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

This paper deals with a coupled chemotaxis–Navier–Stokes system with logistic source and a fractional diffusion of order $\alpha \in (\frac{1}{2},1)$ $\left\{\begin{array}{c}{n}_t+u\cdot \nabla n=-{(-\Delta )}^{\alpha }n-\chi \nabla \cdot (n\nabla c)+an-b{n}^2,\hfill \\ c_t+u\cdot \nabla c=\Delta c-nc,\hfill \\ u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \varphi +f,\nabla \cdot u=0\hfill \end{array}\right.$on three dimensional periodic torus ${\mathbb{T}}^3$. 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 $(\frac{a}{b},0,0).$