Convergence of boundary layers of chemotaxis models with physical boundary conditions~II: Non-degenerate

TL;DR

This paper proves the convergence of boundary-layer solutions in a chemotaxis model with physical boundary conditions as the diffusion parameter approaches zero, extending previous results to non-degenerate initial data.

Contribution

It introduces a new regularization strategy for boundary-layer profiles and establishes convergence for all positive times, overcoming regularity challenges present in prior work.

Findings

Proves convergence of solutions as diffusion parameter tends to zero.

Develops a novel regularization technique for boundary-layer profiles.

Achieves uniform-in-ε estimates for all positive times.

Abstract

This paper establishes the convergence of boundary-layer solutions of the consumption type Keller-Segel model with non-degenerate initial data subject to physical boundary conditions, which is a sequel of \cite{Corrillo-Hong-Wang-vanishing} on the case of degenerate initial data. Specifically, we justify that the solution with positive chemical diffusion rate $\varepsilon>0 $ converges to the solution with zero diffusion $\varepsilon=0 $ (outer-layer solution) plus the boundary-layer profiles (inner-layer solution) for any time $t>0$ as $ \varepsilon \rightarrow 0 $. Compared to \cite{Corrillo-Hong-Wang-vanishing}, the main difficulty in the analysis is the lack of regularity of the outer- and boundary-layer profiles since only the zero-order compatibility conditions for the leading-order boundary-layer profiles can be fulfilled with non-degenerate initial data. Our new strategy is to regularize the boundary-layer profiles with carefully designed corner-corrector functions and approximate the low-regularity leading-order boundary-layer profiles by higher-regularity profiles with regularized boundary conditions. By using delicate weight functions involving boundary-layer profiles to cancel the multi-scaled linear terms in the perturbed equations, we manage to obtain the requisite uniform-in-$ \varepsilon $ estimates for the convergence analysis. This cancellation technique enables us to prove the convergence to boundary-layer solutions for any time $ t >0 $, which is different from the convergence result in \cite{Corrillo-Hong-Wang-vanishing} which holds true only for some finite time depending on the Dirichlet boundary value.