Homogenization of the Hele-Shaw problem in periodic spatiotemporal media

TL;DR

This paper studies the homogenization of the Hele-Shaw problem in media that vary periodically in space and time, showing convergence of solutions and free boundaries to a homogenized problem with nonlinear velocity dependence.

Contribution

It extends viscosity solution theory to inhomogeneous media and proves convergence of solutions and free boundaries in a homogenization setting.

Findings

Solutions converge to a homogenized Hele-Shaw problem

Free boundaries converge uniformly in Hausdorff distance

Homogenized velocity depends nonlinearly on the gradient

Abstract

We consider the homogenization of the Hele-Shaw problem in periodic media that are inhomogeneous both in space and time. After extending the theory of viscosity solutions into this context, we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous Hele-Shaw-type problem with a general, possibly nonlinear dependence of the free boundary velocity on the gradient. Moreover, the free boundaries converge locally uniformly in Hausdorff distance.