Drift-diffusion Limits of Kinetic Models for Chemotaxis: a Generalization

We study a kinetic model for chemotaxis introduced by Othmer, Dunbar, and Alt [22], which was motivated by earlier results of Alt, presented in [1], [2]. In two papers by Chalub, Markowich, Perthame and Schmeiser, it was rigorously shown that, in three dimensions, this kinetic model leads to the classical KellerSegel model as its drift-diffusion limit when the equation of the chemo-attractant is of elliptic type [4], [5]. As an extension of these works we prove that such kinetic models have a macroscopic diffusion limit in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type, which is the original version of the chemotaxis model. Introduction In [16] and [17] Keller and Segel introduced and studied a model for aggregation of the cellular slime mold Dictyostelium discoideum due to cyclic AMP which is an attractive chemical signal for the amoebae. The model is of advection-diffusion type and consists of two coupled parabolic equations (1) ∂ρ ∂t = ∇ · (D(ρ, S)∇ρ− χ(ρ, S)ρ∇S), (2) ∂S ∂t = D0∆S + φ(ρ, S). Here ρ = ρ(x, t) denotes the cell density and S = S(x, t) is the density of the chemo-attractant. The cells are attracted by the chemical and χ denotes their chemotactic sensitivity. The substance S diffuses and is also produced by the amoebae. Typically φ(ρ, S) is given by (3) φ(ρ, S) = αρ− βS, α, β ≥ 0. where −βS is the loss term due to decay or external chemical reactions. The first rigorous derivation of the macroscopic chemotaxis equations from microscopic models, namely interacting stochastic many particle sytems, was given in [26]. In [4] a kinetic model of the equation (1) was discussed with a reduced version of the equation (2) which is the Poisson equation without decay term (4) −∆S = αρ. The following kinetic equation for the oriented cell density f = f(x, v, t) ≥ 0 is considered in [4, page 3] (5) ∂f ∂t + v · ∇xf = ∫ V (T [S]f ′ − T ∗[S]f)dv′, where x, v, and t indicate position, velocity, and time, respectively. Here the abbreviations f ′ = f(x, v′, t), T [S] = T [S](x, v, v′, t) and T ∗[S] = T [S](x, v′, v, t) are used. The cell density ρ fulfills