Finite time singularities for the free boundary incompressible Euler equations

TL;DR

This paper demonstrates that smooth initial conditions in 2D free boundary incompressible Euler equations can lead to finite-time interface singularities, such as splash or splat, highlighting potential breakdowns in fluid interface smoothness.

Contribution

It proves the existence of smooth initial data that evolve into finite-time singularities in the 2D free boundary Euler equations, a significant advance in understanding fluid interface dynamics.

Findings

Finite-time formation of splash singularities

Finite-time formation of splat singularities

Breakdown of interface smoothness in finite time

Abstract

In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.