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

### Daniel Coutand

Department of Mathematics, Heriot–Watt University Edinburgh, EH14 4AS, UK### Steve Shkoller

Department of Mathematics, University of California Davis, CA 95616, USA

## 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.

## Cite this article

Daniel Coutand, Steve Shkoller, On the splash singularity for the free-surface of a Navier–Stokes fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, pp. 475–503

DOI 10.1016/J.ANIHPC.2018.06.004