Regularity of weak solutions and supersolutions to the porous medium equation

TL;DR

This paper investigates the regularity conditions for weak solutions and supersolutions of the porous medium equation, establishing their equivalence under certain conditions using approximation techniques.

Contribution

It proves the equivalence of two key regularity assumptions in the definition of weak solutions to the porous medium equation.

Findings

Equivalence of $u^m \\in L^2_{\\rm loc}(0,T;H^1_{\\rm loc}(\\\Omega))$ and $u^{(m+1)/2} \\in L^2_{\\rm loc}(0,T;H^1_{\\rm loc}(\\\Omega))$.

Use of obstacle problem approximations to establish regularity equivalences.

Abstract

We study the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions $u^m \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(\Omega))$ and $u^\frac{m+1}2 \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(\Omega))$ in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.