On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results

TL;DR

This paper analyzes the Muskat problem with capillary and gravity effects in a periodic setting, proving well-posedness, stability of flat interfaces, and identifying unstable finger-shaped steady states through bifurcation theory.

Contribution

It establishes well-posedness and stability results for the Muskat problem with additional physical effects, and characterizes unstable finger solutions using bifurcation analysis.

Findings

Proves well-posedness of the problem

Shows exponential stability of flat equilibrium

Identifies unstable finger-shaped steady states

Abstract

We consider in this paper the Muskat problem in a periodic geometry and incorporate capillary as well as gravity effects in the modelling. The problem re-writes as an abstract evolution equation and we use this property to prove well-posedness of the problem and to establish exponential stability of some flat equilibrium. Using bifurcation theory we also find finger shaped steady-states which are all unstable.