Convergence of boundary layers for the Keller–Segel system with singular sensitivity in the half-plane

Qianqian Hou, Zhian Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

8 Citations (Scopus)

Abstract

Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63]), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by ε≥0. Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with ε=0) plus the inner layer as ε→0, where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63], and open a new window in the future theoretical study of chemotaxis models.

Original languageEnglish
Pages (from-to)251-287
Number of pages37
JournalJournal des Mathematiques Pures et Appliquees
Volume130
DOIs
Publication statusPublished - Oct 2019

Keywords

  • Boundary layers
  • Chemotaxis
  • Energy estimates
  • Logarithmic singularity

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this