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Kinetic models for chemotaxis and their drift-diffusion limits. (English) Zbl 1052.92005
Summary: 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 H. G. Othmer, S. R. Dunbar and W. Alt, J. Math. Biol. 26, 263–298 (1988; Zbl 0713.92018)], 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 [E. F. Keller and L. A. Segel, J. Theor. Biol. 26, 399–415 (1970)]. 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.

MSC:
92C17 Cell movement (chemotaxis, etc.)
35K90 Abstract parabolic equations
82D99 Applications of statistical mechanics to specific types of physical systems
35K99 Parabolic equations and parabolic systems
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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