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

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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, 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.

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Author notes

  1. Steve Shkoller

    Present address: Department of Mathematics, University of California, Davis, CA, 95616, USA

Authors and Affiliations

  1. CANPDE, Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Daniel Coutand

  2. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Steve Shkoller

Authors
  1. Daniel Coutand
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  2. Steve Shkoller
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Correspondence to Steve Shkoller.

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Communicated by P. Constantin

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Coutand, D., Shkoller, S. On the Impossibility of Finite-Time Splash Singularities for Vortex Sheets. Arch Rational Mech Anal 221, 987–1033 (2016). https://doi.org/10.1007/s00205-016-0977-z

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  • DOI: https://doi.org/10.1007/s00205-016-0977-z

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