Regularity of traveling periodic stratified water waves with vorticity

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 Hölder continuous Bernoulli function and a Hölder continuously differentiable density function. Furthermore, we show that if the Bernoulli function and the density function possess some Gevrey regularity of index ss, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index ss 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.