On the Impossibility of Finite-Time Splash Singularities for Vortex Sheets

Daniel Coutand, Steve Shkoller

Research output: Contribution to journalArticle

10 Citations (Scopus)

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, that is 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.

Original languageEnglish
Pages (from-to)987–1033
Number of pages47
JournalArchive for Rational Mechanics and Analysis
Volume221
Issue number2
Early online date24 Feb 2016
DOIs
Publication statusPublished - Aug 2016

ASJC Scopus subject areas

  • Analysis
  • Mechanical Engineering
  • Mathematics (miscellaneous)

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