The small-convection limit in a two-dimensional Keller–Segel–Navier–Stokes system

Abstract

In this paper, we will investigate the small-convection limit of solutions (nκ,cκ,uκ) to the Keller–Segel–Navier–Stokes system{ntκ+uκ⋅∇nκ=Δnκ−∇⋅(nκ(1+nκ)−α∇cκ),ctκ+uκ⋅∇cκ=Δcκ−cκ+nκ,utκ+κ(uκ⋅∇)uκ=Δuκ−∇Pκ+nκ∇ϕ,∇⋅uκ=0 along with no-flux boundary conditions for nκ and cκ and a no-slip boundary condition for uκ, and with suitable regular initial data in a bounded convex domain Ω⊂R2 with smooth boundary. Our first result asserts that for general large data, (nκ,cκ,uκ) will stabilize to (n0,c0,u0) with an explicit rate and a time dependent coefficient as κ→0+. Our second result further reveals that such a convergence is uniform with respect to κ at an exponential time decay rate provided that the initial data is suitable small. To the best of our knowledge, this seems to be the first rigorous theoretical analysis on the convergence of solutions of the Keller–Segel–Naiver–Stokes system with signal production to the corresponding Keller–Segel–Stokes system as κ→0+.