A blow-up criteria and the existence of 2d gravity water waves with angled crests

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 $\frak E(0)$.