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

The flux-limited Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ 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 $$K_f>0$$ K f > 0 and $$\alpha >0$$ α > 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.