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We consider the two dimensional gravity water wave equation in the regimethat includes free surfaces with angled crests. We assume that the fluid isinviscid, incompressible and irrotational, the air density is zero, and weneglect 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 $\frak E$ wasconstructed and an aprori estimate was proved. In this paper we show that a(generalized) solution of the water wave equation with smooth data will remainsmooth 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 perioddepending only on $\frak E(0)$.

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