A functional framework for the Keller–Segel system: Logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities

Jean Dolbeault; Juan Campos

doi:10.1016/j.crma.2012.10.023

Partial Differential Equations/Functional Analysis

A functional framework for the Keller–Segel system: Logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities
[Un cadre fonctionnel pour le système de Keller–Segel : inégalité logarithmique de Hardy–Littlewood–Sobolev et inégalités de trou spectral reliées]

Présenté par : the Editorial Board

Jean Dolbeault ¹ ; Juan Campos ^1,²

¹ Ceremade (UMR CNRS no. 7534), université Paris-Dauphine, place de-Lattre-de-Tassigny, 75775 Paris 16, France
² Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 949-954.

Résumé
Abstract

Cette Note est consacrée à plusieurs inégalités fonctionnelles déduites dʼune forme particulière de lʼinégalité logarithmique de Hardy–Littlewood–Sobolev, qui est bien adaptée à la caractérisation des solutions stationnaires dʼun système de Keller–Segel écrit en variables auto-similaires, dans le cas dʼune masse sous-critique. Pour le problème dʼévolution correspondant, ces inégalités fonctionnelles jouent un rôle important dans lʼidentification des taux de convergence des solutions vers la solution stationnaire de même masse.

This Note is devoted to several inequalities deduced from a special form of the logarithmic Hardy–Littlewood–Sobolev, which is well adapted to the characterization of stationary solutions of a Keller–Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass.

Reçu le : 2012-06-09
Accepté le : 2012-10-24
Publié le : 2012-11-01

DOI : 10.1016/j.crma.2012.10.023

Affiliations des auteurs :

Jean Dolbeault ¹ ; Juan Campos ^{1, 2}

¹ Ceremade (UMR CNRS no. 7534), université Paris-Dauphine, place de-Lattre-de-Tassigny, 75775 Paris 16, France
² Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

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     author = {Jean Dolbeault and Juan Campos},
     title = {A functional framework for the {Keller{\textendash}Segel} system: {Logarithmic} {Hardy{\textendash}Littlewood{\textendash}Sobolev} and related spectral gap inequalities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {949--954},
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     volume = {350},
     number = {21-22},
     year = {2012},
     doi = {10.1016/j.crma.2012.10.023},
     language = {en},
}

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Jean Dolbeault; Juan Campos. A functional framework for the Keller–Segel system: Logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities. Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 949-954. doi : 10.1016/j.crma.2012.10.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.023/

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