Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data

TL;DR

This paper proves the global existence and decay of Lipschitz solutions for the inhomogeneous Muskat problem, describing internal wave evolution in porous media with discontinuous permeability, under specific initial conditions.

Contribution

It establishes the first rigorous results on global existence and decay for Lipschitz solutions in the inhomogeneous Muskat problem with discontinuous permeability.

Findings

Solutions decay towards equilibrium in Lipschitz norm

Global existence of Lipschitz solutions is proven

Decay rates depend on initial data and physical parameters

Abstract

In this work we study the inhomogeneous Muskat problem, \emph{i.e.} the evolution of an internal wave between two different fluids in a porous medium with discontinuous permeability. In particular, under precise conditions on the initial datum and the physical quantities of the problem, our results ensure the decay of the solutions towards the equilibrium state in the Lipschitz norm. In addition, we establish the global existence and decay of Lipschitz solutions.