Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity

TL;DR

This paper analyzes a generalized Dirichlet-Neumann operator for 3D water waves with vorticity, extending classical results without geometric restrictions, and provides detailed mathematical properties of the operator.

Contribution

It introduces a detailed study of a generalized Dirichlet-Neumann operator for water waves with vorticity, extending classical irrotational results to more general cases.

Findings

Extended Taylor expansion for the generalized operator

Computed the differential of the operator

Established paralinearization results without geometric restrictions

Abstract

In this paper we consider three-dimensional water waves with vorticity, under the action of gravity. We discuss a generalized Zakharov-Craig-Sulem formulation of the problem introduced by Castro and Lannes, which involves a generalized Dirichlet-Neumann operator. We study this operator in detail, extending some well-known results about the classical Dirichlet-Neumann operator for irrotational water waves, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and a paralinearization result. We stress the fact that no geometric condition on either the velocity field or the vorticity is assumed.