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

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.