Asymptotic Stability for some Stratified Stationary Solutions of the Inviscid Incompressible Porous Medium Equation

TL;DR

This paper investigates the stability of stratified stationary solutions of the 2D inviscid incompressible porous medium equation, demonstrating that such solutions are asymptotically stable under small, regular perturbations due to stratification effects.

Contribution

It classifies stationary solutions of the inviscid IPM and proves their asymptotic stability under small perturbations, highlighting stratification as the key stability mechanism.

Findings

Stratified stationary solutions not opposing gravity are asymptotically stable.

Small, regular perturbations of these solutions remain globally regular and converge to a steady state.

Stratification, rather than dispersion or mixing, underpins the stability results.

Abstract

We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability results. We then study solutions of the IPM equation which are sufficiently regular perturbations of linearly stable steady states. We prove that sufficiently regular perturbations which are also small must be globally regular and strongly converge to a steady state. The mechanism behind the stability is \emph{stratification} as opposed to previous stability results based on dispersion and/or mixing \cite{BedrossianMasmoudi}. More or less, we prove that stratified stationary solutions which do not go against gravity are asymptotically stable.