In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, $p>2$)$\{\begin{array}{cc}\hfill & n_t+u\cdot \mathrm{\nabla }n=\mathrm{\nabla }\cdot (|\mathrm{\nabla }n{|}^{p-2}\mathrm{\nabla }n)-\chi \mathrm{\nabla }\cdot (n\mathrm{\nabla }c),\hfill \\ \hfill & c_t+u\cdot \mathrm{\nabla }c-\mathrm{\Delta }c=-cn,\hfill \\ \hfill & u_t+\mathrm{\nabla }\pi =\mathrm{\Delta }u+n\mathrm{\nabla }\phi ,\hfill \\ \hfill & \mathrm{div}u=0\hfill \end{array}$ in a bounded domain Ω of ${\mathbb{R}}^3$ with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem to find an optimal p-value ($p\ge 2$) which ensures that the solution is global bounded. In particular, the closer the value of p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist whenever$p>p^{⁎}(\approx 2.012).$ It improved the result of [21], [22], in which, the authors established the global bounded solutions for $p>\frac{23}{11}$. Moreover, we also consider the large time behavior of solutions, and show that the weak solutions will converge to the spatially homogeneous steady state $({\bar{n}}_0,0,0)$. Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of n is proved in the sense of $L^{\infty }$-norm, not only in $L^p$-norm or weak-* topology.

在本文中，我们处理以下具有非牛顿过滤慢扩散的趋化-斯托克斯模型（即， $p>\mathrm{2个}$）$\{\begin{array}{cc}\hfill & {ñ}_{Ť}+ü\cdot \mathrm{\nabla }ñ=\mathrm{\nabla }\cdot （|\mathrm{\nabla }ñ{|}^{p-\mathrm{2个}}\mathrm{\nabla }ñ）-\chi \mathrm{\nabla }\cdot （ñ\mathrm{\nabla }C），\hfill \\ \hfill & C_{Ť}+ü\cdot \mathrm{\nabla }C-\mathrm{\Delta }C=-Cñ，\hfill \\ \hfill & {ü}_{Ť}+\mathrm{\nabla }\pi =\mathrm{\Delta }ü+ñ\mathrm{\nabla }\phi ，\hfill \\ \hfill & \mathrm{股利}ü=0\hfill \end{array}$ 在的有限域Ω中 ${\mathbb{[R}}^3$具有零通量边界条件和无滑移边界条件。类似于对具有多孔介质扩散的趋化性-Stokes系统的研究，寻找最佳p值（$p\ge \mathrm{2个}$），以确保解决方案具有全局边界。特别是，p的值越接近2，研究就越困难。在本文中，我们证明了只要有$p>p^{⁎}（\approx 2.012）。$ 它改进了[21]，[22]的结果，其中，作者建立了针对 $p>\frac{23}{11}$。此外，我们还考虑了解的长时间行为，并表明弱解将收敛到空间均匀稳态$（{\stackrel{〜}{ñ}}_0，0，0）$。与具有多孔介质扩散的趋化流体系统相比，证明了n的当前收敛性。${\mathrm{大号}}^{\infty }$-规范，不仅在 ${\mathrm{大号}}^p$-norm或weak- *拓扑。