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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Existence of solutions of the hyperbolic Keller-Segel model
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by Benoît Perthame and Anne-Laure Dalibard PDF
Trans. Amer. Math. Soc. 361 (2009), 2319-2335

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
  • MR Author ID: 791274
  • Email: dalibard@ceremade.dauphine.fr, Anne_Laure.Dalibard@ens.fr
  • Received by editor(s): December 18, 2006
  • Published electronically: December 15, 2008
  • © Copyright 2008 by the authors
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 2319-2335
  • MSC (2000): Primary 35D05, 35L60, 92C17
  • DOI: https://doi.org/10.1090/S0002-9947-08-04656-4
  • MathSciNet review: 2471920