Global existence and boundedness in a chemotaxis-Stokes system with slow $p$-Laplacian diffusion

Weirun Tao, Yuxiang Li

Published 2018-09-10Version 1

This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion \begin{equation} \left\{ \begin{aligned} &n_t+u\cdot\nabla n=\nabla\cdot\left(|\nabla n|^{p-2}\nabla n\right)-\nabla\cdot(n\nabla c), &x\in\Omega,\ t>0,\ \ &c_t+u\cdot\nabla c=\Delta c-nc,&x\in\Omega,\ t>0,\ \ &u_t=\Delta u+\nabla P+n\nabla\phi ,&x\in\Omega,\ t>0,\ \ &\nabla\cdot u=0, &x\in\Omega,\ t>0,\ \ \end{aligned} \right. \end{equation} where $\phi\in W^{2,\infty}(\Omega)$ is the gravitational potential. It is proved that global bounded weak solutions exist whenever $p>\frac{23}{11}$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0\geq 0$ and $c_0\geq 0$.