Surface waves on steady perfect-fluid flows with vorticity

Reference:

Burton, G. R. and Toland, J. F., 2011. Surface waves on steady perfect-fluid flows with vorticity. Communications on Pure and Applied Mathematics, 64 (7), pp. 975-1007.

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Official URL:

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

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.

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