# American Institute of Mathematical Sciences

October  2015, 20(8): 2751-2759. doi: 10.3934/dcdsb.2015.20.2751

## Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system

 1 Department of Mathematics, Southeast University, Nanjing 211189, China, China

Received  November 2014 Revised  February 2015 Published  August 2015

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equation \begin{eqnarray*} \left\{\begin{array}{lll} n_t + u \cdot \nabla n = \Delta n - \chi\nabla\cdot(n \nabla c),&{} x\in\Omega,\ t>0,\\ c_t + u \cdot \nabla c = \Delta c - nc, &{} x \in \Omega,\ t>0,\\ u_t + \kappa(u\cdot\nabla)u = \Delta u + \nabla P + n\nabla\phi ,&{} x\in\Omega,\ t>0,\\ \nabla\cdot u=0, &{}x\in\Omega,\ t>0, \end{array}\right. \end{eqnarray*} in a bounded domain $\Omega\subset\mathbb{R}^2$. It is known that if $\chi>0$, $\kappa\in\mathbb{R}$ and $\phi\in C^2(\bar{\Omega})$, for sufficiently smooth initial data, the model possesses a unique global classical solution which satisfies $(n, c, u)\rightarrow(\bar{n}_0, 0, 0)$ as $t\rightarrow\infty$ uniformly with respect to $x\in\Omega$, where $\bar{n}_0:=\frac{1}{|\Omega|}\int_{\Omega}n(x, 0)dx$. In the present paper, we prove this solution converges to $(\bar{n}_0, 0, 0)$ exponentially in time.
Citation: Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751
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