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

Abstract

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 (1981) [2]. It was proved by Guo and Tice (2013) [6] that with certain additional low horizontal frequency assumption of the initial data in 3D an integrable decay rate of the velocity is obtained so that the global unique solution 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 cancellation 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.