An optimal regularity criterion for the Navier–Stokes equations proved by a blow-up argument. (April 2021)

• Record Type: Journal Article
• Authors: Skalak, Zdenek
• Is Part Of: Nonlinear analysis. Volume 58(2021)
• Publication Date: 2021-04
• Subjects: Navier–Stokes equations -- Conditional regularity -- Regularity criteria -- Rescaling procedure; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2020.103207
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We study the conditional regularity for the incompressible Navier–Stokes equations in terms of one directional derivative of the velocity field. We show that if u is a weak solution in the whole three dimensional space and ∂ 3 u ∈ L p ( 0, T ; L q ( R 3 ) ), T > 0, where 2 ∕ p + 3 ∕ q = 2 and q ∈ ( 3, 19 ∕ 6 ], then u is regular on ( 0, T ], crossing so for the first time the barrier q = 3 . The proof is based on a suitable combination of a blow-up argument and mutual estimates of certain integral quantities pertained to the rescaled solution.