Global boundedness of solutions for the chemotaxis-Navier-Stokes system in R2

This paper is concerned with a coupled chemotaxis-Navier-Stokes system{nt+u⋅∇n=Δn−∇⋅(n∇c),x∈R2,t>0,ct+u⋅∇c=Δc−nc,x∈R2,t>0,ut+u⋅∇u=Δu+∇P+n∇ϕ,x∈R2,t>0,∇⋅u=0,x∈R2,t>0, which describes the motion of oxygen-driven bacteria in a fluid. The gravitational potential ϕ is assumed to satisfy some technical conditions. It is proved that global classical bounded solutions exist for regular initial data.