A class of Keller–Segel–Stokes systems generalizing the prototype nt+u·∇n=Δn-∇·n(n+1)-α∇c,ct+u·∇c=Δc-c+n,ut+∇P=Δu+n∇ϕ+f(x,t),∇·u=0,(⋆)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{l} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot \left( n(n+1)^{-\alpha }\nabla c\right) , \\ c_t + u\cdot \nabla c = \Delta c-c+n, \\ u_t +\nabla P = \Delta u + n \nabla \phi + f(x,t), \quad \nabla \cdot u =0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$\end{document}is considered in a bounded domain Ω⊂R3\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset \mathbb {R}^3$$\end{document}, where ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} and f are given sufficiently smooth functions such that f is bounded in Ω×(0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \times (0,\infty )$$\end{document}. It is shown that under the condition that α>13,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha >\frac{1}{3}, \end{aligned}$$\end{document}for all sufficiently regular initial data a corresponding Neumann–Neumann–Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption α>12\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >\frac{1}{2}$$\end{document}. In view of known results on the existence of exploding solutions when α<13\documentclass[12pt]{minimal}
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\begin{document}$$\alpha <\frac{1}{3}$$\end{document}, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate 13\documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{3}$$\end{document}, as previously found for the fluid-free counterpart of (⋆\documentclass[12pt]{minimal}
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\begin{document}$$\star $$\end{document}), remains essentially unaffected by fluid interaction of the type considered here.
