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 $(\frac{a}{b},0,0).$

中文翻译：

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具有逻辑源的分数趋化-纳维-斯托克斯系统中的大时间行为

本文涉及具有逻辑源和有序分数扩散的耦合趋化-纳维-斯托克斯系统 $\alpha \in (\frac{1}{2},1)$ $\left\{\begin{array}{c}{n}_{吨}+你\cdot \nabla n=-{(-\Delta )}^{\alpha }n-\chi \nabla \cdot (n\nabla C)+\mathrm{一种}n-乙n^2,\hfill \\ C_{吨}+你\cdot \nabla C=\Delta C-nC,\hfill \\ {你}_{吨}+(你\cdot \nabla )你=\Delta 你+\nabla 磷+n\nabla \phi +F,\nabla \cdot 你=0\hfill \end{array}\right.$在三维周期环面上 ${\mathbb{吨}}^3$. 由于三维全纳维-斯托克斯方程没有经典解，本文的主要目的是研究上述系统在较弱扩散情况下弱解的全局存在性，经过一段时间的等待，弱解实际上变得平滑并收敛到半平凡的稳态$(\frac{\mathrm{一种}}{乙},0,0).$