Some new regularity criteria for the Navier–Stokes equations in terms of one directional derivative of the velocity field. (December 2021) - Giang, N.V.; Khai, D.Q.
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 < ∞ .