How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?

The chemotaxis-Navier-Stokes system ( ⋆ ) { n t + u ⋅ ∇ n a m p ; = Δ n − ∇ ⋅ ( n χ ( c ) ∇ c ) , c t + u ⋅ ∇ c a m p ; = Δ c − n f ( c ) , u t + ( u ⋅ ∇ ) u a m p ; = Δ u + ∇ P + n ∇ Φ , ∇ ⋅