Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal

The chemotaxis-Navier-Stokes system$\{\begin{array}{cc}& n_t+u\cdot \nabla n=\Delta n-\nabla \cdot \left(n\chi \right(n\left)\nabla c\right),\\ & c_t+u\cdot \nabla c=\Delta c-nc,\\ & u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \varphi ,\\ & \nabla \cdot u=0\end{array}$ is considered in a smoothly bounded domain $\Omega \subset R^2$ under the boundary conditions$(\nabla n-n\chi (n\left)\nabla c\right)\cdot \nu =0,c=c_{\star },u=0,x\in \partial \Omega ,t>0$ with a given nonnegative constant $c_{\star }$. It is shown that if $\chi \in C^2\left(\right[0,\infty \left)\right)$ and $\chi \left(n\right)\to 0$ as $n\to \infty$, then for all suitably regular initial data, an associated initial value problem possesses a globally defined and bounded classical solution. When ${\Vert n_0\Vert }_{L^1\left(\Omega \right)}$ and ${\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)}$ are suitably small and $c_{\star }\equiv 0$, we further obtain the stabilization of the classical solution.