On the existence of steady periodic capillary-gravity stratified water waves

TL;DR

This paper proves the existence of small steady periodic capillary-gravity water waves in stratified flows, including flows with stagnation points, using bifurcation theory and elliptic estimates.

Contribution

It demonstrates the existence of both laminar and non-laminar solutions for stratified water waves with surface tension and stagnation points, expanding prior understanding.

Findings

Existence of small steady periodic waves proven.

Both laminar and non-laminar solutions established.

Bifurcation theory applied to stratified water wave equations.

Abstract

We prove the existence of small steady periodic capillary-gravity water waves for general stratified flows, where we allow for stagnation points in the flow. We establish the existence of both laminar and non-laminar flow solutions for the governing equations. This is achieved by using bifurcation theory and estimates based on the ellipticity of the system, where we regard, in turn, the mass-flux and surface tension as bifurcation parameters.