Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal

Abstract

The chemotaxis-Navier-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(n)∇c),ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇ϕ,∇⋅u=0 is considered in a smoothly bounded domain Ω⊂R2 under the boundary conditions(∇n−nχ(n)∇c)⋅ν=0,c=c⋆,u=0,x∈∂Ω,t>0 with a given nonnegative constant c⋆. It is shown that if χ∈C2([0,∞)) and χ(n)→0 as n→∞, then for all suitably regular initial data, an associated initial value problem possesses a globally defined and bounded classical solution. When ‖n0‖L1(Ω) and ‖c0‖L∞(Ω) are suitably small and c⋆≡0, we further obtain the stabilization of the classical solution.