On the global well‐posedness for the 2D incompressible Keller‐Segel‐Navier‐Stokes equations

The Keller‐Segel‐Navier‐Stokes systemρ t + u · ∇ ρ = Δ ρ − ∇ · ( ρ ∇ c ) − ρ 2 ,c t + u · ∇ c = Δ c − c + ρ ,u t + u · ∇ u + ∇ P = Δ u − ρ ∇ ϕ ,∇ · u = 0 ,is considered in R 2 . It is proved that we obtain the existence and uniqueness of weak solutions for the two dimensional incompressible Keller‐Segel‐Navier‐Stokes equations for a large class of initial data by using Fourier localization technique.