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arXiv · 2606.31266

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

Yoshikazu Giga·Zhongyang Gu
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Abstract

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.

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Yoshikazu Giga, Zhongyang Gu. 2026-06-30. On existence of a collapsed bubble with surface tension in viscous incompressible fluid. https://arxiv.org/abs/2606.31266

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