scholarly journals Discrete Vector Potential Representation of a Divergence-Free Vector Field in Three-Dimensional Domains: Numerical Analysis of a Model Problem

1990 ◽  
Vol 27 (5) ◽  
pp. 1103-1141 ◽  
Author(s):  
François Dubois
Keyword(s):  
Vector Field ◽  
Numerical Analysis ◽  
Vector Potential ◽  
Model Problem ◽  
Three Dimensional ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Potential Representation

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.

Analyticity and metric transitivity on the torus

1998 ◽  
Vol 18 (3) ◽  
pp. 717-723
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Differential Equations ◽  
Entire Plane ◽  
Transitive Flow ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Open Question ◽  
Set Of Points

Suppose we are given an analytic divergence free vector field $(X,Y)$ on the standard torus. We can find constants $a$ and $b$ and a function $F(x,y)$ of period one in both $x$ and $y$ such that $(X,Y)=(a-F_y,b+F_x)$. For a given $F$, let $P$ be the map sending $(x,y)$ into $(F_y(x,y),-F_x(x,y))$. Let $A$ be the image of the torus under this map and let $B$ be the image under this map of the set of points $(x,y)$ at which $F_{xx}F_{yy}-(F_{xy})^2$ vanishes. For any point $(a,b)$ in the complement of the interior of $A$, the flow on the torus arising from the differential equations $dx/dt=a-F_y(x,y)$, $dy/dt=b+F_x(x,y)$ is metrically transitive if and only if $a/b$ is irrational. For any point in $A$ but not in $B$ the flow is not metrically transitive. Moreover, if $a/b$ is irrational but the flow on the torus is not metrically transitive and we use our differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It is an open question whether a point $(a,b)$ in the interior of $A$ can give rise to a metrically transitive flow.

scholarly journals Divergence-free vector-field and reduction

1998 ◽  
Vol 244 (5) ◽  
pp. 377-382 ◽  
Author(s):  
Debin Huang ◽  
Xiaohua Zhao ◽  
Zengrong Liu
Keyword(s):  
Vector Field ◽  
Divergence Free Vector ◽  
Divergence Free

A divergence-free vector field model for imaging applications

2009 ◽  
Author(s):  
O. Skrinjar ◽  
A. Bistoquet ◽  
J. Oshinski ◽  
K. Sundareswaran ◽  
D. Frakes ◽  
...  
Keyword(s):  
Vector Field ◽  
Field Model ◽  
Divergence Free Vector ◽  
Divergence Free

scholarly journals Vector fields satisfying the barycenter property

2018 ◽  
Vol 16 (1) ◽  
pp. 429-436 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Axiom A ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe show that if a vector fieldXhas theC1robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a genericC1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

Basic Technical Outline

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Vector Field ◽  
Bounded Number ◽  
Divergence Free Vector ◽  
Individual Wave ◽  
Error Terms ◽  
Interaction Term ◽  
Divergence Free ◽  
Complex Valued ◽  
Stress Term ◽  
Transport Term

This chapter provides a more technical outline of the construction, starting with a solution to the Euler-Reynolds system and a correction v₁ = v + V, p₁ = p + P. The correction V is a divergence free vector field which oscillates rapidly compared to v. In the construction, there will always be a bounded number of waves Vsubscript I (at most 192) which are nonzero at any given time t. Each individual wave Vsubscript I composing V is a complex-valued, divergence free vector field that oscillates rapidly in only one direction. The chapter introduces several ways in which to represent each Vsubscript I. Finally, it presents five main error terms: the Transport term, the High–Low Interaction term, the High–High Interference terms, the Stress term, and the Mollification terms.

scholarly journals Linear Density Waves in Globular Clusters

1988 ◽  
Vol 126 ◽  
pp. 693-693
Author(s):  
Yousef Sobouti
Keyword(s):  
Vector Field ◽  
Vector Potential ◽  
Globular Clusters ◽  
Scalar Potential ◽  
Stellar System ◽  
Three Dimensional ◽  
Normal Modes ◽  
Successive Approximations ◽  
Density Waves ◽  
Variational Integrals

It is often maintained that Antonov's equation, a linearization of the collisionless Liouville-Boltzmann equation, governs small perturbations of a stellar system. The variational integrals resulting from Antonov's equation are in six dimensional phase space. However, expanding the perturbations in the velocity coordinates and carrying out the integrals over the velocity components gives integrals in the three dimensional configuration space. Solutions in successive approximations lead to standing density waves. The first order equations involve a vector field ξ(x) related to but not identical with the Lagrangian displacements of a volume element of the system. In this respect the problem is analogous to the linear oscillations of a fluid star. The analogy is exploited to provide a classification for the modes of oscillation and to obtain suitable data for variational calculations. The normal modes appear to be trispectral, in the sense that the associated vector field ξ(x) is derived predominantly either from a scalar potential, a toroidal vector potential, or a poloidal vector potential. The eigenfrequencies of the radial (ι = 0) and non-radial (ι = 1) modes are calculated. The associated density waves are analyzed.

Sign in / Sign up

Export Citation Format

Share Document