A remark on the vanishing diffusivity limit of the Keller-Segel equations in Besov spaces

Abstract

It is shown in \cite[J. Differ. Equ., (2022)]{22jde} that given initial data $u_0\in B_{p,r}^s$ and for some $T>0$, the solutions of the parabolic-type Keller-Segel equations converge strongly in $L_T^{\infty }{B}_{p,r}^s$ to the hyperbolic Keller-Segel equations as the diffusivity parameter $\epsilon$ tends to zero. In this paper, we furthermore prove this solution maps do not converge uniformly with respect to the initial data $u_0$ as $\epsilon \to 0$ in the same topology of Besov spaces.