In \cite{LZ4}, the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the classical Navier-Stokes system has a global unique solution. The goal of this paper is to extend this type of result to the 3-D anisotropic Navier-Stokes system $(ANS)$ with only horizontal dissipation. More precisely, given initial data $u_0=(u_0^\h,u_0^3)\in \cB^{0,\f12},$ $(ANS)$ has a unique global solution provided that $|D_\h|^{-1}\pa_3u_0$ is sufficiently small in the scaling invariant space $\cB^{0,\f12}.$

中文翻译：

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具有小单向导数的 3-D 各向异性 Navier-Stokes 系统的全局适定性

在\cite{LZ4}中，作者证明了只要初始速度的单向导数在一些标度不变空间中足够小，那么经典的Navier-Stokes系统就有全局唯一解。本文的目标是将这种类型的结果扩展到只有水平耗散的 3-D 各向异性 Navier-Stokes 系统 $(ANS)$。更准确地说，给定初始数据 $u_0=(u_0^\h,u_0^3)\in \cB^{0,\f12}，$$(ANS)$ 具有唯一的全局解，条件是 $|D_\h| ^{-1}\pa_3u_0$ 在缩放不变空间 $\cB^{0,\f12}.$ 中足够小