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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities


Authors: Vincent Calvez and José Antonio Carrillo
Journal: Proc. Amer. Math. Soc. 140 (2012), 3515-3530
MSC (2010): Primary 35B40, 92C17
Published electronically: February 23, 2012
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Abstract: We analyze the rate of convergence towards self-similarity for the subcritical Keller-Segel system in the radially symmetric two-dimensional case and in the corresponding one-dimensional case for logarithmic interaction. We measure convergence in the Wasserstein distance. The rate of convergence towards self-similarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the Logarithmic Hardy-Littlewood-Sobolev inequality in the one-dimensional and radially symmetric two-dimensional cases based on optimal transport arguments. In addition we prove that the one-dimensional equation is a contraction with respect to Fourier distance in the subcritical case.


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Vincent Calvez
Affiliation: École Normale Supérieure de Lyon, UMR CNRS 5669 “Unité de Mathématiques Pures et Appliquées”, 46 allée d’Italie, F-69364 Lyon Cedex 07, France
Email: vincent.calvez@ens-lyon.fr

José Antonio Carrillo
Affiliation: Institució Catalana de Recerca i Estudis Avançats and Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email: carrillo@mat.uab.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11306-1
PII: S 0002-9939(2012)11306-1
Received by editor(s): July 16, 2010
Received by editor(s) in revised form: April 12, 2011
Published electronically: February 23, 2012
Communicated by: Walter Craig
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.