Regularity of solutions of a fractional porous medium equation
Cyril Imbert
École Normale Supérieure, Paris, FranceRana Tarhini
Université Paris Est, Créteil Cedex, FranceFrançois Vigneron
Université de Reims Champagne-Ardennes, Reims, France
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Abstract
This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely
where , for and . We prove that the 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 , we adapt the proof of Caffarelli, Soria and Vázquez (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.
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