Vortex filament dynamics for Gross-Pitaevsky type equations

Robert L. Jerrard

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 4, page 733-768
  • ISSN: 0391-173X

Abstract

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We study solutions of the Gross-Pitaevsky equation and similar equations in m ≥ 3 space dimensions in a certain scaling limit, with initial data u 0 ϵ for which the jacobian J u 0 ϵ concentrates around an (oriented) rectifiable m - 2 dimensional set, say  Γ 0 , of finite measure. It is widely conjectured that under these conditions, the jacobian at later times t > 0 continues to concentrate around some codimension 2 submanifold, say Γ t , and that the family { Γ t } of submanifolds evolves by binormal mean curvature flow. We prove this conjecture when Γ 0 is a round m - 2 -dimensional sphere with multiplicity 1 . We also prove a number of partial results for more general inital data.

How to cite

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Jerrard, Robert L.. "Vortex filament dynamics for Gross-Pitaevsky type equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.4 (2002): 733-768. <http://eudml.org/doc/84485>.

@article{Jerrard2002,
abstract = {We study solutions of the Gross-Pitaevsky equation and similar equations in $m\ge 3$ space dimensions in a certain scaling limit, with initial data $u_0^\epsilon $ for which the jacobian $Ju_0^\epsilon $ concentrates around an (oriented) rectifiable $m-2$ dimensional set, say $\Gamma _0$, of finite measure. It is widely conjectured that under these conditions, the jacobian at later times $t&gt;0$ continues to concentrate around some codimension $2$ submanifold, say $\Gamma _t$, and that the family $\lbrace \Gamma _t \rbrace $ of submanifolds evolves by binormal mean curvature flow. We prove this conjecture when $\Gamma _0$ is a round $m-2$-dimensional sphere with multiplicity $1$. We also prove a number of partial results for more general inital data.},
author = {Jerrard, Robert L.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {733-768},
publisher = {Scuola normale superiore},
title = {Vortex filament dynamics for Gross-Pitaevsky type equations},
url = {http://eudml.org/doc/84485},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Jerrard, Robert L.
TI - Vortex filament dynamics for Gross-Pitaevsky type equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 4
SP - 733
EP - 768
AB - We study solutions of the Gross-Pitaevsky equation and similar equations in $m\ge 3$ space dimensions in a certain scaling limit, with initial data $u_0^\epsilon $ for which the jacobian $Ju_0^\epsilon $ concentrates around an (oriented) rectifiable $m-2$ dimensional set, say $\Gamma _0$, of finite measure. It is widely conjectured that under these conditions, the jacobian at later times $t&gt;0$ continues to concentrate around some codimension $2$ submanifold, say $\Gamma _t$, and that the family $\lbrace \Gamma _t \rbrace $ of submanifolds evolves by binormal mean curvature flow. We prove this conjecture when $\Gamma _0$ is a round $m-2$-dimensional sphere with multiplicity $1$. We also prove a number of partial results for more general inital data.
LA - eng
UR - http://eudml.org/doc/84485
ER -

References

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