A kinetic chemotaxis model and its diffusion limit in slab geometry

TL;DR

This paper develops a kinetic model for chemotaxis in slab geometry, proves local existence and uniqueness of solutions, and demonstrates their convergence to a Keller-Segel diffusion model with explicit rates.

Contribution

It introduces a kinetic chemotaxis model in slab geometry and rigorously establishes its diffusion limit to the Keller-Segel model, including convergence rates.

Findings

Proved local existence and uniqueness of solutions.

Established convergence to Keller-Segel model in the high tumbling rate limit.

Derived explicit convergence rates under regularity assumptions.

Abstract

Chemotaxis describes the intricate interplay of cellular motion in response to a chemical signal. We here consider the case of slab geometry which models chemotactic motion between two infinite membranes. Like previous works, we are particularly interested in the asymptotic regime of high tumbling rates. We establish local existence and uniqueness of solutions to the kinetic equation and show their convergence towards solutions of a parabolic Keller-Segel model in the asymptotic limit. In addition, we prove convergence rates with respect to the asymptotic parameter under additional regularity assumptions on the problem data. Particular difficulties in our analysis are caused by vanishing velocities in the kinetic model as well as the occurrence of boundary terms.