Well-Posedness and Decay of the Viscous Surface Wave

Abstract

In this paper, we consider an incompressible viscous flow without surface tension in a finite-depth domain of three dimensions, with a free top boundary and a fixed bottom boundary. The system is governed by the Navier--Stokes equations in this moving domain. Traditionally, this problem can be analyzed in the Lagrangian coordinates as a perturbation of linear equations in a fixed domain. In the series of papers [Anal. PDE, 6 (2013), pp. 287--369; Anal. PDE, 6 (2013), pp. 1429--1533; Arch. Ration. Mech. Anal., 207 (2013), pp. 459--531], Tice and Guo introduced a new framework using the geometric structure in the Eulerian coordinates to study both local and global well-posedness and decay of this system. Following this path, we extend their result in local well-posedness from the small data case to the general data case. Also, we give a simpler proof of global well-posedness in the small data case with horizontally infinite cross section. Other than the geometric energy estimates, the time-dependent Galerkin method, and the interpolation estimates with Riesz potential and minimal counts, which are introduced in these papers, we utilize three new techniques: (1) using the $\epsilon$-Poisson integral to construct a diffeomorphism between the fixed domain and the moving domain; (2) using a bootstrapping argument to prove the comparison estimates in the steady Navier--Stokes equations for general data of free surface; and (3) redefining the energy and dissipation to simplify the original complicated bootstrapping argument to show the interpolation estimates.