We consider a viscous incompressible fluid below the air and above a fixed bottom. The fluid dynamics is governed by the gravity-driven incompressible Navier-Stokes equations, and the effect of surface tension is neglected on the free surface. The global well-posedness and long-time behavior of solutions near equilibrium have been intriguing questions since Beale (\emph{Comm. Pure Appl. Math.} 34 (1981), no. 3, 359--392). It had been thought that certain low frequency assumption of the initial data is needed to derive an integrable decay rate of the velocity so that the global solutions in $3D$ can be constructed, while the global well-posedness in $2D$ was left open. In this paper, by exploiting the anisotropic decay rates of the velocity, which are even not integrable, we prove the global well-posedness in both $2D$ and $3D$, without any low frequency assumption of the initial data. One of key observations here is a cancelation in nonlinear estimates of the viscous stress tensor term in the bulk by using Alinhac good unknowns, when estimating the energy evolution of the highest order horizontal spatial derivatives of the solution.

中文翻译：

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没有表面张力的粘性表面波的各向异性衰减和全局适定性

我们考虑在空气下方和固定底部上方的粘性不可压缩流体。流体动力学由重力驱动的不可压缩 Navier-Stokes 方程控制，表面张力对自由表面的影响被忽略。自 Beale (\emph{Comm. Pure Appl. Math.} 34 (1981), no. 3, 359--392) 以来，接近平衡的解的全局适定性和长期行为一直是有趣的问题。曾认为需要初始数据的某些低频假设来导出速度的可积衰减率，以便可以构建 $3D$ 中的全局解，而 $2D$ 中的全局适定性是开放的. 在本文中，通过利用速度的各向异性衰减率，甚至不可积分，我们证明了 $2D$ 和 $3D$ 的全局适定性，没有对初始数据进行任何低频假设。这里的一个关键观察结果是在估计解的最高阶水平空间导数的能量演化时，通过使用 Alinhac 良好的未知数来取消对体中粘性应力张量项的非线性估计。