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

Zhouyu Li (School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi, China)

Daoguo Zhou (School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan, China)

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.

Received 2 September 2020

Published 19 February 2021