This paper concerns the chemotaxis-Stokes system $\left\{\begin{array}{cc}\hfill & n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (n\nabla c)+\mu n(1-n),\hfill \\ \hfill & c_t+u\cdot \nabla c=\Delta c-c{n}^{\alpha },\hfill \\ \hfill & u_t=\Delta u-\nabla \pi +n\nabla \phi ,\hfill \\ \hfill & \nabla \cdot u=0\hfill \end{array}\right.$in a three dimensional bounded domain under no-flux boundary conditions for $n,c$ and no-slip boundary conditions for $u$. The purpose of this paper is to study the global solvability and large time asymptotic behavior of solutions. Here, it is worth mentioning that the nonlinear consumption term $c{n}^{\alpha }$ (when $\alpha >1$) will lead to some higher order nonlinear terms in the proof of some uniformly bounded prior estimates of the approximation solutions, which brings great difficulties to the study of the problem. To overcome these difficulties, we make some very precise analysis, combined with some iterative techniques, and finally establish the uniform boundedness of weak solutions for $m>1$, $0<\alpha <2m-1$. Then, the global solvability of weak solutions is derived for any large initial data. Furthermore, we focus on the convergence of weak solutions, and prove that the solutions will converge to the constant steady state $(1,0,0)$ in the large time limit.

本文涉及趋化-斯托克斯系统 $\left\{\begin{array}{cc}\hfill & n_{吨}+你\cdot \nabla n=\Delta n^{米}-\nabla \cdot (n\nabla C)+\mu n(1-n),\hfill \\ \hfill & C_{吨}+你\cdot \nabla C=\Delta C-C{n}^{\alpha },\hfill \\ \hfill & {你}_{吨}=\Delta 你-\nabla \pi +n\nabla \phi ,\hfill \\ \hfill & \nabla \cdot 你=0\hfill \end{array}\right.$在无通量边界条件下的三维有界域中 $n,C$ 和无滑移边界条件 $你$. 本文的目的是研究解的全局可解性和大时间渐近行为。在这里，值得一提的是非线性消耗项$C{n}^{\alpha }$ （什么时候 $\alpha >1$) 会导致一些高阶非线性项在逼近解的一些一致有界先验估计的证明中，给问题的研究带来很大的困难。为了克服这些困难，我们做了一些非常精确的分析，结合一些迭代技术，最终建立了弱解的一致有界性。$米>1$, $0<\alpha <2米-1$. 然后，导出任何大初始数据的弱解的全局可解性。此外，我们关注弱解的收敛性，并证明这些解会收敛到恒定的稳态$(1,0,0)$ 在很长的时间内。