A class of Keller–Segel–Stokes systems generalizing the prototype \(\begin{aligned} {\left\{ \begin{array}{ll} n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n S(n)\nabla c),& \quad x\in \Omega ,t>0, \\ u\cdot \nabla c=\Delta c-c+n,& \quad x\in \Omega ,t>0, \\ u_{t}=\Delta u-\nabla P+n\nabla \phi ,& \quad x\in \Omega ,t>0, \\ \nabla \cdot u=0,& \quad x\in \Omega ,t>0 \end{array}\right. } \qquad (\star ) \end{aligned}\) in a smoothly bounded domain \(\Omega \subset \mathbb {R}^3\) with smooth boundary, where the gravitational potential \(\phi \in W^{2,\infty }(\Omega )\) , and S(n) referring to the chemotactic sensitivity function is considered to satisfy \(|S(n)| \le C_S(1 + n)^{-\alpha }\) for all \(n\ge 0\) with some \(C_S>0\) and \(\alpha \in \mathbb {R}\) . For all sufficiently regular initial data, the corresponding initial-boundary value problem for \((\star )\) under no-flux/no-flux/Dirichlet conditions is shown to possess a globally bounded classical solution whenever the saturation exponent satisfies \(\alpha >\frac{1}{3}\) . This result is sharp with respect to the allowed range of \(\alpha \) , extending and optimizing earlier works that required stronger saturation assumptions.
