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A characterization of 3D steady Euler flows using commuting zero-flux homologies

Part of: Differential topology Equations of mathematical physics and other areas of application Incompressible inviscid fluids

Published online by Cambridge University Press:  18 March 2020

DANIEL PERALTA-SALAS Open the ORCID record for DANIEL PERALTA-SALAS [Opens in a new window] ,
ANA RECHTMAN  and
FRANCISCO TORRES DE LIZAUR
DANIEL PERALTA-SALAS
Affiliation:
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain email dperalta@icmat.es
ANA RECHTMAN
Affiliation:
IRMA, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France email rechtman@math.unistra.fr
FRANCISCO TORRES DE LIZAUR
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany email ftorresdelizaur@mpim-bonn.mpg.de
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Abstract

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We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

Keywords

smooth dynamicssymplectic dynamicsEuler equationsvolume-preserving flows

MSC classification

Primary: 57R30: Foliations; geometric theory 35Q31: Euler equations 76B99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2166 - 2181
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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