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

• Published in: Journal of mathematical fluid mechanics Vol. 20; no. 4; pp. 1889 - 1909
• Main Author: Winkler, Michael
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2018; Springer Nature B.V
• Subjects: Boundary value problems; Classical and Continuum Physics; Decay rate; Dirichlet problem; Fluid dynamics; Fluid mechanics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Theoretical mathematics; 35B65 (primary); Stokes; 35Q35; Boundedness; Maximal Sobolev regularity; Chemotaxis; 92C17 (secondary); 35Q92
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-018-0395-0

A class of Keller–Segel–Stokes systems generalizing the prototype n t + u · ∇ n = Δ n - ∇ · n ( n + 1 ) - α ∇ c , c t + u · ∇ c = Δ c - c + n , u t + ∇ P = Δ u + n ∇ ϕ + f ( x , t ) , ∇ · u = 0 , ( ⋆ ) is considered in a bounded domain Ω ⊂ R 3 , 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 > 1 3 , 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 > 1 2 . In view of known results on the existence of exploding solutions when α < 1 3 , this indicates that with regard to the occurrence of blow-up the criticality of the decay rate 1 3 , as previously found for the fluid-free counterpart of ( ⋆ ), remains essentially unaffected by fluid interaction of the type considered here.