Fluid interaction does not affect the critical exponent in a three-dimensional Keller–Segel–Stokes model

Abstract

The coupled chemotaxis fluid system

$$\begin{aligned} \left\{ \begin{array}{llc} n_t=\nabla \cdot ( (n+1)^{m-1}\nabla n)-\nabla \cdot (n(n+1)^{m-\alpha -1} \nabla c)-u\cdot \nabla n, &{}(x,t)\in \Omega \times (0,T),\\ \displaystyle c_t=\Delta c-c+n-u\cdot \nabla c, &{}(x,t)\in \Omega \times (0,T),\\ \displaystyle u_t=\Delta u+\nabla P+n\nabla \phi ,\quad \nabla \cdot u=0, &{}(x,t)\in \Omega \times (0,T),\\ \displaystyle \nabla c\cdot \nu =\nabla n\cdot \nu =0, \;\; u=0,&{}(x,t)\in \partial \Omega \times (0,T),\\ n(x,0)=n_{0}(x),\quad c(x,0)=c_{0}(x),\quad u(x,0)=u_0(x) &{} x\in \Omega , \end{array} \right. \end{aligned}$$

is considered in a bounded domain $\Omega \subset \mathbb {R}^3$, with smooth boundary, where $m,\alpha \in \mathbb {R}$. It is known that in the fluid-free case ($u\equiv 0$), the system admits a global bounded solution if $\alpha >\frac{1}{3}$. The purpose of this paper is to overcome difficulties arising from the presence of fluid interaction and to show that the same conclusion holds if $m>-\frac{1}{3}$. In the case $m\le -\frac{1}{3}$ and $\alpha >\frac{1}{3}$, global existence of classical solution will be shown.