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

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.