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

TL;DR

This paper analyzes a generalized Dirichlet-Neumann operator in the context of three-dimensional water waves with vorticity, providing new theoretical insights and applying them to prove the existence of specific steady wave solutions.

Contribution

It extends classical Dirichlet-Neumann operator results to a generalized form for Beltrami flows and introduces a new single-equation formulation for wave profiles.

Findings

Proved existence of doubly periodic gravity-capillary steady waves.

Constructed approximate doubly periodic gravity steady waves.

Extended classical operator results to a generalized setting.

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.