### Abstract

We consider a perfect fluid in periodic motion between parallel vertical walls, above a horizontal bottom and beneath a free boundary at constant atmospheric pressure. Gravity acts vertically downwards. Suppose the underlying flow is two-dimensional in a vertical plane orthogonal to the walls and satisfies the constant-pressure condition on the free boundary where surface tension is neglected. Suppose also that it is symmetric about a plane midway between the walls, and periodic in time. Such motion, which can be extended to give a two-dimensional flow of infinite horizontal extent that is periodic in space as well as in time, is referred to as a standing wave. Unlike progressive (or steady) Stokes waves, standing waves are not stationary relative to a moving reference frame.

The purpose of this paper is to show how the Nash-Moser iteration method can be adapted to give a rigorous proof of the existence of small-amplitude standing waves for which the normal component of pressure gradient on the free surface satisfies additional constraints. These constraints are imposed in advance to facilitate the a priori bounds needed for the Nash-Moser method and only solutions satisfying them have been found.(They have no obvious analogue in the theory of Stokes waves.)

The presentation is self-contained and includes a version of the Nash-Moser theorem tailored for the purpose. The imposed constraints are used to define a manifold upon which iteration is carried out and a detailed account from first principles of the a priori bounds required to implement the method is given. We use the Lagrangian form of the Euler equations throughout.

The purpose of this paper is to show how the Nash-Moser iteration method can be adapted to give a rigorous proof of the existence of small-amplitude standing waves for which the normal component of pressure gradient on the free surface satisfies additional constraints. These constraints are imposed in advance to facilitate the a priori bounds needed for the Nash-Moser method and only solutions satisfying them have been found.(They have no obvious analogue in the theory of Stokes waves.)

The presentation is self-contained and includes a version of the Nash-Moser theorem tailored for the purpose. The imposed constraints are used to define a manifold upon which iteration is carried out and a detailed account from first principles of the a priori bounds required to implement the method is given. We use the Lagrangian form of the Euler equations throughout.

Original language | English |
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Pages (from-to) | 1-83 |

Number of pages | 83 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 159 |

Issue number | 1 |

DOIs | |

Publication status | Published - 31 Aug 2001 |

## Cite this

Plotnikov, P. I., & Toland, J. F. (2001). Nash-Moser theory for standing water waves.

*Archive for Rational Mechanics and Analysis*,*159*(1), 1-83. https://doi.org/10.1007/PL00004246