Does fluid interaction affect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?

A class of Keller-Segel-Stokes systems generalizing the prototype $$\{\begin{array}{rclllll}{n}_t+u\cdot \mathrm{\nabla }n& =& \mathrm{\Delta }n-\mathrm{\nabla }\cdot (n(n+1{)}^{-\alpha }\mathrm{\nabla }c),c_t+u\cdot \mathrm{\nabla }c& =& \mathrm{\Delta }c-c+n,u_t+\mathrm{\nabla }P& =& \mathrm{\Delta }u+n\mathrm{\nabla }\varphi +f(x,t),\mathrm{\nabla }\cdot u=0,\end{array}(\star )$$ is considered in a bounded domain $\Omega\subset R^3$,where$\phi$and$f$aregivensufficientlysmoothfunctionssuchthat$f$isboundedin$\Omega\times (0,\infty)$. It is shown that under the condition that $$\alpha >\frac{1}{3},$$ 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 $\alpha >\frac{1}{2}$. In view of known results on the existence of exploding solutions when $\alpha <\frac{1}{3}$, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate $\frac{1}{3}$, as previously found for the fluid-free counterpart of ($\star$), remains essentially unaffected by fluid interaction of the type considered here.