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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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  • 1. Emmanuel Audusse and Benoît Perthame, Uniqueness for a scalar conservation law with discontinuous flux via adapted entropies, Proc. Royal Soc. of Edinburgh 135A (2005), 1-13.
  • 2. F. Bachmann and J. Vovelle, Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients, Comm. in PDEs 31 (2006), 371-395. MR 2209759 (2008b:35170)
  • 3. Matania Ben-Artzi and Philippe LeFloch, Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Annales I.H.P., Anal. Non Linéaire 24 (2007), 989-1008. MR 2371116
  • 4. Martin Burger, Yasmin Dolak, and Christian Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, (2006), preprint.
  • 5. Vincent Calvez and José A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9) 86 (2006), no. 2, 155-175. MR 2247456
  • 6. Constantine M. Dafermos, Hyperbolic conservation laws in Continuum Physics, Grundlehren Mathematischen Wissenschaften, no. GM 325, Springer-Verlag, Berlin, Heidelberg, New York, 1999. MR 1763936 (2001m:35212)
  • 7. Anne-Laure Dalibard, Kinetic formulation for a parabolic conservation law. Application to homogenization, SIAM J. Math. Anal. 39 (2007), 891-915. MR 2349870
  • 8. -, Kinetic formulation for heterogeneous scalar conservation laws, Annales de l'IHP (C) : Analyse Non Linéaire 23 (2006), 475-498. MR 2245753 (2007h:35223)
  • 9. R. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. (1989), no. 98, 511-547. MR 1022305 (90j:34004)
  • 10. Yasmin Dolak and Christian Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math. 66 (2005), no. 1, 286-308 (electronic). MR 2179753 (2006i:35180)
  • 11. Thomas Hillen and Kevin Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math. (2001), no. 26, 280-301. MR 1826309 (2002c:92008)
  • 12. -, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quarterly 4 (2002), no. 10, 501-543. MR 2052525 (2005a:92008)
  • 13. Dirk Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), no. 3, 103-165. MR 2013508
  • 14. -, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein. 106 (2004), no. 2, 51-69. MR 2073515
  • 15. Pierre-Louis Lions, Benoît Perthame, and Eitan Tadmor, Formulation cinétique des lois de conservation scalaires multidimensionnelles, C.R. Acad. Sci. Paris (1991), no. 312, 97-102, Série I. MR 1086510 (91k:35156)
  • 16. -, A kinetic formulation of multidimensional conservation laws and related equations, J. Amer. Math. Soc. (1994), no. 7, 169-191.
  • 17. J. D. Murray, Mathematical biology. II, third ed., Interdisciplinary Applied Mathematics, vol. 18, Springer-Verlag, New York, 2003, Spatial models and biomedical applications. MR 2004b:92001
  • 18. Hans G. Othmer and Angela Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (1997), no. 4, 1044-1081. MR 1462051 (99b:92001)
  • 19. Felix Otto, $ L\sp 1$-contraction and uniqueness for quasilinear elliptic-parabolic equations, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 8, 1005-1010. MR 1360562 (96i:35093)
  • 20. Benoît Perthame, Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure, J. Math. Pures et Appl. (1998), no. 77, 1055-1064. MR 1661021 (2000e:35141)
  • 21. -, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, no. 21, Oxford University Press, 2002. MR 2064166 (2005d:35005)
  • 22. -, Transport equations arising in biology, Frontiers, Birkhäuser, 2006.
  • 23. Takasi Senba and Takashi Suzuki, Applied analysis, Imperial College Press, London, 2004, Mathematical methods in natural science. MR 2093755
  • 24. Denis Serre, Systèmes de lois de conservation I et II, Diderot Editeur Arts et Sciences, 1996. MR 1459988 (99b:35139); MR 1459989 (99e:35144)
  • 25. Angela Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math. 61 (2000), 183-212. MR 1776393 (2001i:92008)

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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

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

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

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