A new result for the global existence and boundedness of weak solutions to a chemotaxis-Stokes system with rotational flux term

Abstract

In this paper, we consider a three-dimensional chemotaxis-Stokes system

$$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),&\quad&x\in \Omega , t>0,\\&c_t+u\cdot \nabla c=\Delta c-nf(c),&\quad&x\in \Omega ,t>0,\\&u_t+\nabla P=\Delta u+n\nabla \phi ,&\quad&x\in \Omega ,t>0,\\&\nabla \cdot u=0,&\quad&x\in \Omega ,t>0,\\&(\nabla n^m- n S(x,n,c)\cdot \nabla c)\cdot \nu =\frac{\partial c}{\partial \nu }=0,u=0,&\quad&x\in \partial \Omega ,t>0,\\&n(x,0)=n_0(x),c(x,0)=c_0(x),u(x,0)=u_0(x),&\quad&x\in \Omega \end{aligned} \right. \end{aligned}$$

(0.1)

in a bounded domain $\Omega \subset {\mathbb {R}}^3$ with smooth boundary, where $\phi$, f and S are given functions with values in $\Omega$, $[0,\infty )$ and ${\mathbb {R}}^{3\times 3}$, respectively, under the no-flux boundary condition for n, c and Dirichlet boundary condition for u. It is also required that $f\in C^1([0,\infty ))$ is locally bounded in $[0,\infty )$, that S satisfies $|S(x,n,c)|\le n^{l-2}S_0(c)$ with some nondecreasing $S_0:[0,\infty )\rightarrow [0,\infty )$, and that m fulfills

$$\begin{aligned} m>\frac{5}{3}l-\frac{20}{9}\quad \text {and}\quad m>-\frac{3}{4}l+\frac{21}{8}\quad \text {with}\quad \frac{25}{12}\ge l>2. \end{aligned}$$

(0.2)

Then, with any reasonably regular initial data, the corresponding initial-boundary problem for (0.1) processes a global in time and bounded solution. This result extends the previous global boundedness result with $m>m_*(l)$, where

$$\begin{aligned} m_*(l)=\left\{ \begin{aligned}&l-\frac{5}{6},&\quad&\text {if }\,\frac{31}{12}\ge l>2,\\&\frac{7}{5}l-\frac{28}{15},&\quad&\text {if }\,l>\frac{31}{12}, \end{aligned} \right. \end{aligned}$$

([15]) since both

$$\begin{aligned} l-\frac{5}{6}\ge \frac{5}{3}l-\frac{20}{9} \end{aligned}$$

and

$$\begin{aligned} l-\frac{5}{6}\ge -\frac{3}{4}l+\frac{21}{8} \end{aligned}$$

hold on $\frac{25}{12}\ge l>2$.