Local wellposedness for the free boundary incompressible Euler equations with interfaces that exhibit cusps and corners of nonconstant angle

TL;DR

This paper proves local well-posedness for the free boundary incompressible Euler equations allowing interfaces with evolving corners and cusps, including non-constant angles, expanding understanding of complex water wave behaviors.

Contribution

It establishes local well-posedness for Euler equations with interfaces that have non-smooth, evolving geometries, including corners and cusps with changing angles.

Findings

Interfaces can develop and evolve corners and cusps over time.

Angles of crests are allowed to change dynamically.

The solutions are well-posed despite non-$C^1$ regularity.

Abstract

We prove that free boundary incompressible Euler equations are locally well posed in a class of solutions in which the interfaces can exhibit corners and cusps. Contrary to what happens in all the previously known non-$C^1$ water waves, the angle of these crests can change in time.