Flatness implies smoothness for solutions of the porous medium equation

TL;DR

This paper proves that flatness in solutions to the porous medium equation leads to smoothness of both the solution and free boundary after a finite time, removing longstanding non-degeneracy assumptions.

Contribution

It establishes that flatness implies smoothness for solutions of the porous medium equation, including free boundary regularity, without requiring initial non-degeneracy.

Findings

Solutions become smooth after finite time T

Free boundary is smooth after small time

Removes non-degeneracy condition on initial data

Abstract

One of the major problems in the theory of the porous medium equation is the regularity of the solutions and the free boundaries. Here we assume flatness of the solution in space time cylinder and derive smoothness of the interface after a small time, as well as smoothness of the solution in the positivity set and up to the free boundary for some time interval. We use these facts to prove the following eventual regularity result: solutions with compactly supported initial data are smooth after a finite time T that depends on mass and the size of the initial support. This result eliminates the condition of non-degeneracy on the initial data that has been carried on for decades in the literature.