Boundary layer analysis for a 2-D Keller-Segel model

Linlin Meng,Shu Wang,Wen-Qing Xu

doi:10.1515/math-2020-0093

Abstract

Abstract We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.

Highlights

Chemotaxis is the directed movement of cells in response to certain chemical substances in their environment, and it plays an important role in many biological processes
In 1970, Keller and Segel [1] proposed the following system to describe the aggregation of cellular slime mold toward higher concentration of the chemical signal nt = ∇⋅(D(n)∇n) − ∇⋅(ψ(n)∇c), x ∈ Ω, t > 0
N(x, 0) = n0(x), c(x, 0) = c0(x), x ∈ Ω, where Ω is a bounded domain in N with smooth boundary ∂Ω, ∂ν denotes the directional derivative in the outer normal direction of ∂Ω, n(x, t) denotes the bacterial population density and c(x, t) denotes the chemical concentration

Summary

Introduction

Chemotaxis is the directed movement of cells in response to certain chemical substances in their environment, and it plays an important role in many biological processes. Under suitable growth conditions on D(n) and S(x, n, c), Wang [12] showed that the system possesses at least one global bounded weak solution for any sufficiently smooth non-negative initial data. Due to the mismatch of the boundary conditions for systems (2) and (3), an important boundary layer correction term at the O( ε ) order has to be added This is a pivotal observation and the starting point of our current work. Hou et al [36,37] have made some very important contributions to the study of boundary layers of chemotaxis systems related to the Keller-Segel model.

Asymptotic expansions

Error equations and main result