Large time behavior in a fractional chemotaxis–Navier–Stokes system with logistic source

Abstract

This paper deals with a coupled chemotaxis–Navier–Stokes system with logistic source and a fractional diffusion of order α∈(12,1)nt+u⋅∇n=−(−Δ)αn−χ∇⋅(n∇c)+an−bn2,ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇ϕ+f,∇⋅u=0on three dimensional periodic torus T3. Since there is no classical solution in the three-dimensional full Navier–Stokes equations, our main purpose of this paper is to investigate the global existence of weak solutions to the above system in the case of a weaker diffusion, and after some waiting time, the weak solutions in fact become smooth and converge to the semi-trivial steady state (ab,0,0).