[Un cadre fonctionnel pour le système de Keller–Segel : inégalité logarithmique de Hardy–Littlewood–Sobolev et inégalités de trou spectral reliées]
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.
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@article{CRMATH_2012__350_21-22_949_0,
author = {Dolbeault, Jean and Campos, Juan},
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},
publisher = {Elsevier},
volume = {350},
number = {21-22},
year = {2012},
doi = {10.1016/j.crma.2012.10.023},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.10.023/}
}
TY - JOUR AU - Dolbeault, Jean AU - Campos, Juan TI - A functional framework for the Keller–Segel system: Logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities JO - Comptes Rendus. Mathématique PY - 2012 SP - 949 EP - 954 VL - 350 IS - 21-22 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.10.023/ DO - 10.1016/j.crma.2012.10.023 LA - en ID - CRMATH_2012__350_21-22_949_0 ER -
%0 Journal Article %A Dolbeault, Jean %A Campos, Juan %T A functional framework for the Keller–Segel system: Logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities %J Comptes Rendus. Mathématique %D 2012 %P 949-954 %V 350 %N 21-22 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.10.023/ %R 10.1016/j.crma.2012.10.023 %G en %F CRMATH_2012__350_21-22_949_0
Dolbeault, Jean; Campos, Juan. A functional framework for the Keller–Segel system: Logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities. Comptes Rendus. Mathématique, Tome 350 (2012) no. 21-22, pp. 949-954. doi : 10.1016/j.crma.2012.10.023. http://www.numdam.org/articles/10.1016/j.crma.2012.10.023/
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