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).

References in corpus (1)

• Scaling invariant Serrin criterion via one velocity component for the Navier-Stokes equations