Surface waves on steady perfect-fluid flows with vorticity

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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 languageEnglish
Pages (from-to)975-1007
Number of pages33
JournalCommunications on Pure and Applied Mathematics
Volume64
Issue number7
Early online date14 Mar 2011
DOIs
Publication statusPublished - Jul 2011

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