Regularity of traveling periodic stratified water waves with vorticity

TL;DR

This paper proves the real analyticity and Gevrey regularity of streamlines and the stream function in steady stratified water waves with vorticity, covering capillary, gravity, and capillary-gravity regimes.

Contribution

It establishes the analyticity and Gevrey regularity of streamlines and the stream function in stratified water waves with vorticity, extending regularity results to multiple physical regimes.

Findings

Streamlines are real analytic under given conditions.

Stream function inherits Gevrey regularity from Bernoulli and density functions.

Results apply to capillary, gravity, and capillary-gravity water waves.

Abstract

We prove real analyticity of all the streamlines, including the free surface, of a steady stratified flow of water over a flat bed in the absence of stagnation points, with a H\"older continuous Bernoulli function and a H\"older continuously differentiable density function. Furthermore, we show that if the Bernoulli function and the density function possess 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 three distinct physical regimes: capillary, capillary-gravity, and gravity water waves.