On existence of a collapsed bubble with surface tension in viscous incompressible fluid

We consider the one-phase free boundary problem for the incompressible Navier-Stokes equations in $\mathbb{R}^d$ ($d\ge2$).

The surface tension is taken into account.

The initial domain, which is the outside of a bubble, is an exterior domain.

We prove that there exists a bubble evolving by this free boundary problem which collapses in a finite time without blowing up of principal curvatures of its boundary.

In other words, what is called a splash singularity is formed in a finite time.

This type of result is also valid for a bounded initial domain.

To construct such an example, we introduce the notion of a domain with $\delta$-wing which is a flat Riemannian manifold that is not embedded in $\mathbb{R}^d$, but it covers the $\delta$-neighborhood of the original domain whose boundary is self-intersected.