Angled crested like water waves with surface tension: Wellposedness of the problem

TL;DR

This paper proves local well-posedness for the 2D capillary-gravity water wave equations without the Taylor sign condition, accommodating angled crests when surface tension is zero and large curvature initial data when surface tension is positive.

Contribution

It introduces a new energy functional that ensures well-posedness without the Taylor sign condition, extending the understanding of water wave interfaces with and without surface tension.

Findings

Well-posedness proven without Taylor sign condition

Allows angled crest interfaces when surface tension is zero

Supports initial data with large curvature for positive surface tension

Abstract

We consider the capillary-gravity water wave equation in two dimensions. We assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. We construct an energy functional and prove a local wellposedness result without assuming the Taylor sign condition. When the surface tension $\sigma$ is zero, the energy reduces to a lower order version of the energy obtained by Kinsey and Wu [23] and allows angled crest interfaces. For positive surface tension, the energy does not allow angled crest interfaces but admits initial data with large curvature of the order of $\sigma^{-\frac{1}{3} + \epsilon} $ for any $\epsilon >0$.