Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

The coupled chemotaxis fluid systemequation(⋆){nt=Δn−∇⋅(nS(x,n,c)⋅∇c)−u⋅∇n,(x,t)∈Ω×(0,T),ct=Δc−nc−u⋅∇c,(x,t)∈Ω×(0,T),ut=Δu−κ(u⋅∇)u+∇P+n∇ϕ,(x,t)∈Ω×(0,T),∇⋅u=0,(x,t)∈Ω×(0,T), is considered under the no-flux boundary conditions for n,cn,c and the Dirichlet boundary condition for u   on a bounded smooth domain Ω⊂RNΩ⊂RN (N=2,3N=2,3), κ∈{0,1}κ∈{0,1}. We assume that S(x,n,c)S(x,n,c) is a matrix-valued sensitivity under a mild assumption such that |S(x,n,c)|