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A note on the stationary Euler equations of hydrodynamics

Published online by Cambridge University Press:  06 October 2015

K. CIELIEBAK  and
E. VOLKOV
K. CIELIEBAK
Affiliation:
Institute of Mathematics, University of Augsburg, 86135 Augsburg, Germany email kai.cieliebak@math.uni-augsburg.de, evgeny.volkov@math.uni-augsburg.de
E. VOLKOV
Affiliation:
Institute of Mathematics, University of Augsburg, 86135 Augsburg, Germany email kai.cieliebak@math.uni-augsburg.de, evgeny.volkov@math.uni-augsburg.de
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Abstract

This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 454 - 480
Copyright
© Cambridge University Press, 2015 

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