Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations

TL;DR

This paper demonstrates the stability of explicit self-similar blow-up solutions for a class of infinite energy solutions to the inviscid incompressible porous medium equations, linking finite-time blow-up to global solutions of a reduced equation.

Contribution

It introduces a stability analysis of a special class of infinite energy solutions, revealing a sharp regularity threshold and connecting blow-up behavior to the Proudman-Johnson equation.

Findings

Explicit self-similar blow-up solution is stable under smooth perturbations.

A sharp regularity threshold determines stability versus instability.

Stability results extend to solutions of Euler and primitive equations.

Abstract

We consider a special class of infinite energy solutions to the inviscid incompressible porous medium equations (IPM), introduced in Castro-C\'ordoba-Gancedo-Orive [9]. The (IPM) equations then reduce to a one-dimensional nonlocal nonlinear equation, for which an explicit self-similar blow-up solution is found in [9]. We show the stability of this explicit blow-up solution by smooth enough perturbations, and identity a sharp regularity threshold below which it is unstable. The heart of our proof is a change of variables that transforms the study of finite time blow-up solutions, to the study of global-in-time solutions to the Proudman-Johnson equation, which is a reduced equation that appears for special classes of solutions of the two-dimensional Euler equations, and of the inviscid primitive equations (or hydrostatic Euler equations). Our main result is in fact the asymptotic stability with decay estimates for a family of steady states of this reduced equation. It thus also implies the corresponding stability of steady states for the associated classes of solutions to the Euler equations and inviscid primitive equations.