A maximum principle for the Muskat problem for fluids with different densities

TL;DR

This paper establishes a maximum principle for the free boundary in the Muskat problem involving two incompressible fluids with different densities, under stable conditions, using Darcy's law in a two-dimensional setting.

Contribution

It proves a maximum principle for the $L^ Infty$ norm of the free boundary in the Muskat problem with different densities, extending understanding of interface stability.

Findings

Maximum principle holds in the stable case

The free boundary's $L^ Infty$ norm is bounded over time

Results are specific to two-dimensional Darcy flow

Abstract

We consider the fluid interface problem given by two incompressible fluids with different densities evolving by Darcy's law. This scenario is known as the Muskat problem for fluids with the same viscosities, being in two dimensions mathematically analogous to the two-phase Hele-Shaw cell. We prove in the stable case (the denser fluid is below) a maximum principle for the $L^\infty$ norm of the free boundary.