Convergence Rate of Zero Viscosity Limit on Large Amplitude Solution to a Conservation Laws Arising in Chemotaxis

TL;DR

This paper analyzes the convergence rates of solutions to a chemotaxis-related conservation law system as the diffusion parameter approaches zero, establishing specific rates for both Cauchy and boundary value problems.

Contribution

It provides the first rigorous analysis of the zero viscosity limit for large amplitude solutions in a chemotaxis model, including explicit convergence rates.

Findings

Convergence rate in $L^ abla$-norm is $O(\epsilon)$ for the Cauchy problem.

Convergence rate in $L^ abla$-norm is $O(\epsilon^{3/4})$ for the initial-boundary value problem.

Global unique solvability is established using the energy method.

Abstract

In this paper, we investigate large amplitude solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. For the Cauchy problem and initial-boundary value problem, the global unique solvability is proved based on the energy method. In particular, our main purpose is to investigate the convergence rates as the diffusion parameter $\epsilon$ goes to zero. It is shown that the convergence rates in $L^\infty$-norm are of the order $(\epsilon)$ and $O(\epsilon^{3/4})$ corresponding to the Cauchy problem and the initial-boundary value problem respectively.