Sharp asymptotic stability of the incompressible porous media equation

TL;DR

This paper establishes the sharp asymptotic stability of the incompressible porous media equation near a stable stratified density in higher-order Sobolev spaces, revealing a variational structure and controlling Sobolev norms.

Contribution

It proves stability in Sobolev spaces with k>2, where instability was previously known in H^2, using new energy and commutator estimates.

Findings

Long-time convergence from potential energy decay

Stability threshold established for Sobolev spaces with k>2

Refined commutator estimates for higher Sobolev norms

Abstract

In this paper, we prove the asymptotic stability of the incompressible porous media (IPM) equation near a stable stratified density, for initial perturbations in the Sobolev space $H^k$ with any $2<k \in\mathbb{R}$. While it is known that such a steady state is unstable in $H^2$, our result establishes a sharp stability threshold in higher-order Sobolev spaces. The key ingredients of our proof are twofold. First, we extract long-time convergence from the decay of a potential energy functional$-$despite its non-coercive nature$-$thereby revealing a variational structure underlying the dynamics. Second, we derive refined commutator estimates to control the evolution of higher Sobolev norms throughout the full range of $k>2$.