Local regularity criteria of the 3D Navier–Stokes and related equations. (July 2016)

• Record Type: Journal Article
• Authors: Wang, Yanqing; Wu, Gang
• Is Part Of: Nonlinear analysis. Volume 140(2016)
• Page Start: 130
• Page End: 144
• Publication Date: 2016-07
• Subjects: 76D03 -- 76D05 -- 35B33 -- 35Q35; Navier–Stokes equations -- Suitable weak solutions -- Regularity; 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.2016.03.014
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we are concerned with the local regularity of suitable weak solutions to the 3D Navier–Stokes equations and its generalized case with fractional power of the Laplacian ( − Δ ) α ( 3 / 4 ≤ α < 1 ). In the first part, we derive some ϵ -regularity criteria in terms of the deformation tensor D ( u ) for the classical Navier–Stokes equations. In the second part, for the fractional case, we obtain some regularity conditions for suitable weak solutions including the velocity u, the gradient of the velocity ∇ u, the rotation tensor Ω ( u ) and the deformation tensor D ( u ) . This generalizes the results obtained by Gustafson et al. (2007).