A two-phase problem for capillary-gravity waves and theBenjamin-Ono equation

Kenta Ohi; Tatsuo Iguchi

doi:10.3934/dcds.2009.23.1205

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2009, Volume 23, Issue 4: 1205-1240. Doi: 10.3934/dcds.2009.23.1205

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A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation

Kenta Ohi^1, and
Tatsuo Iguchi^2,

1.
Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551
2.
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522

Received: July 31, 2007

Revised: September 30, 2007

Published: October 2009

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Abstract

The Benjamin-Ono equation is known as a model of internal long waves in stratified fluids or two-fluid systems. In this paper, we consider the validity of this type of modeling of a two-phase problem for capillary-gravity waves, which is a free boundary problem for the incompressible Euler equation with the irrotational condition. We show that the solutions of the free boundary problem split up into two waves and the shape of each wave is governed by the Benjamin-Ono equation in a slow time scale.

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Mathematics Subject Classification: Primary: 35Q35, 35Q53; Secondary: 76B55, 76B07.

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