open access

https://doi.org/10.14943/2115.100003

https://hdl.handle.net/2115/100003

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

英語

2026-07-02

Department of Mathematics,Hokkaido University

Hokkaido University Preprint Series in Mathematics

1166(1)

1-49

We consider the one-phase free boundary problem for the incompressible Navier-Stokes equations in Rd (d ≥ 2). The surface tension is taken into account. The initial domain, which is the outside 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 also is valid for a bounded initial domain. To construct such an example, we introduce the notion of a domain with δ-wing which is a flat Riemannian manifold that is not embedded in Rd, but it covers the δ-neighborhood of the original domain whose boundary is self-intersected.

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