ON THE REGULARITY OF DEEP-WATER WAVES WITH GENERAL VORTICITY DISTRIBUTIONS

We prove that the streamlines and the profile of traveling deep-water waves with Holder continuous vorticity function are smooth, provided there are no stagnation points in the flow. In addition, if the vorticity function is real analytic, then so is the profile of both solitary and periodic traveling deep-water waves. Finally, by choosing appropriate weighted Sobolev spaces, we show that the streamlines beneath the surface of a periodic traveling water wave are in fact real analytic, provided the vorticity function is merely integrable against a cubic weight.