Avoiding critical mass phenomena by arbitrarily mild saturation of cross-diffusive fluxes in two-dimensional Keller-Segel-Navier-Stokes systems

Abstract

The Keller-Segel-Navier-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(n)∇c),ct+u⋅∇c=Δc−c+n,ut+(u⋅∇)u=Δu+∇P+n∇Φ,∇⋅u=0,(⋆) is considered in a smoothly bounded domain Ω⊂R2, with a given function χ∈C2([0,∞)).It is shown that ifχ(n)→0as n→∞, then for all suitably regular initial data an associated no-flux/no-flux/Dirichlet problem for (⋆) possesses a globally defined and bounded classical solution.This particularly indicates that actually arbitrarily weak saturation of cross-diffusive fluxes at large population densities is sufficient to suppress critical mass phenomena with respect to the occurrence of unbounded solutions, as known to be present when χ≡const. even in the fluid-free case.