  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.
