  This is the sequel of the recent work (Du, Huang, Pu, Commun. Math. Phys,
2023, doi: 10.1007/s00220-023-04651-7) 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 point. 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 dedicated to De
Silva (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 (Caffarelli, Ann.
Scuola Norm. Sup. Pisa, 15, 583-602, 1988), we provide an alternative proof of
the existence of viscosity solutions for a large class of semilinear free
boundary problems.
