A Sharp Global Boundedness Result for Keller--Segel--(Navier--)Stokes Systems with Rapid Diffusion and Saturated Sensitivities

We investigate the Keller--Segel--(Navier--)Stokes system posed in a smooth bounded domain $\Omega \subset \mathbb{R}^N$ with $N = 2,3$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot \big( n S(n)\nabla c \big), \2mm] u \cdot \nabla c = \Delta c - c + n, \\[2mm] u_t + \kappa (u \cdot \nabla) u = \Delta u - \nabla P + n \nabla \phi, \\[2mm] \nabla \cdot u = 0, \end{cases} \end{equation*} where \(\kappa \in \left \{0,1 \right \} \), the given gravitational potential \(\phi \in W^{2, \infty}(\Omega)\), and the chemotactic sensitivity function \(S \in C^2([0,\infty))\). Under no-flux boundary conditions for \(n\) and \(c\), together with the Dirichlet boundary condition for \(u\), we show that, provided the initial data satisfy suitable regularity assumptions, the following results hold: \begin{itemize} \item If \(N = 2\), \(\kappa = 1\), and the sensitivity function satisfies \(\lim_{\xi \to \infty} S(\xi) = 0\), then the Keller--Segel--Navier--Stokes system admits a global classical solution that remains uniformly bounded in time. \item If \(N = 3\), \(\kappa = 0\), and \(S\) satisfies \[ |S(\xi)| \le K_S (\xi + 1)^{-\alpha} \quad \text{for all } \xi \ge 0, with some constants $K_S > 0$ and $\alpha > \frac{1}{3}$, then the Keller--Segel--Stokes system possesses a global bounded classical solution. \end{itemize} Our results are optimal, since it is well established that, in the absence of fluid effects, blow-up can occur when $S \equiv \mathrm{const}$ in two dimensions, or when $\alpha < \tfrac{1}{3}$ in three dimensions.