Barenblatt profiles for a nonlocal porous medium equation

Piotr Biler; Cyril Imbert; Grzegorz Karch

doi:10.1016/j.crma.2011.06.003

Partial Differential Equations

Barenblatt profiles for a nonlocal porous medium equation
[Solutions auto-similaires pour une équation des milieux poreux non locale]

Présenté par : Pierre-Louis Lions

Piotr Biler ¹ ; Cyril Imbert ² ; Grzegorz Karch ¹

¹ Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
² Université Paris-Dauphine, CEREMADE (UMR CNRS 7534), place de Lattre de Tassigny, 75775 Paris cedex 16, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 641-645.

Abstract (VO)
Résumé

We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit self-similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the $L^{p}$ decay of weak solutions of the corresponding Cauchy problem.

Cette Note est consacrée à lʼétude dʼune généralisation non locale de lʼéquation des milieux poreux. Plus précisément, on obtient des formules explicites de solutions auto-similaires à support compact qui ressemblent fortement aux solutions de type Barenblatt. On donne aussi un argument formel qui permet dʼobtenir des estimations $L^{p}$ des solutions faibles du problème de Cauchy.

Reçu le : 2010-02-07
Accepté le : 2011-06-01
Publié le : 2011-06-01

DOI : 10.1016/j.crma.2011.06.003

Affiliations des auteurs :

Piotr Biler ¹ ; Cyril Imbert ² ; Grzegorz Karch ¹

¹ Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
² Université Paris-Dauphine, CEREMADE (UMR CNRS 7534), place de Lattre de Tassigny, 75775 Paris cedex 16, France

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Piotr Biler; Cyril Imbert; Grzegorz Karch. Barenblatt profiles for a nonlocal porous medium equation. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 641-645. doi : 10.1016/j.crma.2011.06.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.06.003/

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