Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation

TL;DR

This paper demonstrates the mild ill-posedness and unexpected instability of the 2D incompressible porous media equation in the critical Sobolev space, even for physically stable density profiles, challenging traditional stability assumptions.

Contribution

It provides the first rigorous proof of instability for the IPM equation with vertically nonlinear density profiles, regardless of the sign of the density gradient.

Findings

Instability occurs even when density gradient is negative and smooth.

Results hold for arbitrary stratified density profiles without sign restrictions.

First rigorous demonstration of IPM instability for such profiles.

Abstract

In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\infty}$ when the initial data are small perturbations of stable profile $g(x_2).$ Consequently, instability can be inferred. Notably, our results are valid for arbitrary vertically stratified density profiles $g(x_2)$ without imposing any restrictions on the sign of $g'(x_2).$ From a physical perspective, since gravity acts downward, density profiles satisfying $g'(x_2) < 0$ typically correspond to stable configurations, whereas those with $g '(x_2) > 0$ are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when $g'(x_2) < 0$ and $g'(x_2)\in W^{2,\infty}(\mathbb{R})$. To the best of our knowledge, this work provides the first rigorous demonstration of IPM instability for vertically nonlinear density profiles, marking a significant departure from conventional physical expectations.