Some new regularity criteria for the Navier–Stokes equations in terms of one directional derivative of the velocity field. (December 2021)

• Record Type: Journal Article
• Authors: Giang, N.V.; Khai, D.Q.
• Is Part Of: Nonlinear analysis. Volume 62(2021)
• Publication Date: 2021-12
• Subjects: Navier–Stokes equations -- Regularity criterion -- Energy inequality -- Weak solution -- One velocity component; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2021.103379
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We establish some regularity criteria for the solutions to the Navier–Stokes equations in the full three-dimensional space in terms of one directional derivative of the velocity field. Revising the method used by Zujin Zhang (2018), we show that a weak solution u is regular on (0, T] provided that ∂ u ∂ x 3 ∈ L p ( 0, T ; L q ( R 3 ) ) with s = 2 for 3 ≤ q ≤ 6, 11 6 < s ≤ 2 for 6 ≤ q ≤ 6 6 s − 11 where s = 2 p + 3 q . They improve the known results 2 p + 3 q = 3 2 for 2 ≤ q ≤ ∞, 2 p + 3 q ≤ 8 5 + 9 11 q for 5 2 ≤ q < ∞ and 2 p + 3 q ≤ 14 11 + 3 5 q for 4 ≤ q < ∞ .