Stability and large-time behavior are essential properties of solutions to many partial differential equations (PDEs) and play crucial roles in many practical applications. When there is full Laplacian, many techniques such as the Fourier splitting method have been created to obtain the large-time decay rates. However, when a PDE is anisotropic and involves only partial dissipation, these methods no longer apply and no effective approach is currently available. This paper aims at the stability and large-time behavior of the 3D anisotropic Navier-Stokes equations. We present a systematic approach to obtain the optimal decay rates of the stable solutions emanating from a small data. We establish that, if the initial velocity is small in the Sobolev space $H^4({\mathbb{R}}^3)\cap H_h^{-\sigma }({\mathbb{R}}^3)$, then the anisotropic Navier-Stokes equations have a unique global solution, and the solution and its first-order derivatives all decay at the optimal rates. Here $H_h^{-\sigma }$ with $\sigma >0$ denotes a Sobolev space with negative horizontal index.

稳定性和长时间行为是许多偏微分方程（PDE）解的基本属性，并且在许多实际应用中起着至关重要的作用。当存在完整的拉普拉斯算子时，已经创建了许多技术（例如傅立叶分裂方法）来获得长时间的衰减率。但是，当PDE是各向异性的并且仅涉及部分耗散时，这些方法将不再适用，并且目前没有有效的方法。本文旨在研究3D各向异性Navier-Stokes方程的稳定性和长时间行为。我们提出了一种系统的方法，以获取由小数据产生的稳定解的最佳衰减率。我们确定，如果Sobolev空间中的初始速度较小$H^4（{\mathbb{[R}}^3）\cap H_H^{-\sigma }（{\mathbb{[R}}^3）$，则各向异性的Navier-Stokes方程具有唯一的整体解，并且该解及其一阶导数都以最佳速率衰减。这里$H_H^{-\sigma }$ 和 $\sigma >0$ 表示水平索引为负的Sobolev空间。