On the splash singularity for the free-surface of a Navier-Stokes fluid

TL;DR

This paper proves that the free surface of a viscous fluid governed by Navier-Stokes equations can develop a finite-time splash singularity, where the interface self-intersects, starting from smooth initial conditions.

Contribution

It demonstrates the occurrence of splash singularities in viscous fluid flows modeled by Navier-Stokes equations in two and three dimensions, which was previously known mainly for inviscid flows.

Findings

Finite-time self-intersection of the free surface occurs in viscous flows.

Smooth initial conditions can lead to splash singularities.

The result applies to both 2D and 3D Navier-Stokes flows.

Abstract

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for $d$-dimensional flows, $d=2$ or $3$, the free-surface of a viscous water wave, modeled by the incompressible Navier-Stokes equations with moving free-boundary, has a finite-time splash singularity. In particular, we prove that given a sufficiently smooth initial boundary and divergence-free velocity field, the interface will self-intersect in finite time.