Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Fabio A. C. C. Chalub (), Peter A. Markowich (), Benoît Perthame () and Christian Schmeiser ()
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Fabio A. C. C. Chalub: Universität Wien, Institut für Mathematik
Peter A. Markowich: Universität Wien, Institut für Mathematik
Benoît Perthame: École Normale Supérieure, Département de Mathématiques et Applications (UMR8553)
Christian Schmeiser: Technische Universität Wien, Institut für Angewandte und Numerische Mathematik

A chapter in Nonlinear Differential Equation Models, 2004, pp 123-141 from Springer

Abstract: 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.

Keywords: Chemotaxis; drift-diffusion limits; kinetic models (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-3-7091-0609-9_10

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