Large time behavior in a fractional chemotaxis–Navier–Stokes system with logistic source. (February 2022) - Lei, Yuzhu; Liu, Zuhan; Zhou, Ling
Abstract: This paper deals with a coupled chemotaxis–Navier–Stokes system with logistic source and a fractional diffusion of order α ∈ ( 1 2, 1 ) n t + u ⋅ ∇ n = − ( − Δ ) α n − χ ∇ ⋅ ( n ∇ c ) + a n − b n 2, c t + u ⋅ ∇ c = Δ c − n c, u t + ( u ⋅ ∇ ) u = Δ u + ∇ P + n ∇ ϕ + f, ∇ ⋅ u = 0 on three dimensional periodic torus T 3 . 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 ( a b, 0, 0 ) .