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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with $N=2,3$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n = Δn - \nabla \cdot \big( n S(n)\nabla c \big), \[2mm] u \cdot \nabla c = Δc - c + n, \[2mm] u_t + κ(u \cdot \nabla) u = Δu - \nabla P + n \nabla φ, \[2mm] \nabla \cdot u = 0, \end{cases} \end{equation*} where $\kappa \in \{0,1\}$, the given gravitational potential $\phi \in W^{2,\mathrm{\infty }}(\mathrm{\Omega })$, and the chemotactic sensitivity function $S\in C^2([0,\mathrm{\infty }))$. 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 ${\lim }_{\xi \to \mathrm{\infty }}S(\xi )=0$, 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 $$\mathrm{\mid }S(\xi )\mathrm{\mid }\le K_S(\xi +1{)}^{-\alpha }\text{for all }\xi \ge 0,$$ 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 $S\equiv \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ in two dimensions, or when $\alpha <\frac{1}{3}$ in three dimensions.