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

### Wendong Wang

Dalian University of Technology, China### Di Wu

South China University of Technology, Guangzhou, China### Zhifei Zhang

Peking University, Beijing, China

## Abstract

The classical Ladyzhenskaya–Prodi–Serrin regularity criterion states that if the Leray weak solution $u$ of the Navier–Stokes equations satisfies $u\in L^q(0,T; L^p(\mathbb{R}^3))$ with $\frac{2}{q} +\frac{3}{p} \leq 1$, $p>3$, then it is regular in $\mathbb{R}^3\times (0,T)$. In this paper, we prove that the Leray weak solution is also regular in $\mathbb{R}^3\times (0,T)$ under the scaling-invariant Serrin condition imposed on one component of the velocity, i.e., $u_3\in L^{q,1}(0,T; L^p(\mathbb{R}^3))$ with $\frac{2}{q} +\frac{3}{p} \leq 1$, $3<p<+\infty$. This result means that if the solution blows up at a time, then all three components of the velocity have to blow up simultaneously.

## Cite this article

Wendong Wang, Di Wu, Zhifei Zhang, Scaling-invariant Serrin criterion via one velocity component for the Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire (2023),

DOI 10.4171/AIHPC/77