Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller-Segel System with Saturated Sensitivity?

A class of Keller-Segel-Stokes systems generalizing the prototype nt+u·∇n=Δn-∇·n(n+1)-α∇c,ct+u·∇c=Δc-c+n,ut+∇P=Δu+n∇ϕ+f(x,t),∇·u=0,(⋆)is considered in a bounded domain Ω⊂R3, where ϕ and f are given sufficiently smooth functions such that f is bounded in Ω×(0,∞). It is shown that under the condition that α>13,for all sufficiently regular initial data a corresponding Neumann-Neumann-Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption α>12. In view of known results on the existence of exploding solutions when α<13, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate 13, as previously found for the fluid-free counterpart of (⋆), remains essentially unaffected by fluid interaction of the type considered here. [ABSTRACT FROM AUTHOR]