Global solutions of nonlinear transport equations for chemosensitive movement

Hyung Ju Hwang, Kyungkeun Kang, and Angela Stevens

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Submission date: 19. Jul. 2003
Pages: 25
published as: Erratum: 'Global solutions of nonlinear transport equations for chemosensitive movement' [SIAM J. Math. Anal. 36 (2005), no. 4, 1177--1199].
In: SIAM journal on mathematical analysis, 39 (2007) 3, p. 1018-1021 
DOI number (of the published article): 10.1137/070680813
published as: Global solutions of nonlinear transport equations for chemosensitive movement.
In: SIAM journal on mathematical analysis, 36 (2005) 4, p. 1177-1199 
DOI number (of the published article): 10.1137/S0036141003431888
Bibtex
MSC-Numbers: 35K55, 45K05, 82C70, 92C17
Keywords and phrases: chemosensitive movement, sensing of gradient fields, nonlinear transport equations, global solutions, drift-diffusion limit, keller-segel model
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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.Erratum published under the same title in SIAM Journal of Mathematical Analysis 39(2007)3, p. 1018-1021