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MiS Preprint
65/2003
Global solutions of nonlinear transport equations for chemosensitive movement
Hyung Ju Hwang, Kyungkeun Kang and Angela Stevens
Abstract
A widespread phenomenon in moving microorganisms and cells is their ability to orient themselves in dependence of chemical signals. In this paper we discuss kinetic models for chemosensitive movement, which take into account also evaluations of gradient fields of chemical stimuli which subsequently influence the motion of the respective microbiological species.
The basic type of model was discussed by Alt [J. Math. Biol. 9 (1980), 147--177, J. Reine Angew. Math. 322 (1981), 15--41] and in Othmer, Dunbar, and Alt [J. Math. Biol. 26 (1988), no. 3, 263--298].
Chalub, Markowich, Perthame and Schmeiser rigorously proved that, in three dimensions, these kind of kinetic models lead to the classical Keller-Segel model as its drift-diffusion limit when the equation for the chemo-attractant is of elliptic type [ANUM preprint 5/02, ANUM preprint 14/02].
In [MPI MIS, Leipzig, Preprint 19 (2003)] it was proved that the macroscopic diffusion limit exists in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type. So far in the rigorous derivations only the density of the chemo-attractant was supposed to influence the motion of the chemosensitive species. Here we are concerned with the effects of evaluations of gradient fields of the chemical stimulus on the behavior of the chemosensitive species. In the macroscopic limit some effects result in a change of the classical parabolic Keller-Segel model for chemotaxis. Under suitable structure conditions global solutions for the kinetic models can be shown.
