Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with gradient-dependent flux limitation

In a bounded domain Ω⊂R3, we are concerned with the evolution system (⋆)nt+u⋅∇n=Δn−∇⋅(nf(|∇c|2)∇c),ct+u⋅∇c=Δc−c+n,ut+(u⋅∇)u=Δu+∇P+n∇Φ,∇⋅u=0,coupling the incompressible Navier–Stokes equations to a class of flux-limited Keller–Segel systems which has received noticeable attention in the recent biomathematical literature. When considered without such fluid interaction, no-flux boundary value problems for chemotaxis systems of the latter type are known to admit global bounded solutions for widely arbitrary initial data whenever f is a suitably smooth function fulfilling (⋆⋆)|f(ξ)|≤Kf⋅(ξ+1)−α2for allξ≥0with some Kf>0 and α>12, while if here the converse inequality holds with some Kf>0 and α<12, then blow-up occurs at least in some simplified parabolic–elliptic counterpart. The present work now asserts that the former condition remains sufficient to ensure global solvability in a corresponding initial–boundary value problem for the fully coupled system (⋆), within a natural weak solution concept consistent with those underlying well-established theories for the Navier–Stokes equations. This indicates that the saturation exponent α=12 in (⋆⋆) continues to play the role of a critical flux limitation parameter also in the presence of fluid interaction.