Non-existence of cusps for a Free-boundary Problem for Water Waves

TL;DR

This paper proves that cusps do not form in a broad class of free-boundary water wave problems, extending previous results to more general domain shapes.

Contribution

It establishes the non-existence of cusps in natural settings for free-boundary water wave problems, including non strip-like domains, advancing prior work.

Findings

Cusps do not exist in the natural setting for these water wave problems.

The result applies to non strip-like domains, broadening previous scope.

Builds upon recent non-existence results for cusps in water wave theory.

Abstract

In arXiv:0908.1031, Varvaruca and Weiss eliminate the existence of cusps for a free-boundary problem for two-dimensional water waves under assumptions that hold for solutions such that $\{u>0\}$ is a "strip-like" domain in the sense of arXiv:0708.4371. In this paper it is proven that cusps do not exists in the natural setting for these free-boundary problems. In particular, non strip-like domains are also allowed. This builds upon recent work on non-existence of cusps in arXiv:2202.00616.