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

Hantaek Bae

doi:10.3934/dcds.2011.29.769

Article Contents

2011, Volume 29, Issue 3: 769-801. Doi: 10.3934/dcds.2011.29.769

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

Hantaek Bae^1,

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

Received: January 2010

Revised: June 2010

Published: July 2011

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Abstract

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.

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Mathematics Subject Classification: 35K51, 76D05.

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