Finite time singularities for water waves with surface tension

TL;DR

This paper demonstrates that surface tension does not prevent finite time singularities in 2D water waves, showing that the free boundary can still develop splash or splat singularities despite surface tension effects.

Contribution

It proves that surface tension cannot prevent finite time singularities in 2D water waves, extending understanding of singularity formation in fluid dynamics.

Findings

Surface tension does not prevent splash or splat singularities.

A transformation and energy estimates are used to prove singularity formation.

Finite time singularities can occur even with surface tension present.

Abstract

Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. To do so, the main ingredients of the proof are a transformation to desingularize the curve and a priori energy estimates.