# American Institute of Mathematical Sciences

July  2011, 29(3): 769-801. doi: 10.3934/dcds.2011.29.769

## Solvability of the free boundary value problem of the Navier-Stokes equations

 1 Center For Scientific Computation And Mathematical Modeling, University of Maryland, 4125 CSIC Building, Paint Branch Drive, College Park, MD 20742, United States

Received  January 2010 Revised  June 2010 Published  November 2010

In this paper, we study the incompressible Navier-Stokes equations on a moving domain in $\mathbb{R}^{3}$ of finite depth, bounded above by the free surface and bounded below by a solid flat bottom. We prove that there exists a unique, global-in-time solution to the problem provided that the initial velocity field and the initial profile of the boundary are sufficiently small in Sobolev spaces.
Citation: Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
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