Bounds for solutions to the problem of steady water waves with vorticity

TL;DR

This paper establishes bounds for solutions of steady water waves with vorticity, analyzing free-surface profiles and flow characteristics, including counter-currents, under specific vorticity conditions.

Contribution

It provides new bounds for stream functions and free-surface profiles in steady water waves with vorticity, and identifies conditions for counter-current flows.

Findings

Bounds for stream functions and free-surface profiles are derived.

Counter-currents occur when the free surface exceeds a critical height.

Wave flows exhibit specific flow structures under vorticity constraints.

Abstract

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the total head are obtained under the assumption that the vorticity distribution is a locally Lipschitz function. It is also shown that wave flows have counter-currents in the case when the infimum of the free surface profile exceeds a certain critical value.