Regularity of solutions of a fractional porous medium equation

被引:2
|
作者
Imbert, Cyril [1 ]
Tarhini, Rana [2 ]
Vigneron, Francois [3 ]
机构
[1] Ecole Normale Super, CNRS, Dept Math & Applicat, 45 Rue Ulm, F-75005 Paris, France
[2] Univ Paris Est, Lab Anal & Math Appl, UMR 8050, 61 Ave Gen Gaulle, F-94010 Creteil, France
[3] Univ Reims Champagne Ardennes, Lab Math Reims, UMR 9008, BP 1039, F-51687 Reims 2, France
关键词
Parabolic regularity; De Giorgi method; porous medium equation (PME); Holder regularity; non local operators; fractional derivatives; BARENBLATT PROFILES;
D O I
10.4171/IFB/445
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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.
引用
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页码:401 / 442
页数:42
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