Vortex filament dynamics for Gross-Pitaevsky type equations

Robert L. Jerrard

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Submission date: 11. Jul. 2000
Pages: 32
published in: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 1 (2002) 4, p. 733-768 
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Abstract:
We study solutions of the Gross-Pitaevsky equation and similar equations in space dimensions in a certain scaling limit, with initial data for which the Jacobian concentrates as around an (oriented) rectifiable m-2 dimensional set, say , 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 , and that the family of submanifolds evolves by binormal mean curvature flow. We prove this conjecture when is a round m-2-dimensional sphere with multiplicity 1. We also prove a number of partial results for more general inital data.