  We consider the two dimensional gravity water wave equation in a regime where
the free interface is allowed to be non-$C^1$. In this regime, only a
degenerate Taylor inequality $-\frac{\partial P}{\partial \bf n}\ge 0$ holds,
with degeneracy at the singularities. In \cite{kw} an energy functional
$\mathcal E(t)$ was constructed and an a-prori estimate was proved. The energy
functional $\mathcal E(t)$ is not only finite for interfaces and velocities in
Sobolev spaces, but also finite for a class of non-$C^1$ interfaces with angled
crests. In this paper we prove the existence, uniqueness and stability of the
solution of the 2d gravity water wave equation in the class where $\mathcal
E(t)<\infty$, locally in time, for any given data satisfying $\mathcal
E(0)<\infty$.
