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Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

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Nonlinear Differential Equation Models

Abstract

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemoattractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.

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Author information

Author notes

  1. Fabio A. C. C. Chalub

    Present address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

Authors and Affiliations

  1. Institut für Mathematik, Universität Wien, Nordbergstrasse 15, 1090, Wien, Austria

    Fabio A. C. C. Chalub & Peter A. Markowich

  2. Département de Mathématiques et Applications (UMR8553), École Normale Supérieure, 75230, Paris, France

    Benoît Perthame

  3. Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, Hauptstraße 8-10, 1040, Wien, Austria

    Christian Schmeiser

Authors
  1. Fabio A. C. C. Chalub
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  2. Peter A. Markowich
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  3. Benoît Perthame
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  4. Christian Schmeiser
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Editor information

Editors and Affiliations

  1. FB Mathematik und Informatik, Johannes Gutenberg Universität Mainz, Hong Kong

    Ansgar Jüngel

  2. Centro de Modelamiento Matemático, Santiago de Chile

    Raul Manasevich

  3. Fakultät fur Mathematik, Universität Wien, Germany

    Peter A. Markowich

  4. Institute of Mathematics, KTH, Stockholm

    Henrik Shahgholian

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© 2004 Springer-Verlag Wien

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Chalub, F.A.C.C., Markowich, P.A., Perthame, B., Schmeiser, C. (2004). Kinetic Models for Chemotaxis and their Drift-Diffusion Limits. In: Jüngel, A., Manasevich, R., Markowich, P.A., Shahgholian, H. (eds) Nonlinear Differential Equation Models. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0609-9_10

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  • DOI: https://doi.org/10.1007/978-3-7091-0609-9_10

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  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-7208-7

  • Online ISBN: 978-3-7091-0609-9

  • eBook Packages: Springer Book Archive

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