Regularity of solutions of a fractional porous medium equation

TL;DR

This paper proves local Hölder continuity of weak solutions to a nonlinear, nonlocal fractional porous medium equation using adapted De Giorgi techniques, extending regularity results to a broader class of equations.

Contribution

It introduces a novel regularity proof for solutions of a fractional porous medium equation with nonlinear and nonlocal pressure law, extending classical methods.

Findings

Weak solutions are locally Hölder continuous in time and space.

The proof adapts De Giorgi techniques to a fractional, nonlinear PDE.

Local energy estimates and an intermediate value lemma are key tools.

Abstract

This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely $\partial_t u = { \nabla \cdot} \left(u \nabla(-\Delta)^{\frac{\alpha}{2}-1}u^{m-1} \right)$ where $u:\mathbb{R}_+\times \mathbb{R}^N \to \mathbb{R}_+$, for $0<\alpha<2$ and $m\geq2$. We prove that the $L^1\cap L^\infty$ weak solutions constructed by Biler, Imbert and Karch (2015) are locally H{\"o}lder-continuous in time and space. In this article, the classical parabolic De Giorgi techniques for the regularity of PDEs are tailored to fit this particular variant of the PME equation. In the spirit of the work of Caffarelli, Chan and Vasseur (2011), the two main ingredients are the derivation of local energy estimates and a so-called ''intermediate value lemma''. For $\alpha\leq1$, we adapt the proof of Caffarelli, Soria and V{\'a}zquez (2013), who treated the case of a linear pressure law. We then use a non-linear drift to cancel out the singular terms that would otherwise appear in the energy estimates.