Regularity of traveling free surface water waves with vorticity

TL;DR

This paper proves that the streamlines and free surface of steady water waves with vorticity are real analytic under certain conditions, and establishes the regularity of the stream function based on the vorticity's smoothness.

Contribution

It demonstrates the real analyticity of all streamlines, including the free surface, for steady water waves with vorticity, extending regularity results to both periodic and solitary waves.

Findings

Streamlines and free surface are real analytic if wave speed exceeds horizontal velocity.

Stream function inherits Gevrey regularity from vorticity, including analyticity for Gevrey index 1.

Regularity results apply to both periodic and solitary water waves.

Abstract

We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a H\"{o}lder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index $s$, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index $s$ equals to 1, then we obtain analyticity of the stream function. The regularity results hold for both periodic and solitary water waves.