Local solvability and turning for the inhomogeneous Muskat problem

TL;DR

This paper investigates the evolution and stability of the free boundary between two fluids in a porous medium with variable permeability, establishing local existence results and demonstrating finite-time regime changes.

Contribution

It extends local solvability results to inhomogeneous media and proves finite-time turning from stable to unstable regimes, a phenomenon previously shown only in homogeneous cases.

Findings

Local existence in Sobolev spaces for the stable regime

Finite-time transition from stable to unstable regime

Extension of turning results to inhomogeneous media

Abstract

In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane $\mathbb{R}^2$ or a bounded strip $S=\mathbb{R}\times(-\pi/2,\pi/2)$. The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This change of regime (turning) was first proven in \cite{ccfgl} for the homogeneus Muskat problem with infinite depth.