Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion

Abstract

This paper is concerned with the two-dimensional chemotaxis-fluid model \begin{equation*} \begin{cases} 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{cases} \end{equation*} 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 $\mu\geq 0$. 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.