Two-dimensional magnetohydrodynamic equilibria with prescribed topology

Laurence, Peter; Stredulinsky, Edward W.

doi:10.1002/1097-0312(200009)53:9<1177::aid-cpa5>3.0.co;2-a

Cited by 9 publications

(3 citation statements)

References 11 publications

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“…There is a considerable body of literature devoted to the analysis of topological and geometrical structures that appear in fluid flows, which has led to significant results e.g. on particle trajectories and vortex lines [2,13,21,11,25], on the relationship between the Euler equation and the group of volume-preserving diffeomorphisms [2,10,19,6], and on the connection of the helicity with the energy functional and the asymptotic linking number [3,16,17,29]. However, Kelvin's conjecture remains wide open, and indeed has been included as a major open problem in topological fluid mechanics in the surveys [27,23].…”

Section: Introductionmentioning

confidence: 99%

Existence of knotted vortex tubes in steady Euler flows

2015

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We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R 3 . More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R 3 , we show that they can be transformed with a C m -small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin.

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Section: Introductionmentioning

confidence: 99%

Existence of knotted vortex tubes in steady Euler flows

2015

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“…The second result, by Laurence and Stredulinsky [16], utilizes variational techniques to analyze steady solutions of the Euler equation with axial symmetry. This symmetry makes the problem effectively two-dimensional and allows to formulate it in terms of the scalar stream function.…”

Section: Introductionmentioning

confidence: 99%

Knots and links in steady solutions of the Euler equation

Enciso¹,

2012

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“…In [15] we used weighted rearrangements, that is, Lebesgue measure was replaced by measures m defined by dm = rdrdz, dm = r −1 drdz (0 < r 0 ≤ r ≤ R 0 < ∞) in polar coordinates. Such variations have no effect on the basic analysis if for a positive measure m one has m(E) = 0 ⇒ |E| = 0, |E| ≥ c m(E), c > 0 for sets E.…”

Section: Initial Technical Definitionsmentioning

confidence: 99%

Variational problems with topological constraints

Laurence

Stredulinsky

2000

Calculus of Variations and Partial Differential Equations

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We introduce a decomposition that captures much of the topology of level sets for functions in certain Sobolev spaces, and allows the definition of an analog of the decreasing rearrangement of a function which respects the topology of level sets. In a variety of settings this decomposition is preserved under weak limits, and so is useful in establishing existence of minimizers of various variational problems with prescribed topological properties. These include variational problems in which 'topological rearrangements' are fixed, and ones in which the functional depends on derivatives of rearrangements.

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Two-dimensional magnetohydrodynamic equilibria with prescribed topology

Cited by 9 publications

References 11 publications

Existence of knotted vortex tubes in steady Euler flows

Existence of knotted vortex tubes in steady Euler flows

Knots and links in steady solutions of the Euler equation

Variational problems with topological constraints

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