论文标题
在3D Navier-Stokes方程的端点规律性标准下
On Endpoint Regularity Criterion of the 3D Navier-Stokes equations
论文作者
论文摘要
令$（u，π）$带有$ u =（u_1，u_2，u_3）$是$ \ mathbb {r}^3 \ times [0，t] $的三维navier-stokes方程的合适解决方案。用$ \ dot {\ mathcal {b}}^{ - 1} _ {\ infty，\ infty} $表示$ c_0^\ infty $ in $ \ dot {b}^{ - 1} _ { - 1} _ {\ infty，iffty，infty，\ infty，\ infty，\ infty，\ infty} $。我们证明，如果$ u \ in l^\ infty（0，t; \ dot {b}^{ - 1} _ {\ infty，\ infty}）$，$ u（x，x，x，t）\ in \ dot {\ dot {\ mathcal {\ mathcal {b}}} l^\ infty（0，t; l^{3，\ infty}）$或$ u_3 \ in l^\ infty（0，t; \ dot {b}^{ - 1+3/p} _ {p，q，q} $，带有$ 3 <p，q <p，q <\ iffty $，$ unty $ $ $ $ $ $ $ us fmain t] $。我们的结果改善了Wang和Zhang [Sci的先前结果。中国数学。 60，637-650（2017）]。
Let $(u, π)$ 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 $C_0^\infty$ 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 $3<p, q< \infty$, 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)].
