Steady periodic water waves with constant vorticity: regularity and local bifurcation

TL;DR

This paper investigates the existence and regularity of steady periodic water waves with constant vorticity, employing conformal mappings and bifurcation theory to establish the presence of small amplitude waves, including those with stagnation points.

Contribution

It introduces a new formulation transforming the free-boundary problem into a pseudodifferential equation, enabling regularity analysis and bifurcation-based existence results for waves with constant vorticity.

Findings

Regularity of solutions established

Existence of small amplitude waves proven

Waves with stagnation points included

Abstract

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.