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

We consider the capillary–gravity water wave equation in two dimensions. assume that fluid is inviscid, incompressible, irrotational and air density zero. construct an energy functional prove a local wellposedness result without assuming Taylor sign condition. When surface tension $$\sigma $$ zero, reduces to lower order version of obtained by Kinsey Wu (Camb J Math 6(2):93–181, 2018) allows angled crest interfaces. For positive tension, does not allow interfaces but admits initial data with large curvature ^{-\frac{1}{3}+ \epsilon } for any $$\epsilon >0$$ .