Finite-time singularity formation for angled-crested water waves

TL;DR

This paper demonstrates that water wave systems with corners can develop finite-time singularities, showing the evolution of corners and their angles, with implications for understanding wave behavior near singular points.

Contribution

It establishes local well-posedness of water waves with corners in weighted Sobolev spaces and proves finite-time singularity formation without symmetry assumptions.

Findings

Corners persist over time with changing angles.

Finite-time singularities can occur from smooth initial data.

The behavior near corners is characterized by energy estimates.

Abstract

We show that the water waves system is locally wellposed in weighted Sobolev spaces which allow for interfaces with corners. No symmetry assumptions are required. These singular points are not rigid: if the initial interface exhibits a corner, it remains a corner but generically its angle changes. Using a characterization of the asymptotic behavior of the fluid near a corner that follows from our a priori energy estimates, we show the existence of initial data in these spaces for which the fluid becomes singular in finite time.