Anisotropic decay and global well-posedness of viscous surface waves without surface tension

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.