This paper deals with the following chemotaxis–Stokes system nt+u·∇n=Δnm-∇·(nS(x,n,c)·∇c),x∈Ω,t>0,ct+u·∇c=Δc-nf(c),x∈Ω,t>0,ut=Δu+∇P+n∇ϕ,x∈Ω,t>0,∇·u=0,x∈Ω,t>0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c), &{}\quad x\in \Omega ,\,\, t>0,\\ c_t+u\cdot \nabla c=\Delta c-nf(c),&{}\quad x\in \Omega , \,\,t>0,\\ u_t=\Delta u+\nabla P+n\nabla \phi ,&{}\quad x\in \Omega , \,\,t>0,\\ \nabla \cdot u=0,&{}\quad x\in \Omega , \,\,t>0\\ \end{array} \right. \end{aligned}$$\end{document}under no-flux boundary conditions in a bounded domain Ω⊂R3\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset \mathbb {R}^{3}$$\end{document} with smooth boundary, where m≥1\documentclass[12pt]{minimal}
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\begin{document}$$m\ge 1$$\end{document}, ϕ∈W1,∞(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in W^{1,\infty }(\Omega )$$\end{document}, f and S are given functions with values in [0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$[0,\,\infty )$$\end{document} and R3×3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^{3\times 3}$$\end{document}, respectively. Here S satisfies |S(x,n,c)|<S0(c)(1+n)-α\documentclass[12pt]{minimal}
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\begin{document}$$|S(x,n,c)|<S_0(c)(1+n)^{-\alpha }$$\end{document} with α≥0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \ge 0$$\end{document} and some nonnegative nondecreasing function S0\documentclass[12pt]{minimal}
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\begin{document}$$S_0$$\end{document}. With the tensor-valued sensitivity S, this system does not possess energy-type functionals which seem to be available only when S is a scalar function. We can establish a priori estimation to overcome this difficulty and explore a relationship between m and α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, i.e., m+α>76\documentclass[12pt]{minimal}
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\begin{document}$$m+\alpha >\frac{7}{6}$$\end{document}, which insures the global existence of bounded weak solution. Our result covers completely and improves the recent result by Wang and Cao (Discrete Contin Dyn Syst Ser B 20:3235–3254, 2015) which asserts, just in the case m=1\documentclass[12pt]{minimal}
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\begin{document}$$m=1$$\end{document}, the global existence of solutions, but without boundedness, and that by Winkler (Calc Var Partial Differ Equ 54:3789–3828, 2015) which only involves the case of α=0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =0$$\end{document} and requires the convexity of the domain.
