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{eqnarray*} \left\{\begin{array}{lll} \medskip n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n S(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0, \medskip c_t+u\cdot\nabla c=\Delta c-nf(c),&x\in\Omega,\ \ t>0, \medskip u_t+\nabla P=\Delta u +n\nabla\phi,&x\in\Omega,\ \ t>0, \nabla\cdot u=0, &x\in\Omega,\ \ t>0,, \end{array}\right. \end{eqnarray*} 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 present work, it is shown that the weak solution is global in time and bounded while $m>m^\star(l)$, where \begin{eqnarray*} m^\star(l)= \left\{\begin{array}{lll} \medskip l-\frac{5}{6},\ &\mathrm{if}\ \frac{31}{12}\geq l>2, \medskip \frac{7}{5}l-\frac{28}{15},\ &\mathrm{if}\ l>\frac{31}{12}. \end{array}\right. \end{eqnarray*}