This is the sequel of the recent work Du et al. (Commun Math Phys 400:2137–2179, 2023) on axially symmetric gravity water waves with general vorticities, which has investigated the singular wave profile of the free boundary near the degenerate points. In this companion paper, we are interested in the regularity of the free surface of the water wave near the non-degenerate points. Precisely, we showed that the free boundary is \(C^{1,\gamma }\) smooth for some \(\gamma \in (0,1)\) near all non-degenerate points. The problem is intrinsically intertwined with the regularity theory of the semilinear Bernoulli-type free boundary problem. Our approach is closely related to the monotonicity formula developed by Weiss in his celebrated work Weiss (J Geom Anal 9:317–326, 1999), and to a partial boundary Harnack inequality for the one-phase free boundary problem, which is due to De Silva (Interfaces Free Bound 13:223–238, 2011). Mathematically, we associate the existence of viscosity solutions with the Weiss boundary-adjusted energy. Compared to the classical approach of Caffarelli (Ann Sc Norm Super Pisa Cl Sci 15:583–602, 1988), we provide an alternative proof of the existence of viscosity solutions for a large class of semilinear free boundary problems.
