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

TL;DR

This paper proves the local well-posedness of the 2D gravity water wave equation allowing non-$C^1$ interfaces with angled crests, using an energy functional that remains finite in this singular regime.

Contribution

It establishes existence, uniqueness, and stability of solutions for the water wave equation with non-$C^1$ interfaces, extending previous results to more singular interface geometries.

Findings

Existence and uniqueness of solutions with non-$C^1$ interfaces.

Stability of solutions in the class with finite energy functional.

Extension of well-posedness results to interfaces with angled crests.

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