Regularity conditions of 3D Navier–Stokes flow in terms of large spectral components. (April 2015)

• Record Type: Journal Article
• Authors: Kim, Namkwon; Kwak, Minkyu; Yoo, Minha
• Is Part Of: Nonlinear analysis. Volume 116(2015)
• Page Start: 75
• Page End: 84
• Publication Date: 2015-04
• Subjects: Navier–Stokes equation -- Serrin condition -- Large spectral components; 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.2014.12.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We develop Ladyzhenskaya–Prodi–Serrin type spectral regularity criteria for 3D incompressible Navier–Stokes equations in a torus. Concretely, for any N > 0, let w N be the sum of all spectral components of the velocity fields whose wave numbers | k i | > N for all i = 1, 2, 3 . Then, we show that for any N > 0, the finiteness of the Serrin type norm of w N implies the regularity of the flow. It implies that if the flow breaks down in a finite time, the energy of the velocity fields cascades down to the arbitrarily large spectral components of w N and corresponding energy spectrum, in some sense, roughly decays slower than κ − 2 .