Boundedness for a 3D chemotaxis–Stokes system with porous medium diffusion and tensor-valued chemotactic sensitivity

Abstract

This paper deals with the following chemotaxis–Stokes system

$$\begin{aligned} \left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c), &{}\quad x\in \Omega ,\,\, t>0,\\ c_t+u\cdot \nabla c=\Delta c-nf(c),&{}\quad x\in \Omega , \,\,t>0,\\ u_t=\Delta u+\nabla P+n\nabla \phi ,&{}\quad x\in \Omega , \,\,t>0,\\ \nabla \cdot u=0,&{}\quad x\in \Omega , \,\,t>0\\ \end{array} \right. \end{aligned}$$

under no-flux boundary conditions in a bounded domain $\Omega \subset \mathbb {R}^{3}$ with smooth boundary, where $m\ge 1$, $\phi \in W^{1,\infty }(\Omega )$, f and S are given functions with values in $[0,\,\infty )$ and $\mathbb {R}^{3\times 3}$, respectively. Here S satisfies $|S(x,n,c)|<S_0(c)(1+n)^{-\alpha }$ with $\alpha \ge 0$ and some nonnegative nondecreasing function $S_0$. With the tensor-valued sensitivity S, this system does not possess energy-type functionals which seem to be available only when S is a scalar function. We can establish a priori estimation to overcome this difficulty and explore a relationship between m and $\alpha$, i.e., $m+\alpha >\frac{7}{6}$, which insures the global existence of bounded weak solution. Our result covers completely and improves the recent result by Wang and Cao (Discrete Contin Dyn Syst Ser B 20:3235–3254, 2015) which asserts, just in the case $m=1$, the global existence of solutions, but without boundedness, and that by Winkler (Calc Var Partial Differ Equ 54:3789–3828, 2015) which only involves the case of $\alpha =0$ and requires the convexity of the domain.