On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak–Keller–Segel model

Didier Bresch; Pierre-Emmanuel Jabin; Zhenfu Wang

doi:10.1016/j.crma.2019.09.007

Partial differential equations/Mathematical physics

On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak–Keller–Segel model
[Limites de champ moyen pour des noyaux singuliers et applications au modèle de Patlak–Keller–Segel]

Présenté par : Laure Saint-Raymond

Didier Bresch ¹ ; Pierre-Emmanuel Jabin ² ; Zhenfu Wang ³

¹ LAMA – UMR5127 CNRS, Bât. Le Chablais, Campus scientifique, 73376 Le Bourget-du-Lac, France
² CSCAMM and Dept. of Mathematics, University of Maryland, College Park, MD 20742, USA
³ Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA

Comptes Rendus. Mathématique, Volume 357 (2019) no. 9, pp. 708-720.

Résumé
Abstract

Dans cette note, on propose une énergie libre modulée combinant les méthodes développées par P.-E. Jabin et Z. Wang [Inventiones (2018)] et par S. Serfaty [voir l'article de revue Proc. Int. Cong. Math. (2018) et ses références] pour traiter des noyaux plus généraux en théorie de la limite champ moyen. Cette énergie libre modulée consiste en l'introduction d'une famille de poids appropriés dans l'entropie relative développée par P.-E. Jabin et Z. Wang (dans le même esprit que dans le travail récent de D. Bresch et P.-E. Jabin [Ann. of Math. (2) (2018)]) pour compenser les termes les plus singuliers qui font intervenir la divergence du champ de vitesse. Comme exemple, une preuve avec estimation quantitative de la limite champ moyen vers le modèle de Patlak–Keller–Segel en régime sous-critique est obtenue. Notre méthode permet également de couvrir des potentiels singuliers qui peuvent combiner une partie réguliere, une petite partie singulière attractive et une grande partie singulière répulsive.

In this note, we propose a modulated free energy combination of the methods developed by P.-E. Jabin and Z. Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. Math. (2018) and references therein] to treat more general kernels in mean-field limit theory. This modulated free energy may be understood as introducing appropriate weights in the relative entropy developed by P.-E. Jabin and Z. Wang (in the spirit of what has been recently developed by D. Bresch and P.-E. Jabin [Ann. of Math. (2) (2018)]) to cancel the most singular terms involving the divergence of the flow. Our modulated free energy allows us to treat singular potentials that combine large smooth part, small attractive singular part, and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as the Patlak–Keller–Segel system in subcritical regimes, is obtained.

Reçu le : 2019-06-11
Accepté le : 2019-09-19
Publié le : 2019-09-01

DOI : 10.1016/j.crma.2019.09.007

Affiliations des auteurs :

Didier Bresch ¹ ; Pierre-Emmanuel Jabin ² ; Zhenfu Wang ³

¹ LAMA – UMR5127 CNRS, Bât. Le Chablais, Campus scientifique, 73376 Le Bourget-du-Lac, France
² CSCAMM and Dept. of Mathematics, University of Maryland, College Park, MD 20742, USA
³ Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA

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     author = {Didier Bresch and Pierre-Emmanuel Jabin and Zhenfu Wang},
     title = {On mean-field limits and quantitative estimates with a large class of singular kernels: {Application} to the {Patlak{\textendash}Keller{\textendash}Segel} model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {708--720},
     publisher = {Elsevier},
     volume = {357},
     number = {9},
     year = {2019},
     doi = {10.1016/j.crma.2019.09.007},
     language = {en},
}

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Didier Bresch; Pierre-Emmanuel Jabin; Zhenfu Wang. On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak–Keller–Segel model. Comptes Rendus. Mathématique, Volume 357 (2019) no. 9, pp. 708-720. doi : 10.1016/j.crma.2019.09.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.09.007/

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