Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

This paper deals with an initial-boundary value problem for the system n t + u · ∇ n = Δ n - ∇ · ( n χ ( c ) ∇ c ) , x ∈ Ω , t > 0 , c t + u · ∇ c = Δ c - n f ( c ) , x ∈ Ω , t > 0 , u t + κ ( u · ∇ ) u = Δ u + ∇ P + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ · u = 0 , x ∈ Ω , t > 0 , which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains Ω ⊂ R 2 and under appropriate assumptions on the parameter functions χ, f and ϕ, for each κ ∈ R and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium ( n 0 ¯ , 0 , 0 ) , where n 0 ¯ : = 1 | Ω | ∫ Ω n ( x , 0 ) d x , in the sense that as t →∞, n ( · , t ) → n 0 ¯ , c ( · , t ) → 0 and u ( · , t ) → 0 hold with respect to the norm in L ∞ ( Ω ) .