Contents Online

# Dynamics of Partial Differential Equations

## Volume 18 (2021)

### Number 1

### On endpoint regularity criterion of the 3D Navier–Stokes equations

Pages: 71 – 80

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n1.a5

#### Authors

#### Abstract

Let $(u,\pi)$ with $u = (u_1, u_2, u_3)$ be a suitable weak solution of the three-dimensional Navier–Stokes equations in $\mathbb{R}^3 \times (0, T)$. Denote by $\dot{\mathcal{B}}^{-1}_{\infty,\infty}$ the closure of $C^\infty_0$ in $\dot{B}^{-1}_{\infty,\infty}$. We prove that if $u \in L^\infty (0, T; \dot{B}^{-1}_{\infty,\infty}), u(x, T) \in \dot{\mathcal{B}}^{-1}_{\infty,\infty})$, and $u_3 \in L^\infty (0, T; L^{3,\infty})$ or $u_3 \in L^\infty (0, T; \dot{B}^{-1+3/p}_{p,q})$ with $3 \lt p, q \lt \infty$, then $u$ is smooth in $\mathbb{R}^3 \times (0, T]$. Our result improves a previous result established by Wang and Zhang [*Sci. China Math.* **60**, 637-650 (2017)].

#### Keywords

Navier–Stokes equations, regularity criterion, endpoint space

#### 2010 Mathematics Subject Classification

Primary 35Q30. Secondary 76D05.

Z. Li was partially supported by the National Natural Science Foundation of China (No. 11601423) and the Natural Science Foundation of Shaanxi Province (No. 2020JQ-120). D. Zhou was partially supported by the National Natural Science Foundation of China (No. 11971446).

Received 2 September 2020

Published 19 February 2021