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

Abstract

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 $\mathcal E(0)<\infty$.