Global existence and boundedness of weak solutions to a chemotaxis–Stokes system with rotational flux term

Abstract

In this paper, the three-dimensional chemotaxis–Stokes system

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

posed in a bounded domain $\Omega \subset {\mathbb {R}}^3$ with smooth boundary is considered under the no-flux boundary condition for n, c and the Dirichlect boundary condition for u under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity S(x, n, c) satisfies $S(x,n,c)<n^{l-2}\widetilde{S}(c)$ with $l>2$ for some non-decreasing function $\widetilde{S}\in C^{2}([0,\infty ))$. In the present work, it is shown that the weak solution is global in time and bounded while $m>m^\star (l)$, where

$$\begin{aligned} m^\star (l)= \left\{ \begin{array}{ll} l-\frac{5}{6},&{}\quad \mathrm {if}\ \ \frac{31}{12}\ge l>2,\\ \frac{7}{5}l-\frac{28}{15}, &{}\quad \mathrm {if}\ \ l>\frac{31}{12}. \end{array}\right. \end{aligned}$$