Global existence and large time behavior for a two-dimensional chemotaxis-Navier–Stokes system

Abstract

This paper concerns the coupled chemotaxis-Navier–Stokes system in the two-dimensional setting. Such a system was proposed in [19] to describe the collective effects arising in bacterial suspensions in fluid drops. Under some basic assumptions on the parameter functions χ(⋅), k(⋅) and the potential function ϕ, which are consistent with those used by the experimentalists but weaker than those appeared in the known mathematical works, we establish the global existence of weak solutions and classical solutions for both the Cauchy problem and the initial-boundary value problem supplemented with some initial data. For the initial-boundary value problem, we also assert that the solution converges in large time to the spatially homogeneous equilibrium (n0‾,0,0) with n0‾:=1|Ω|∫Ωn0(x)dx. Our results also show that the large diffusion of the cell density or the chemical concentration can rule out the finite-time blow-up even though the Navier–Stokes fluid is included.