Scaling invariant Serrin criterion via one velocity component for the Navier-Stokes equations

Abstract

In this paper, we prove that the Leray weak solution u : R 3 × (0, T ) → R 3 of the Navier-Stokes equations is regular in R 3 × (0, T ) under the scaling invariant Serrin condition imposed on one component of the velocity u3 ∈ L q,1 (0, T ; L p (R 3 )) withThis result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

Quantitative transfer of regularity of the incompressible Navier-Stokes equations from $\mathbb{R}^3$ to the case of a bounded domain

“…In fact in that case M ≤ C( u L ∞ ((0,T );L ∞ ) , u 0 H 5 , u 0 W 5,∞ ), which can be shown by an iteration (with respect to the order of the derivatives) of Gronwall inequalities (see [11] or [32,Theorem 7.1] for details). There are a number of well-known sufficient conditions that guarantee that a given Leray weak solution is in fact strong [7,10,21,24,27,29,32,33,34,39]. One of them is the Ladyzhenskaya-Prodi-Serrin [24,29,33] condition, u ∈ L p ((0, T ); L q (R 3 )), where p ∈ [2, ∞), q ∈ (3, ∞] are such that…”

“…+ ∇v 2 N εe 2N 4 C 2 tat each time, where we used(39) and(23) in the second inequality, as well as (41) and (42) in the last. Thus(18) gives∇π L p ((0,t);L 2 (B R ) p F − (v • ∇)r − (r • ∇)v L p ((0,t);L 2 (B R )) t N εe 2N 4 C 2 t…”

Serrin-type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component

“…if the weak solution u satisfies (1.3) u ∈ L p (0, T ; L q (R 3 )), 2 p + 3 q = 1, 3 q ∞, then the weak solution is regular in (0, T ]. There are several notable results [3,4,5,6,11,24] to weaken the above criteria by imposing constraints only on partial components or directional derivatives of velocity field. In particular, D. Chae and J. Wolf [3] made an important progress and obtained the regularity of solution under the condition…”

“…W. Wang, D. Wu and Z. Zhang [24] improved to u 3 ∈ L p,1 0, T ; L q R 3 , 2 p + 3 q = 1, 3 < q < ∞. (1.5) Throughout this paper, L p,1 denotes the Lorentz space with respect to the time variable.…”

“…In this paper, inspired by [3,24], we focus on the regularity criteria to the MHD equation (1.1) involving the terms u 3 and b 3 only. Theorem 1.2.…”