Critical regularity criteria for Navier–Stokes equations in terms of one directional derivative of the velocity. (7th December 2020)

• Record Type: Journal Article
• Authors: Chen, Hui; Fang, Daoyuan; Zhang, Ting
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 6(2021)
• Page Start: 5123
• Page End: 5132
• Publication Date: 2020-12-07
• Subjects: Navier–Stokes equations -- one directional derivative -- Prodi–Serrin conditions -- regularity -- the decomposition of velocity; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.7097
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we consider the 3D Navier–Stokes equations in the whole space. We investigate some new inequalities and a priori estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely, ∂ 3 u ∈ L p ( ( 0, T ) ; L q ( ℝ 3 ) ), 2 p + 3 q = 2, 3 2 < q ≤ 6 . Moreover, we extend the range of q while the solution is axisymmetric, that is, the axisymmetric solution u is regular in (0, T ], if ∂ 3 u 3 ∈ L p ( ( 0, T ) ; L q ( ℝ 3 ) ), 2 p + 3 q = 2, 3 2 < q < ∞ .