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

2010/10/20 by Norbert Pozar, Pozar, Norbert
Mathematics · #35R35 35D40 35B40 35B27 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B27 #msc:35B40 #msc:35D40 #msc:35R35

paper · pdf · doi:10.48550/arxiv.1010.4275

24 pages

arxiv created 2010/10/20 · arxiv updated 2010/10/21

Abstract

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