Regularity criteria for 3D Navier–Stokes equations in terms of finite frequency parts of velocity in Ḃ∞, ∞−1. (January 2020)

• Record Type: Journal Article
• Authors: Ri, Myong-Hwan
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: 35Q30 -- 76D05; Navier–Stokes equations -- Regularity criterion -- Critical space; 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.2019.111619
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular if L ∞ ( 0, T ; B ̇ ∞, ∞ − 1 ) -norm of a suitable low frequency part of u is bounded by a scaling invariant constant depending on the kinematic viscosity ν and initial value u 0 . Moreover, we prove that a Leray–Hopf weak solution is regular if its medium frequency part with Fourier modes between k ∕ 2 and k for a sufficiently high wavenumber k has small B ̇ ∞, ∞ − 1 -norm. Our results imply that energy concentration at sufficiently high wavenumber bands bringing about singularity of the incompressible Navier–Stokes flow can be prevented by an "energy threshold" at a lower wavenumber band.