Stokes waves with vorticity

TL;DR

This paper proves the existence of symmetric, monotone periodic water waves with vorticity using advanced bifurcation and topological methods, establishing a global continuum of solutions for a nonlinear elliptic boundary value problem.

Contribution

It introduces a novel application of singular bifurcation theory and topological methods to establish global solutions for water waves with vorticity, extending previous results to a broader class of vorticities.

Findings

Existence of a global continuum of solutions with symmetric, monotone profiles

Application of generalized degree and bifurcation theories to nonlinear elliptic problems

Solutions characterized by a fixed semi-infinite strip boundary value problem

Abstract

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep water under gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite strip with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. Each solution on the continuum has a symmetric and monotone wave profile. The proof uses a generalized degree theory, global bifurcation theory and Wyburn's lemma in topology, combined with the Schauder theory for elliptic problems and the maximum principle.