Well-posedness of a model for water waves with viscosity

David M. Ambrose; Jerry L. Bona; David P. Nicholls

doi:10.3934/dcdsb.2012.17.1113

Article Contents

2012, Volume 17, Issue 4: 1113-1137. Doi: 10.3934/dcdsb.2012.17.1113

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Well-posedness of a model for water waves with viscosity

1.
Department of Mathematics, Drexel University, Philadelphia, PA 19104
2.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607
3.
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607

Received: December 31, 2010

Revised: July 31, 2011

Published: May 2012

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Abstract

The water wave equations of ideal free-surface fluid mechanics are a fundamental model of open ocean movements with a surprisingly subtle well-posedness theory. In consequence of both theoretical and computational difficulties with the full water wave equations, various asymptotic approximations have been proposed, analyzed and used in practical situations. In this essay, we establish the well-posedness of a model system of water wave equations which is inspired by recent work of Dias, Dyachenko, and Zakharov (Phys. Lett. A, 372:2008). The model in question includes dissipative effects and is weakly nonlinear. The present contribution is a first step in a larger program centered around the Dias-Dychenko-Zhakharov system.

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Mathematics Subject Classification: Primary: 35Q35, 76B15; Secondary: 76B03, 76B07.

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