Vortex filament dynamics for Gross-Pitaevsky type equations

Jerrard, Robert L. 

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 4, pp. 733-768

Résumé

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 $u_0^{ϵ}$ for which the jacobian $J{u}_0^{ϵ}$ 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 $t>0$ continues to concentrate around some codimension $2$ submanifold, say ${\Gamma }_t$, and that the family $\left\{{\Gamma }_t\right\}$ 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.

Détail

Classification : 35B25, 35Q55