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Monatshefte für Mathematik

, Volume 142, Issue 1–2, pp 123–141 | Cite as

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

  • Fabio A. C. C. Chalub
  • Peter A. Markowich
  • Benoît Perthame
  • Christian Schmeiser
Article

Abstract.

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant 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.

2000 Mathematics Subject Classification: 92B05, 82B40 
Key words: Chemotaxis, drift-diffusion limits, kinetic models 

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Copyright information

© Springer-Verlag/Wien 2004

Authors and Affiliations

  • Fabio A. C. C. Chalub
    • 1
  • Peter A. Markowich
    • 1
  • Benoît Perthame
    • 2
  • Christian Schmeiser
    • 3
  1. 1.Universität WienAustria
  2. 2.École Normale SupérieureParisFrance
  3. 3.Technische Universität WienAustria

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