Global Results for the Inhomogeneous Muskat Problem

TL;DR

This paper proves the global existence, uniqueness, and instant analyticity of solutions for the inhomogeneous Muskat problem with a permeability jump, showing solutions decay to a flat interface over time.

Contribution

It establishes the first global well-posedness and decay results for the inhomogeneous Muskat problem with permeability discontinuities.

Findings

Global existence and uniqueness of solutions

Solutions become instantly analytic

Solutions decay to a flat interface over time

Abstract

The inhomogeneous Muskat problem models the dynamics of an interface between two fluids of differing characteristics inside a non-uniform porous medium. We consider the case of a porous media with a permeability jump across a horizontal boundary away from an interface between two fluids of different viscosities and densities. For initial data of explicit medium size, depending on the characteristics of the fluids and porous media, we will prove the global existence and uniqueness of a solution which is instantly analytic and decays in time to the flat interface.