Regularity for steady periodic capillary water waves with vorticity

In the following, we prove new regularity results for two-dimensional steady periodic capillary water waves with vorticity, in the absence of stagnation points. Firstly, we prove that if the vorticity function has a Hölder-continuous first derivative, then the free surface is a smooth curve and the streamlines beneath the surface will be real analytic. Furthermore, once we assume that the vorticity function is real analytic, it will follow that the wave surface profile is itself also analytic. A particular case of this result includes irrotational fluid flow where the vorticity is zero. The property of the streamlines being analytic allows us to gain physical insight into small-amplitude waves by justifying a power-series approach.