The small‐convection limit in a two‐dimensional chemotaxis‐Navier‐Stokes system with singular sensitivity

Abstract

In this paper, we consider the two‐dimensional chemotaxis‐Navier‐Stokes system with singular sensitivity {left left centerarraynt+u·∇n=Δn−χ∇·(nc∇c),arrayx∈Ω,t>0,arrayct+u·∇c=Δc−c+n,arrayx∈Ω,t>0,arrayut+κ(u·∇)u=Δu+∇P+n∇ϕ,arrayx∈Ω,t>0,array∇·u=0,arrayx∈Ω,t>0 in a bounded convex domain Ω ⊂ R2 with smooth boundary, with κ ∈ R and a given smooth potential ϕ : Ω → R. It is known that for each κ ∈ [0, 1) and 0 < χ < 1 this problem possesses a unique global classical solution (nκ, cκ, uκ). Our main result asserts that under the assumption of 0 < χ < 1, (nκ, cκ, uκ) stabilizes to (n0, c0, u0) with an explicit rate and a time dependent coefficient as κ → 0+.