Vortex filament dynamics for Gross-Pitaevsky type equations

Jerrard, Robert L.

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 $m \geq 3$ 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.

MR | 1 citation dans Numdam

Classification : 35B25, 35Q55

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     author = {Jerrard, Robert L.},
     title = {Vortex filament dynamics for {Gross-Pitaevsky} type equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {733--768},
     year = {2002},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {4},
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     language = {en},
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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, Série 5, Tome 1 (2002) no. 4, pp. 733-768. https://www.numdam.org/item/ASNSP_2002_5_1_4_733_0/

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