# Regularity of solutions of a fractional porous medium equation

### Cyril Imbert

École Normale Supérieure, Paris, France### Rana Tarhini

Université Paris Est, Créteil Cedex, France### François Vigneron

Université de Reims Champagne-Ardennes, Reims, France

## Abstract

This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely

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 Hölder-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\'azquez~(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. \keywords{Parabolic regularity, De Giorgi method, porous medium equation (PME), H\"older regularity, non local operators, fractional derivatives.

## Cite this article

Cyril Imbert, Rana Tarhini, François Vigneron, Regularity of solutions of a fractional porous medium equation. Interfaces Free Bound. 22 (2020), no. 4, pp. 401–442

DOI 10.4171/IFB/445