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We consider the two dimensional gravity water wave equation in a regime wherethe free interface is allowed to be non-$C^1$. In this regime, only adegenerate 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 energyfunctional $\mathcal E(t)$ is not only finite for interfaces and velocities inSobolev spaces, but also finite for a class of non-$C^1$ interfaces with angledcrests. In this paper we prove the existence, uniqueness and stability of thesolution of the 2d gravity water wave equation in the class where $\mathcalE(t)<\infty$, locally in time, for any given data satisfying $\mathcalE(0)<\infty$.

Wellposedness of the 2D full water wave equation in a regime that allows for non-$C^1$ interfaces