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 $\partial_3 u,$ satisfies $\partial_3 u \in L^{p_0,1}(0,T; L^{q_0}(\mathbb{R}^3))$ with $\frac{2}{p_{0}}+\frac{3}{q_{0}}=2$ and $\frac{3}{2}<q_0< +\infty,$ then the weak solution is regular on $(0,T].$ The proof is based on the new local energy estimates introduced by Chae-Wolf (arXiv:1911.02699) and Wang-Wu-Zhang (arXiv:2005.11906).