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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

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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.

Information

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

References

An, J. and Wang, Z.. Curve Selection Lemma for semianalytic sets and conjugacy classes of finite order in Lie groups. Sci. China Ser. A 51(3) (2008), 383–388.CrossRefGoogle Scholar
Arnold, V. I.. Sur la géométrie differentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16 (1966), 316–361.CrossRefGoogle Scholar
Arnold, V. I. and Khesin, B.. Topological Methods in Hydrodynamics. Springer, Berlin, 1998.CrossRefGoogle Scholar
Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K. and Zehnder, E.. Compactness results in symplectic field theory. Geom. Topol. 7 (2003), 799–888.CrossRefGoogle Scholar
Bruhat, F. and Whitney, H.. Quelques propriétés fondamentales des ensembles analytiques-réels. Comment. Math. Helv. 33 (1959), 132–160.Google Scholar
Cieliebak, K. and Volkov, E.. First steps in stable Hamiltonian topology. J. Eur. Math. Soc. (JEMS) 17(2) (2015), 321–404.CrossRefGoogle Scholar
Etnyre, J. and Ghrist, R.. Contact topology and hydrodynamics I: Beltrami fields and the Seifert conjecture. Nonlinearity 13(2) (2000), 441–458.CrossRefGoogle Scholar
Etnyre, J. and Ghrist, R.. Generic hydrodynamic instability for curl eigenfields. SIAM J. Appl. Dyn. Syst. 4(2) (2005), 377–390.CrossRefGoogle Scholar
Friedlander, S. and Vishik, M.. Instability criteria for steady flows of a perfect fluid. Chaos 2(3) (1992), 455–460.CrossRefGoogle ScholarPubMed
Goresky, M. and MacPherson, R.. Stratified Morse Theory. Springer, Berlin, 1988.CrossRefGoogle Scholar
Hasselblatt, B. and Katok, A.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.Google Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. The dynamics on a strictly convex energy surface in ℝ4 . Ann. of Math. (2) 148 (1998), 197–289.CrossRefGoogle Scholar
Hummel, C.. Gromov’s Compactness Theorem for Pseudo-holomorphic Curves. Birkhäuser, Basel, 1997.CrossRefGoogle Scholar
Hutchings, M. and Taubes, C.. The Weinstein conjecture for stable Hamiltonian structures. Geom. Topol. 13(2) (2009), 901–941.CrossRefGoogle Scholar
Sullivan, D.. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36 (1976), 225–255.CrossRefGoogle Scholar
Sullivan, D.. A foliation of geodesics is characterized by having no tangent homologies. J. Pure Appl. Algebra 13 (1978), 101–104.CrossRefGoogle Scholar