An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion

This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\nabla c),\quad x\in \Omega, t>0, \\ c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0, \quad x\in \Omega, t>0 \end{array}\right.$$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where $\phi\in W^{1,\infty}(\Omega),m>0$. It is proved that if $m>\frac{4}{3}$, then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system $(KSF)$, which in view of the known results for the fluid-free system mentioned below (see Introduction) is an optimal restriction on $m$.