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Finite-Time Singularity Formation for Incompressible Euler Moving Interfaces in the Plane

  • Daniel Coutand
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Abstract

This paper provides a new general method for establishing a finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. This methodology is applied to two different problems. The first problem considered is the two-phase vortex sheet problem with surface tension, for which, under suitable assumptions of smallness of the initial height of the heaviest phase and velocity fields, is proved the finite-time singularity of the natural norm of the problem. This is in striking contrast with the case of finite-time splash and splat singularity formation for the one-phase Euler equations of [4] and [8], for which the natural norm (in the one-phase fluid) stays finite all the way until contact. The second problem considered involves the presence of a heavier rigid body moving in the inviscid fluid. For a very general set of geometries (essentially the contact zone being a graph) we first establish that the rigid body will hit the bottom of the fluid domain in finite time. Compared to the previous paper [20] for the rigid body case, the present paper allows for small square integrable vorticity and provides a characterization of acceleration at contact. A surface energy is shown to blow up and acceleration at contact is shown to oppose the motion: it is either strictly positive and finite if the contact zone is of non zero length, or infinite otherwise.

Notes

Acknowledgements

The authorwants to thank the anonymous referee for the careful reading of this manuscript, for identifying imprecisions and confusing points and for making useful suggestions that have improved the clarity of this work. The author also wants to thank the editor for useful comments that have further improved the clarity of this work.

Compliance with ethical standards

Ethical Responsibilities and Ethical Standard

The author satisfies all required ethical responsibilities and standards required by Archive for Rational Mechanics and Analysis.

Conflict of interest

The author declares that he has no conflict of interest.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghUK

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