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

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 for a large class of specially prepared initial data. In particular, we prove that given a sufficiently smooth initial boundary (which is close to self-intersection) and a divergence-free velocity field designed to push the boundary towards self-intersection, the interface will indeed self-intersect in finite time.