  In this paper, we consider an incompressible viscous flow without surface
tension in a finite-depth domain of three dimensions, with free top boundary
and fixed bottom boundary. This system is governed by a Naiver-Stokes equation
in above moving domain and a transport equation for the top boundary.
Traditionally, we consider this problem in Lagrangian coordinates with
perturbed linear form. In the series of papers [1], [2] and [3], I. Tice and Y.
Guo introduced a new framework using geometric structure in Eulerian
coordinates to study both local and global wellposedness of this system.
Following this path, we extend their result in local wellposedness from small
data case to arbitrary data case. Also, we give a simpler proof for global
wellposedness in infinite domain. Other than the geometric energy estimates,
time-dependent Galerkin method, and interpolation estimate with Riesz potential
and minimum count, which are introduced in these papers, we utilize three new
techniques: (1) using \epsilon-Poisson integral to construct a diffeomorphism
between fixed domain and moving domain; (2) using bootstrapping argument to
prove a comparison result for steady Navier-Stokes equation for arbitrary data
of free surface; (3) redefining the energy and dissipation to replace the
original complicated bootstrapping argument to show interpolation estimate.
