On the existence of extreme waves and the Stokes conjecture with vorticity

TL;DR

This paper investigates the existence and geometric properties of extreme gravity water waves with vorticity, proving convergence of regular waves to extreme waves and characterizing their crest profiles.

Contribution

It demonstrates the convergence of regular waves to extreme waves with stagnation points and characterizes the crest profile angles for waves with vorticity.

Findings

Regular waves converge to extreme waves with stagnation points.

Extreme wave profiles have either a 120° corner or a horizontal tangent at stagnation points.

If vorticity is nonnegative near the surface, the crest has a 120° corner.

Abstract

This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of $120^\circ$ or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of $120^\circ$ if the vorticity is nonnegative near the free surface.