On the impossibility of finite-time splash singularities for vortex sheets

Daniel Coutand; Steve Shkoller

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arXiv · 1407.1479

On the impossibility of finite-time splash singularities for vortex sheets

Daniel Coutand·Steve Shkoller

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Abstract

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, i.e. for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.

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Daniel Coutand, Steve Shkoller. 2014-07-06. On the impossibility of finite-time splash singularities for vortex sheets. https://arxiv.org/abs/1407.1479

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