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Steady Euler flows and Beltrami fields in high dimensions

Part of: Differential topology Smooth dynamical systems: general theory Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  07 December 2020

ROBERT CARDONA
ROBERT CARDONA*
Affiliation:
Laboratory of Geometry and Dynamical Systems, Department of Mathematics, Universitat Politècnica de Catalunya and BGSMath Barcelona Graduate School of Mathematics, Avinguda del Doctor Marañon 44–50, 08028Barcelona, Spain
*
e-mail: robert.cardona@upc.edu
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Abstract

Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.

Keywords

volume-preserving flowssmooth dynamicsEuler equationsperiodic orbits

MSC classification

Primary: 57R30: Foliations; geometric theory 35Q31: Euler equations 37C27: Periodic orbits of vector fields and flows
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3610 - 3633
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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