  We consider the two dimensional gravity water wave equation in the regime
that includes free surfaces with angled crests. We assume that the fluid is
inviscid, incompressible and irrotational, the air density is zero, and we
neglect the surface tension. In \cite{kw} it was shown that in this regime,
only a degenerate Taylor inequality $-\frac{\partial P}{\partial\bold{n}}\ge 0$
holds, with degeneracy at the singularities; an energy functional $\frak E$ was
constructed and an aprori estimate was proved. In this paper we show that a
(generalized) solution of the water wave equation with smooth data will remain
smooth so long as $\frak E(t)$ remains finite; and for any data satisfying
$\frak E(0)<\infty$, the equation is solvable locally in time, for a period
depending only on $\frak E(0)$.
