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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