The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system with density-dependent motion

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This paper deals with the chemotaxis-(Navier-)Stokes system{nt+u⋅∇n=∇⋅(φ(c)∇n)−∇⋅(nχ(c)∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+κ(u⋅∇)u=Δu−∇P+n∇ψ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, subject to no-flux boundary conditions in a bounded domain Ω⊂R2 with smooth boundary, with a given smooth potential ψ:Ω→R and κ∈R. The density-dependent motility φ(c) and the chemotactic function χ(v) are assumed to satisfy some further technical conditions. It is shown that the problem possesses a global classical solution (nκ,cκ,uκ). Furthermore, we prove that these solutions stabilize uniformly to (n0‾,0,0) exponentially with respect to t, where n0‾:=1|Ω|∫Ωn0. Finally, it is also shown that there exists a constant μ>0 such that for all κ∈(0,1)‖n(κ)(⋅,t)−n(0)(⋅,t)‖L∞(Ω)+‖c(κ)(⋅,t)−c(0)(⋅,t)‖L∞(Ω)+‖u(κ)(⋅,t)−u(0)(⋅,t)‖L∞(Ω)≤Cκ13e−μtfor allt>0, where C>0 is a constant independent of κ and t.