Continuous alignment of vorticity direction prevents the blow-up of the Navier–Stokes flow under the no-slip boundary condition. (December 2019)

• Record Type: Journal Article
• Authors: Giga, Yoshikazu; Gu, Zhongyang; Hsu, Pen-Yuan
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: Vorticity direction -- L∞ theory -- Liouville type theorem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111579
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is concerned with a regularity criterion based on vorticity direction for Navier–Stokes equations in a three-dimensional bounded domain under the no-slip boundary condition. It asserts that if the vorticity direction is uniformly continuous in space uniformly in time, there is no type I blow-up. A similar result has been proved for a half space by Y. Maekawa and the first and the last authors (2014). The result of this paper is its natural but non-trivial extension based on L ∞ theory of the Stokes and the Navier–Stokes equations recently developed by K. Abe and the first author.