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.
Cite this article
Norbert Požár, Long-time behavior of a Hele-Shaw type problem in random media. Interfaces Free Bound. 13 (2011), no. 3, pp. 373–395DOI 10.4171/IFB/263