Some new regularity criteria for the Navier–Stokes equations in terms of one directional derivative of the velocity field

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 < ∞ .