### Abstract

In this theory of two-dimensional steady periodic surface waves on flows under gravity, the functional dependence of vorticity on the stream function is a priori unknown. It is shown that when the vorticity distribution function is given, weak solutions arise from minimization of the total energy. The fact that vorticity is indeed a function of the stream function is then an infinite-dimensional Lagrange multiplier rule, the consequence of minimizing energy subject to the vorticity distribution function being prescribed.

To illustrate the idea with a minimum of technical difficulties, the existence of non-trivial waves with a prescribed distribution of vorticity on the surface of a fluid confined beneath an elastic sheet is proved. The theory does not distinguish between irrotational waves and waves with locally square-integrable vorticity.

Original language | English |
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Pages (from-to) | 975-1007 |

Number of pages | 33 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 64 |

Issue number | 7 |

Early online date | 14 Mar 2011 |

DOIs | |

Publication status | Published - Jul 2011 |

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**Surface waves on steady perfect-fluid flows with vorticity.** / Burton, Geoffrey R; Toland, John F.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 64, no. 7, pp. 975-1007. https://doi.org/10.1002/cpa.20365

}

TY - JOUR

T1 - Surface waves on steady perfect-fluid flows with vorticity

AU - Burton, Geoffrey R

AU - Toland, John F

PY - 2011/7

Y1 - 2011/7

N2 - In this theory of two-dimensional steady periodic surface waves on flows under gravity, the functional dependence of vorticity on the stream function is a priori unknown. It is shown that when the vorticity distribution function is given, weak solutions arise from minimization of the total energy. The fact that vorticity is indeed a function of the stream function is then an infinite-dimensional Lagrange multiplier rule, the consequence of minimizing energy subject to the vorticity distribution function being prescribed. To illustrate the idea with a minimum of technical difficulties, the existence of non-trivial waves with a prescribed distribution of vorticity on the surface of a fluid confined beneath an elastic sheet is proved. The theory does not distinguish between irrotational waves and waves with locally square-integrable vorticity.

AB - In this theory of two-dimensional steady periodic surface waves on flows under gravity, the functional dependence of vorticity on the stream function is a priori unknown. It is shown that when the vorticity distribution function is given, weak solutions arise from minimization of the total energy. The fact that vorticity is indeed a function of the stream function is then an infinite-dimensional Lagrange multiplier rule, the consequence of minimizing energy subject to the vorticity distribution function being prescribed. To illustrate the idea with a minimum of technical difficulties, the existence of non-trivial waves with a prescribed distribution of vorticity on the surface of a fluid confined beneath an elastic sheet is proved. The theory does not distinguish between irrotational waves and waves with locally square-integrable vorticity.

UR - http://www.scopus.com/inward/record.url?scp=79953316611&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1002/cpa.20365

U2 - 10.1002/cpa.20365

DO - 10.1002/cpa.20365

M3 - Article

VL - 64

SP - 975

EP - 1007

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 7

ER -