Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity

In this paper, we consider the following Keller-Segel-Navier-Stokes system{nt=Δn−∇⋅(nS(x,n,c)⋅∇c)−u⋅∇ninΩ×(0,T),ct=Δc−c+n−u⋅∇cinΩ×(0,T),ut=Δu−(u⋅∇)u+∇P+n∇Φ,∇⋅u=0inΩ×(0,T) subject to the boundary condition ∇c⋅ν=(∇n−nS(x,n,c)⋅∇c)⋅ν=0, u=0, and the initial data (n0(x),c0(x),u0(x)), where Ω⊂RN is a smooth bounded domain with N∈{2,3}, ν denotes the unit outer normal of ∂Ω, S∈C2(Ω¯×[0,∞)2)N×N and Φ∈C1+δ(Ω¯) with δ∈(0,1). We establish global classical solutions decaying to the constant steady state (n¯0,n¯0,0) exponentially with n¯0:=1|Ω|∫Ωn0(x)dx, whenever ‖n0‖LN2(Ω), ‖∇c0‖LN(Ω) and ‖u0‖LN(Ω) small enough.