This paper is concerned with the two-dimensional chemotaxis-fluid model \(\begin{aligned} {\left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta (n\phi (v))+\mu n(1-n),\\ v_t+u\cdot \nabla v=\Delta v-nv,\\ u_t+ \kappa (u\cdot \nabla ) u=\Delta u+n\nabla \Phi -\nabla P, \quad \nabla \cdot u=0, \end{array}\right. } \end{aligned}\) accounting for signal-dependent motilities of microbial populations interacting with an incompressible liquid through transport and buoyancy, where the suitably smooth function \(\phi \) satisfies \(\phi >0\) on \((0,\infty )\) with \(\phi (0)=0\) and \(\phi '(0)>0\) , and the parameter \(\mu \ge 0\) . For all reasonably regular initial data, if \(\mu =0\) , the corresponding initial boundary value problem possesses global classical solutions with a smallness condition on \(\int _\Omega n_0\) ; whereas if \(\mu >0\) , this problem possesses global bounded classical solutions, which can converge toward (1,0,0) as time tends to infinity when a certain small mass is imposed on the initial data \(v_0\) . These results extend recent results for the fluid-free system to one in a Navier-Stokes fluid environment.
