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

TL;DR

This paper investigates the long-term evolution of a Hele-Shaw problem in random media, demonstrating homogenization and convergence to a spherical shape through a limit obstacle problem analysis.

Contribution

It introduces a homogenization approach for the free boundary velocity in a Hele-Shaw problem within random media and proves convergence to a self-similar spherical profile.

Findings

Rescaled solutions converge uniformly to a self-similar profile.

The free boundary approaches a sphere uniformly over time.

Homogenization of the free boundary velocity is established.

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 $n \geq 2$. 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.