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

Abstract

In this paper, we consider the conditional regularity of weak solution to the 3D Navier–Stokes equations. More precisely, we prove that if one directional derivative of velocity, say ∂ 3 u , satisfies ∂ 3 u ∈ L p 0 , 1 ( 0 , T ; L q 0 ( R 3 ) ) with 2 p 0 + 3 q 0 = 2 and 3 2 < q 0 < + ∞ , then the weak solution is regular on ( 0 , T ] . The proof is based on the new local energy estimates introduced by Chae-Wolf (2019) [4] and Wang-Wu-Zhang (2020) [21] .