Kinetic models for chemotaxis and their drift-diffusion limits. (English) Zbl 1052.92005

Chalub, Fabio A. C. C.; Markowich, Peter A.; Perthame, Benoît; Schmeiser, Christian

Monatsh. Math. 142, No. 1-2, 123-141 (2004).

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

Keywords:

chemotaxis; drift-diffusion; limits; kinetic models

Citations:

Zbl 0713.92018