Asymptotic behavior of classical solutions of a three-dimensional Keller–Segel–Navier–Stokes system modeling coral fertilization

We are concerned with the Keller–Segel–Navier–Stokes system {ρt+u·∇ρ=Δρ-∇·(ρS(x,ρ,c)∇c)-ρm,(x,t)∈Ω×(0,T),mt+u·∇m=Δm-ρm,(x,t)∈Ω×(0,T),ct+u·∇c=Δc-c+m,(x,t)∈Ω×(0,T),ut+(u·∇)u=Δu-∇P+(ρ+m)∇ϕ,∇·u=0,(x,t)∈Ω×(0,T)subject to the boundary condition (∇ ρ- ρS(x, ρ, c) ∇ c) · ν= ∇ m· ν= ∇ c· ν= 0 , u= 0 in a bounded smooth domain Ω⊂ R³. It is shown that this problem admits a global classical solution with exponential decay properties when S∈C2(Ω¯×[0,∞)2)3×3 satisfies | S(x, ρ, c) | ≤ C_S for some C_S> 0 , and the initial data satisfy certain smallness conditions.