Local-existence for the Inhomogeneous Muskat problem

TL;DR

This paper proves local existence results for the interface evolution between two fluids with different densities and viscosities in porous media with varying permeabilities, using Sobolev space techniques.

Contribution

It establishes local existence for the inhomogeneous Muskat problem with discontinuous free boundaries in Sobolev spaces, addressing a more complex scenario than homogeneous cases.

Findings

Proves local existence in Sobolev spaces for the inhomogeneous Muskat problem.

Handles discontinuities in densities and viscosities at the interface.

Addresses porous media with different permeabilities.

Abstract

In this work we study the evolution of the interface between two different fluids in a porous media with two different permeabilities. We prove local existence in Sobolev spaces, when the free boundary is given by the discontinuity among the densities and viscosities of the fluids.