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=Δ(nϕ(v))+μn(1-n),\\ v_t+u\cdot\nabla v=Δv-nv,\\ u_t+ κ(u\cdot\nabla) u=Δu+n\nablaΦ-\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 $\ddot{I}•$ satisfies $\ddot{I}•>0$ on $(0,\infty )$ with $\ddot{I}•(0)=0$ and $\ddot{I}{•}^{\prime }(0)>0$, and the parameter $\hat{I}¼\ge 0$. For all reasonably regular initial data, if $\hat{I}¼=0$, the corresponding initial boundary value problem possesses global classical solutions with a smallness condition on ${\int }_{\hat{I}}\text{c◯}{n}_0$; whereas if $\hat{I}¼>0$, 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.

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