Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T14:09:40.216Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady Euler flows and Beltrami fields in high dimensions

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

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albers, P. and Hofer, H.. On the Weinstein conjecture in higher dimensions. Comment. Math. Helv. 84 (2009), 429–436.CrossRefGoogle Scholar
Arnold, V. I. and Khesin, B.A.. Topological Methods in Hydrodynamics. Springer-Verlag, New York, 1998.10.1007/b97593CrossRefGoogle Scholar
Asimov, D.. Round handles and non-singular Morse-Smale flows. Ann. of Math. 102 (1975), 41–54.10.2307/1970972CrossRefGoogle Scholar
Cardona, R., Miranda, E. and Peralta-Salas, D.. Euler flows and singular geometric structures. Philos. Trans. Roy. Soc. A 377 (2019), 20190034.10.1098/rsta.2019.0034CrossRefGoogle Scholar
Cardona, R., Miranda, E., Peralta-Salas, D. and Presas, F.. Universality of Euler flows and flexibility of Reeb embeddings. Preprint, 2019, arXiv:1911.01963.Google Scholar
Cieliebak, K. and Volkov, E.. A note on the stationary Euler equations of hydrodynamics. Ergod. Th. & Dynam. Sys. 37(2) (2017), 454–480.CrossRefGoogle Scholar
Cieliebak, K. and Volkov, E.. First steps in stable Hamiltonian topology. J. Eur. Math. Soc. 17(2) (2015), 321–404. CrossRefGoogle Scholar
Colin, V., Presas, F. and Vogel, T.. Notes on open book decompositions for Engel structures. Algebr. Geom. Topol. 18 (2018), 4275–4303. CrossRefGoogle Scholar
Etnyre, J. B.. Contact structures on $5$ -manifold. Preprint, 2012, arXiv:1210.5208.Google 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.. Contact topology and hydrodynamics III. Knotted orbits. Trans. Amer. Math. Soc. 352 (2000), 5781–5794.CrossRefGoogle Scholar
Ghrist, R.. Steady nonintegrable high-dimensional fluids . Lett. Math. Phys. 55(3) (2001), 193–204.CrossRefGoogle Scholar
Ginzburg, V. L.. Hamiltonian dynamical systems without periodic orbits. Northern California Symplectic Geometry Seminar ( American Mathematical Society Translations: Series 2 , 196). American Mathematical Society, Providence, RI, 1999, pp. 35–48.Google Scholar
Ginzburg, V. L. and Khesin, B. A.. Steady fluid flows and symplectic geometry. J. Geom. Phys. 14(2) (1994), 195 – 210.10.1016/0393-0440(94)90006-XCrossRefGoogle Scholar
Gluck, H.. Dynamical behavior of geodesic fields. Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) ( Lecture Notes in Mathematics , 819). Springer, Berlin, 1980, pp. 190–215.CrossRefGoogle Scholar
Hajduk, B. and Walczak, R.. On vector fields having properties of Reeb fields. Topol. Methods Nonlinear Anal. 41(2) (2013), 401–408.Google Scholar
Hutchings, M. and Taubes, C. H.. The Weinstein conjecture for stable Hamiltonian structures. Geom. Topol. 13 (2009), 901–941.CrossRefGoogle Scholar
Kuperberg, G.. A volume-preserving counterexample to the Seifert conjecture. Comment. Math. Helv. 71(1) (1996), 70–97.CrossRefGoogle Scholar
Lutz, R.. Structures de contact sur les fibrés principaux en cercles de dimension trois. Ann. Inst. Fourier (Grenoble) 27(3) (1977), 1–15.10.5802/aif.658CrossRefGoogle Scholar
Martinet, J.. Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium II ( Lecture Notes in Mathematics , 209). Springer, Berlin, 1971, pp. 142–163.CrossRefGoogle Scholar
Meigniez, G.. Regularization and minimization of codimension-one Haefliger structures. J. Differential Geom. 107 (2017), 157–202.CrossRefGoogle Scholar
Miyoshi, S.. Foliated round surgery of codimension-one foliated manifolds. Topology 21 (1982), 245–261.CrossRefGoogle Scholar
Peralta-Salas, D., Pino del, A. and Presas, F.. Foliated vector fields without periodic orbits. Israel J. Math. 214 (2016), 443.CrossRefGoogle Scholar
Peralta-Salas, D., Rechtman, A., Torres de Lizaur, F.. A characterization of 3D Euler flows using commuting zero-flux homologies. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2020.25. Published online 18 March 2020.CrossRefGoogle Scholar
Plante, J. F.. Diffeomorphisms without periodic points. Proc. Amer. Math. Soc. 88 (1983), 716–718.10.1090/S0002-9939-1983-0702306-5CrossRefGoogle Scholar
Quinn, F.. Open book decompositions, and the bordism of automorphisms. Topology 18(1) (1979), 55–73.10.1016/0040-9383(79)90014-4CrossRefGoogle Scholar
Rechtman, A.. Use and disuse of plugs in foliations. PhD Thesis, ENS Lyon, 2009.Google Scholar
Rechtman, A.. Existence of periodic orbits for geodesible vector fields on closed 3-manifolds. Ergod. Th. & Dynam. Sys. 30 (2010) 1817–1841.10.1017/S0143385709000807CrossRefGoogle Scholar
Sullivan, D.. A foliation of geodesics is characterized by having no “tangent homologies”. J. Pure Appl. Algebra 13(1) (1978), 101–104.CrossRefGoogle Scholar
Sullivan, D.. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36 (1976), 225–255.CrossRefGoogle Scholar
Taubes, C. H.. The Seiberg-Witten equations and the Weinstein conjecture. Geom. Topol. 11 (2007), 2117–2202.CrossRefGoogle Scholar
Tao, T.. On the universality of the incompressible Euler equation on compact manifolds, II. Nonrigidity of Euler flows. Preprint, 2019, arXiv:1902.0631.Google Scholar
Thurston, W.. Existence of codimension-one foliations. Ann. of Math. 104 (1976), 249–268.CrossRefGoogle Scholar
Watanabe, N.. Existence of volume preserving diffeomorphisms without periodic points on three-dimensional manifolds. Proc. Amer. Math. Soc. 97(4) (1986), 724–726.Google Scholar
Wilson, F. W. Jr. On the minimal sets of non-singular vector fields. Ann. of Math. 84 (1966), 529–536.10.2307/1970458CrossRefGoogle Scholar