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Existence of solutions of the hyperbolic Keller-Segel model


Authors: Benoît Perthame and Anne-Laure Dalibard
Journal: Trans. Amer. Math. Soc. 361 (2009), 2319-2335
MSC (2000): Primary 35D05, 35L60, 92C17
Published electronically: December 15, 2008
MathSciNet review: 2471920
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Abstract: We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities.

This is a first order quasilinear equation, its flux depends on space and time via the solution to an elliptic PDE in which the right-hand side is the solution to the hyperbolic equation. This model lacks strong compactness or contraction properties. Our purpose is to prove the existence of an entropy solution obtained, as usual, in passing to the limit in a sequence of solutions to the parabolic approximation.

The method consists in the derivation of a kinetic formulation for the weak limit. The specific structure of the limiting kinetic equation allows for a `rigidity theorem', which identifies some property of the solution (which might be nonunique) to this kinetic equation. This is enough to deduce a posteriori the strong convergence of a subsequence.


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Additional Information

Benoît Perthame
Affiliation: UMR 7598 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie-Paris 6, F-75005 Paris, France
Email: perthame@ann.jussieu.fr

Anne-Laure Dalibard
Affiliation: Ceremade, Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris cedex 16, France
Address at time of publication: Département de mathématiques et applications, UMR 8553, Ecole normale supérieure, 45 rue d’Ulm, F-75005 Paris, France
Email: dalibard@ceremade.dauphine.fr, Anne_Laure.Dalibard@ens.fr

DOI: https://doi.org/10.1090/S0002-9947-08-04656-4
Keywords: Keller-Segel system, kinetic formulation, compactness, entropy inequalities.
Received by editor(s): December 18, 2006
Published electronically: December 15, 2008
Article copyright: © Copyright 2008 by the authors