Splash singularity for water waves

TL;DR

This paper demonstrates the formation of splash singularities in finite time for the 2D water wave equation, showing that smooth initial interfaces can evolve into self-intersecting curves, with both theoretical and numerical support.

Contribution

It provides the first rigorous proof and numerical evidence of splash singularities forming in finite time for the 2D water wave equation from smooth initial data.

Findings

Existence of initial data leading to splash singularities

Finite-time breakdown of interface smoothness

Numerical simulations confirming theoretical results

Abstract

We exhibit smooth initial data for the 2D water wave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water wave equation that start from a graph, turn over and collapse in a splash singularity (self intersecting curve in one point) in finite time.