 
Summary: DISCRETE AND CONTINUOUS doi:10.3934/dcds.2011.29.xx
DYNAMICAL SYSTEMS
Volume 29, Number 3, March 2011 pp. 1XX
SOLVABILITY OF THE FREE BOUNDARY VALUE PROBLEM
OF THE NAVIERSTOKES EQUATIONS
Hantaek Bae
Center For Scientific Computation And Mathematical Modeling
University of Maryland
4125 CSIC Building, Paint Branch Drive
College Park, MD 20742, USA
(Communicated by Fanghua Lin)
Abstract. In this paper, we study the incompressible NavierStokes equations
on a moving domain in R3 of finite depth, bounded above by the free surface
and bounded below by a solid flat bottom. We prove that there exists a unique,
globalintime solution to the problem provided that the initial velocity field
and the initial profile of the boundary are sufficiently small in Sobolev spaces.
1. Introduction. In this paper, we study a viscous free boundary value problem
with surface tension. The NavierStokes equations describe the evolution of the
velocity field in the fluid body. With boundary conditions stated below, we have
the following system of equations:
