Yuan-yuan KE, Jia-Shan ZHENG
In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed, $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, \\ c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, \ \ t>0, \\ u_t+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, \ \ t>0, \\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array} \right. (KSS) $$ subject to the boundary conditions $(\nabla n-nS(x,n,c)\nabla c)\cdot\nu=\nabla c\cdot\nu=0$ and $u=0$, and suitably regular initial data $(n_0 (x),c_0 (x),u_0 (x))$, where $\Omega\subset \mathbb{R}^3$ is a bounded domain with smooth boundary $\partial\Omega$. Here $S$ is a chemotactic sensitivity satisfying $|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$ with some $C_S> 0$ and $\alpha> 0$. The greatest contribution of this paper is to consider the large time behavior of solutions for the system (KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system (KSS) decays to $(\frac{1}{|\Omega|}\int_{\Omega}n_0,\frac{1}{|\Omega|}\int_{\Omega}n_0,0)$ exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient $C_S$ of chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.
