On the Serrin-type condition on one velocity component for the Navier-Stokes equations

Abstract

In this paper we consider the regularity problem of the Navier-Stokes equations in R 3 . We show that the Serrin-type condition imposed on one component of the velocity u 3 ∈ L p (0, T ; L q (R 3 )) for with 2 p + 3 q < 1, 3 < q ≤ +∞ implies the regularity of the weak Leray solution u : R 3 × (0, T ) → R 3 with the initial data belonging to L 2 (R 2 ) ∩ L 3 (R 3 ). The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

Prodi–Serrin condition for 3D Navier–Stokes equations via one directional derivative of velocity

“…Remark 1.2. Theorem 1.1 is a direct consequence of Theorem 1.3 and Remark 1.4 via a similar compactness argument in [4]. Moreover, the initial data u 0 ∈ L 2 (R 3 ) ∩ L 3 (R 3 ) implies the local-in-time regularity of weak solution, thus the weak solution is actually suitable weak solution.…”

“…We also shall use the same notation as that in Chae-Wolf [4]. For x = (x 1 , x 2 , x 3 ) ∈ R 3 , we denote x ′ = (x 1 , x 2 ) the horizontal variable.…”

“…[10] proved that if u 3 ∈ L p (0, T ; Ḣ1/2+2/p (R 3 )) with 2 ≤ p < +∞, the solution u is regular in (0, T ]. Very recently, D. Chae and J. Wolf [4] made an important progress and obtained the regularity of solution to (1.1) under the condition u 3 ∈ L p 0, T ; L q R 3 , 2 p + 3 q < 1, 3 < q ≤ ∞. (1.4) W. Wang, D. Wu and Z. Zhang [20] improved to u 3 ∈ L p,1 0, T ; L q R 3 , 2 p + 3 q = 1, 3 < q < ∞.…”

“…The proof of Theorem 1.3 follows from the ideal of local energy estimates introduced in [2,4,20]. The non-trivial part is to establish the necessary a priori estimates of the quantity…”

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…”