scholarly journals Steady Euler flows and Beltrami fields in high dimensions

2020 ◽  
pp. 1-24
Author(s):  
ROBERT CARDONA
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
High Dimensions ◽  
Beltrami Fields ◽  
Open Books ◽  
Euler Flows ◽  
Reeb Field ◽  
Beltrami Field ◽  
Volume Preserving

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.

scholarly journals Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems

2012 ◽  
Vol 33 (5) ◽  
pp. 1550-1583 ◽  
Author(s):  
STEFAN MÜLLER ◽  
PETER SPAETH
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Contact Dynamics ◽  
Dimensional Manifold ◽  
Contact Form ◽  
Hamiltonian Flows ◽  
Higher Dimensional ◽  
Calabi Invariant ◽  
Regular Contact ◽  
Volume Preserving

AbstractWe compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results of Gambaudo and Ghys [Enlacements asymptotiques. Topology 36(6) (1997), 1355–1379] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the isotopy. Based on these results, we provide positive answers to two questions posed by Arnold in [The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5(4) (1986), 327–345]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume-preserving homeomorphisms, and is invariant under conjugation by volume-preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed by Banyaga and Spaeth [On the uniqueness of generating Hamiltonians for topological strictly contact isotopies.Preprint, 2012], Buhovsky and Seyfaddini [Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symplectic Geom. to appear, arXiv:1003.2612v2], Müller [The group of Hamiltonian homeomorphisms in the$L^\infty $-norm. J. Korean Math. Soc.45(6) (2008), 1769–1784], Müller and Oh [The group of Hamiltonian homeomorphisms and$C^0$-symplectic topology. J. Symplectic Geom. 5(2) (2007), 167–219], Müller and Spaeth [Topological contact dynamics I: symplectization and applications of the energy-capacity inequality.Preprint, 2011, arXiv:1110.6705v2] and Viterbo [On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. (2006), 34028; Erratum,Int. Math. Res. Not.(2006), 38748]. Moreover, we generalize an example of Furstenberg [Strict ergodicity and transformation of the torus. Amer. J. Math. 83(1961), 573–601] on topologically conjugate but not$C^1$-conjugate area-preserving diffeomorphisms of the two-torus to trivial$T^2$-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not$C^1$-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.

When is an Anosov flow geodesic?

1992 ◽  
Vol 12 (2) ◽  
pp. 227-232
Author(s):  
Leon W. Green
Keyword(s):  
Flow Field ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Anosov Flow ◽  
Unit Tangent Bundle

AbstractLet X, H+, H− be vector fields tangent, respectively, to an Anosov flow and its expanding and contracting foliations in a compact three-dimensional manifold, with γ, α+, α− one forms dual to them. If α+([H+, H−]) = α−([H+, H−]) and γ([H+, H−]) = α−([X, H−]) − α+([X, H+]), then the manifold has the structure of the unit tangent bundle of a Riemannian orbifold with geodesic flow field X.

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

2020 ◽  
pp. 1-16
Author(s):  
DANIEL PERALTA-SALAS ◽  
ANA RECHTMAN ◽  
FRANCISCO TORRES DE LIZAUR
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Higher Dimensions ◽  
Riemannian Metric ◽  
Euler Flows ◽  
Homological Characterization ◽  
Steady Solutions ◽  
Volume Preserving

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.

A pasting lemma and some applications for conservative systems

2007 ◽  
Vol 27 (5) ◽  
pp. 1399-1417 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
CARLOS MATHEUS
Keyword(s):  
Vector Field ◽  
Compact Manifold ◽  
Vector Fields ◽  
Three Dimensional ◽  
Invariant Set ◽  
Dominated Splitting ◽  
Divergence Free Vector ◽  
Conservative Systems ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

WHICH ARE THE FUNDAMENTAL MAGNETIC FIELD CONFIGURATIONS IN NATURE?

1987 ◽  
Vol 01 (05n06) ◽  
pp. 1329-1349 ◽  
Author(s):  
N.A. SALINGAROS
Keyword(s):  
Vector Fields ◽  
Minimum Energy ◽  
Scalar Function ◽  
Field Energy ◽  
Beltrami Fields ◽  
Laboratory Plasmas ◽  
Ferromagnetic Domains ◽  
Beltrami Field ◽  
Energy Plane ◽  
Limiting Case

The dipole field represents a magnetic paradigm. It occurs in a hierarchy of scale, from elementary particles, to ferromagnetic domains, to laboratory magnets, up to the earth itself. Nevertheless, the present magnetic paradigm for plasma physics and astrophysics is not the dipole, but the Beltrami field [Formula: see text], k a constant or scalar function. Such fields are regarded as the most stable, and are extensively used in models of cosmic and laboratory plasmas. The problem is that these two classes of supposedly paradigmatic vector fields are incompatible, as it is impossible to transform one into the other via a continuous limiting process. This paper attempts to resolve this basic contradiction by identifying the configurations corresponding to extrema of the field energy. It is found that, contrary to a long-held belief, Beltrami fields correspond to a point on an energy plane (the flat limiting case of a saddle) and not to a minimum. The classes of energy minima derived here include axisymmetric toroidal and axisymmetric poloidal fields. The latter naturally reduce to dipole fields in the limit of a microscopic current, in agreement with the magnetic paradigm. Some of the situations which have traditionally used Beltrami fields as a model can be explained with the minimum-energy fields derived here. The observed instability of axisymmetric magnetoplasmas follows from a separate energy equipartitioning mechanism.

scholarly journals Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Z. Ok Bayrakdar ◽  
T. Bayrakdar
Keyword(s):  
Vector Fields ◽  
Integral Manifold ◽  
Integrability Condition ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Jet Bundle ◽  
First Order ◽  
Burgers Equations ◽  
Integral Curves ◽  
Metric Connection

We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

scholarly journals Three dimensional steady subsonic Euler flows in bounded nozzles

2014 ◽  
Vol 256 (11) ◽  
pp. 3684-3708 ◽  
Author(s):  
Chao Chen ◽  
Chunjing Xie
Keyword(s):  
Three Dimensional ◽  
Euler Flows

Reconstructing Three-Dimensional Fluid Velocity Vector Fields from Acoustic Transmission Measurements

1977 ◽  
pp. 307-326 ◽  
Author(s):  
S. A. Johnson ◽  
J. F. Greenleaf ◽  
C. R. Hansen ◽  
W. F. Samayoa ◽  
M. Tanaka ◽  
...  
Keyword(s):  
Velocity Vector ◽  
Vector Fields ◽  
Fluid Velocity ◽  
Three Dimensional ◽  
Acoustic Transmission ◽  
Transmission Measurements
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