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

This paper is concerned with the two-dimensional chemotaxis-fluid model (Formula presented.) accounting for signal-dependent motilities of microbial populations interacting with an incompressible liquid through transport and buoyancy, where the suitably smooth function ϕ satisfies ϕ>0 on (0,∞) with ϕ(0)=0 and ϕ′(0)>0, and the parameter μ≥0. For all reasonably regular initial data, if μ=0, the corresponding initial boundary value problem possesses global classical solutions with a smallness condition on ∫Ωn0; whereas if μ>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 v0. These results extend recent results for the fluid-free system to one in a Navier-Stokes fluid environment.