Regularity of solutions of a fractional porous medium equation

This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely partial derivative(t)u =del.(u del(-Delta)(alpha/2)-1u(m-1)) where u : R+ x R-N -> R+, for 0 < alpha < 2 and m >= 2. We prove that the L-1 boolean AND L-infinity weak solutions constructed by Biler, Imbert and Karch (2015) are locally Holder-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 <= 1, 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.