Two-dimensional water waves with constant vorticity and general bottom topography

TL;DR

This paper extends the mathematical framework for two-dimensional water waves with constant vorticity over arbitrary bottom topographies, including analysis of key operators and proving local well-posedness.

Contribution

It generalizes existing formulations to include general bottom topography and analyzes the associated operators, establishing foundational results for this class of water wave problems.

Findings

Generalized Zakharov-Craig-Sulem formulation for variable bottom topography.

Extended analysis of the Dirichlet-Neumann operator and related operators.

Proved local well-posedness for the water wave problem with constant vorticity and general bottom.

Abstract

In this paper we consider two-dimensional water waves with constant vorticity, under the action of gravity and surface tension, in a fluid domain with finite depth and general bottom topography. We present a formulation which generalizes the one by Zakharov-Craig-Sulem for irrotational water waves, and the one by Constantin-Ivanov-Prodanov for water waves with constant vorticity and flat bottom topography. We study in detail an operator which appears in such formulation, extending well-known results for the classical Dirichlet-Neumann operator, such as an analiticity result, the Taylor expansion in homogeneous powers of the wave profile, and a paralinearization formula. As an application, we prove a local well-posedness result.