Inviscid limit for the hyperbolic Keller--Segel equations in $B_{\infty ,1}^1$

Abstract

In this paper, we study the inviscid limit for the hyperbolic Keller--Segel equations with logistic sensitivity $1-u$. We establish a common existence time and a priori estimates uniform with respect to the diffusion coefficient in the critical Besov spaces $B_{p,1}^{1+d/p}(R^d)$ with $1\le p\le \infty$. For $1<p<\infty$, we obtain explicit convergence rates in $B_{p,1}^0$, and we also give estimates for the case $p=1$. For initial data in $B_{\infty ,1}^1$, we prove convergence at rate $O({\epsilon }^{1/2})$ in $L^{\infty }$, a bound of order $O({\epsilon }^{1/2}\log (e/\epsilon ))$ in $B_{\infty ,1}^0$, and strong convergence in $L^{\infty }(0,T;B_{\infty ,1}^1)$. Additional regularity of the initial data yields improved convergence rates. We construct a fixed initial datum showing that the exponent $1/2$ in $L^{\infty }$ is optimal.