Singularity Formation in the Incompressible Porous Medium Equation without Boundary Mass

TL;DR

This paper proves that solutions to the inviscid porous medium equation can develop finite-time singularities even when solutions are Lipschitz continuous and vanish at the boundary, highlighting complex boundary effects.

Contribution

It demonstrates finite-time singularity formation for Lipschitz solutions of the inviscid porous medium equation with boundary vanishing density, a novel boundary behavior analysis.

Findings

Finite-time singularities occur in Lipschitz solutions.

Solutions are smooth away from the origin.

Density can be compactly supported.

Abstract

We prove finite-time singularity formation for Lipschitz continuous solutions of the inviscid porous medium equation which vanish on the boundary of the domain. As the density vanishes on the boundary of the domain, the full regularizing effect of transport is present and must be overcome. The solutions are smooth away from the origin and the density can be made compactly supported.