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 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 $\alpha\leq1$, we adapt the proof of Caffarelli, Soria and Vazquez (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.

中文翻译：

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分数阶多孔介质方程解的正则性

本文关注的是一个多孔介质方程，其压力定律既是非线性又是非局部的，即 $\partial_t u = { \nabla \cdot} \left(u \nabla(-\Delta)^{\frac{\alpha}{ 2}-1}u^{m-1} \right)$ 其中 $u:\mathbb{R}_+\times \mathbb{R}^N \to \mathbb{R}_+$，对于 $0< \alpha<2$ 和 $m\geq2$。我们证明了由 Biler、Imbert 和 Karch (2015) 构建的 $L^1\cap L^\infty$ 弱解在时间和空间上是局部持有者连续的。在本文中，用于 PDE 正则性的经典抛物线 De Giorgi 技术经过定制，以适合 PME 方程的这一特定变体。本着 Caffarelli、Chan 和 Vasseur (2011) 的工作精神，两个主要成分是局部能量估计的推导和所谓的“中间值引理”。对于 $\alpha\leq1$，我们采用 Caffarelli 的证明，Soria 和 Vazquez (2013)，他们处理了线性压力定律的情况。然后，我们使用非线性漂移来抵消可能出现在能量估计中的奇异项。