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On endpoint regularity criterion of the 3D Navier–Stokes equations
Dynamics of Partial Differential Equations ( IF 1.1 ) Pub Date : 2021-03-01 , DOI: 10.4310/dpde.2021.v18.n1.a5 Zhouyu Li 1 , Daoguo Zhou 2
Dynamics of Partial Differential Equations ( IF 1.1 ) Pub Date : 2021-03-01 , DOI: 10.4310/dpde.2021.v18.n1.a5 Zhouyu Li 1 , Daoguo Zhou 2
Affiliation
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)].
更新日期：2021-02-19
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