We investigate the Keller–Segel–(Navier–)Stokes system posed in a smooth bounded domain \(\Omega \subset \mathbb {R}^N\) with \(N = 2,3\) : \(\begin{aligned} {\left\{ \begin{array}{ll} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot \big ( n S(n)\nabla c \big ), \\ u \cdot \nabla c = \Delta c - c + n, \\ u_t + \kappa (u \cdot \nabla ) u = \Delta u - \nabla P + n \nabla \phi , \\ \nabla \cdot u = 0, \end{array}\right. } \end{aligned}\) 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: If \(N = 2\) , \(\kappa = 1\) , and the sensitivity function satisfies \(\lim _{\xi \rightarrow \infty } S(\xi ) = 0\) , then the Keller–Segel–Navier–Stokes system admits a global classical solution that remains uniformly bounded in time.
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.
Our results expected to be optimal, since it is well established that, in the absence of fluid effects, blow-up can occur when \(S \equiv \textrm{const}\) in two dimensions, or when \(\alpha < \tfrac{1}{3}\) in three dimensions.
