The coupled chemotaxis fluid system nt=∇·((n+1)m-1∇n)-∇·(n(n+1)m-α-1∇c)-u·∇n,(x,t)∈Ω×(0,T),ct=Δc-c+n-u·∇c,(x,t)∈Ω×(0,T),ut=Δu+∇P+n∇ϕ,∇·u=0,(x,t)∈Ω×(0,T),∇c·ν=∇n·ν=0,u=0,(x,t)∈∂Ω×(0,T),n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x)x∈Ω,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \left\{ \begin{array}{llc} n_t=\nabla \cdot ( (n+1)^{m-1}\nabla n)-\nabla \cdot (n(n+1)^{m-\alpha -1} \nabla c)-u\cdot \nabla n, &{}(x,t)\in \Omega \times (0,T),\\ \displaystyle c_t=\Delta c-c+n-u\cdot \nabla c, &{}(x,t)\in \Omega \times (0,T),\\ \displaystyle u_t=\Delta u+\nabla P+n\nabla \phi ,\quad \nabla \cdot u=0, &{}(x,t)\in \Omega \times (0,T),\\ \displaystyle \nabla c\cdot \nu =\nabla n\cdot \nu =0, \;\; u=0,&{}(x,t)\in \partial \Omega \times (0,T),\\ n(x,0)=n_{0}(x),\quad c(x,0)=c_{0}(x),\quad u(x,0)=u_0(x) &{} x\in \Omega , \end{array} \right. \end{aligned}$$\end{document}is considered in a bounded domain Ω⊂R3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega \subset \mathbb {R}^3$$\end{document}, with smooth boundary, where m,α∈R\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$m,\alpha \in \mathbb {R}$$\end{document}. It is known that in the fluid-free case (u≡0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$u\equiv 0$$\end{document}), the system admits a global bounded solution if α>13\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha >\frac{1}{3}$$\end{document}. The purpose of this paper is to overcome difficulties arising from the presence of fluid interaction and to show that the same conclusion holds if m>-13\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$m>-\frac{1}{3}$$\end{document}. In the case m≤-13\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$m\le -\frac{1}{3}$$\end{document} and α>13\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha >\frac{1}{3}$$\end{document}, global existence of classical solution will be shown.
