Angled Crested Like Water Waves with Surface Tension: Wellposedness of the Problem

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 (Camb J Math 6(2):93–181, 2018) 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 $$\epsilon >0$$ .