Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity

TL;DR

This paper establishes the equivalence of three weak formulations for steady periodic water waves with discontinuous vorticity, connecting classical Euler and stream function approaches to a novel height-based formulation in Hölder spaces.

Contribution

It proves the equivalence of multiple weak formulations for water waves with discontinuous vorticity, including a new height-based approach, in Hölder spaces.

Findings

Proves equivalence between Euler and stream function formulations and a height formulation.

Extends the theory of water waves to cases with discontinuous vorticity.

Uses Hölder space regularity for weak solutions.

Abstract

In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and stream function formulation of the governing equations are equivalent to a weak version of the recently introduced modified-height formulation. The weak solutions of these formulations are considered in H\"older spaces.