Interface evolution: the Hele-Shaw and Muskat problems

Abstract

We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy’s law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two-dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.

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Authors

Antonio Córdoba

Departamento de Matemáticas
Universidad Autónoma de Madrid
28049 Madrid
Spain

Diego Córdoba

Instituto de Ciencias Matemáticas
Consejo Superior de Investigaciones Científicas
28006 Madrid
Spain

Francisco Gancedo

Department of Mathematics
University of Chicago
Chicago, IL 60637