point source, we are able to show the uniform convergence of the rescaled solution to a self-similar limit profile and we deduce that the rescaled free boundary uniformly approaches a sphere.

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Požár Norbert: Long-time behavior of a Hele-Shaw type problem in random media. Interfaces Free Bound. 13 (2011), 373-395. doi: 10.4171/IFB/263