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

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} onumber left{ begin{aligned} &n_t+ucdot abla n= ablacdotleft(| abla n|^{p-2} abla nright)- ablacdot(n abla c), &xinOmega, t>0, &c_t+ucdot abla c=Delta c-nc,&xinOmega, t>0, &u_t=Delta u+ abla P+n ablaphi ,&xinOmega, t>0, & ablacdot u=0, &xinOmega, t>0, end{aligned} right. end{equation} where $phiin W^{2,infty}(Omega)$ 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 $n_0geq 0$ and $c_0geq 0$.