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arXiv · 2307.01702

Analytical Study of a generalised Dirichlet-Neumann operator and application to three-dimensional water waves on Beltrami flows

Mark D. Groves·Dag Nilsson·Stefano Pasquali·Erik Wahlén
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Abstract

In this paper we consider three-dimensional steady water waves with vorticity, under the action of gravity and surface tension; in particular we consider so-called Beltrami flows, for which the velocity field and the vorticity are collinear. We discuss a recent variational formulation of the problem which involves a generalisation of the classical Dirichlet-Neumann operator. We study this operator in detail, extending some well-known results for the classical Dirichlet-Neumann operator, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and the asymptotic expansion of its associated symbol. A new formulation of the problem as a single equation for the wave profile is also presented and discussed in a similar vein. As an application of these results we rigorously prove existence of doubly periodic gravity-capillary steady waves and construct approximate doubly periodic gravity steady waves.

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Mark D. Groves, Dag Nilsson, Stefano Pasquali, Erik Wahlén. 2023-07-04. Analytical Study of a generalised Dirichlet-Neumann operator and application to three-dimensional water waves on Beltrami flows. https://arxiv.org/abs/2307.01702

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