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The flux-limited Keller–Segel–Navier–Stokes system    $$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &amp;{}=&amp;{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &amp;{}=&amp;{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &amp;{}=&amp;{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$  n t + u · ∇ n = Δ n - ∇ · ( n f ( | ∇ c | 2 ) ∇ c ) , c t + u · ∇ c = Δ c - c + n , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , ( ⋆ ) is considered in a smoothly bounded domain    $$\Omega \subset {\mathbb {R}}^2$$  Ω ⊂ R 2 . It is shown that whenever the suitably smooth function  f  models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that    $$\begin{aligned} |f(\xi )| \le K_f\cdot (\xi +1)^{-\frac{\alpha }{2}} \end{aligned}$$  | f ( ξ ) | ≤ K f · ( ξ + 1 ) - α 2 holds for all    $$\xi \ge 0$$  ξ ≥ 0 with some    $$K_f&gt;0$$  K f &gt; 0 and    $$\alpha &gt;0$$  α &gt; 0 , for any all reasonably regular initial data a corresponding no-flux/no-flux/Dirichlet problem admits a globally defined classical solution which is bounded, inter alia, in    $$L^\infty (\Omega \times (0,\infty ))$$  L ∞ ( Ω × ( 0 , ∞ ) ) with respect to all its components. By extending a corresponding result known for a fluid-free counterpart of (   $$\star $$  ⋆ ), this confirms that with regard to the possible emergence of blow-up phenomena, the choice    $$f\equiv const.$$  f ≡ c o n s t . retains some criticality also in the presence of fluid interaction.

Suppressing blow-up by gradient-dependent flux limitation in a planar Keller–Segel–Navier–Stokes system