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MiS Preprint
45/2000
Vortex filament dynamics for Gross-Pitaevsky type equations
Robert L. Jerrard
Abstract
We study solutions of the Gross-Pitaevsky equation and similar equations in $m\geq 3$ space dimensions in a certain scaling limit, with initial data $u^\epsilon_0$ for which the Jacobian $J u^\epsilon_0$ concentrates as $\epsilon \to 0$ 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>0 continues to concentrate around some codimension 2 submanifold, say $\Gamma_\imath$, and that the family $\{\Gamma_\imath \}$ 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.
