On Schrödinger maps from $T^1$ to $S^2$

[À propos des Schrödinger maps de $T^1$ dans $S^2$]

Jerrard, Robert L. ; Smets, Didier

Annales scientifiques de l'École Normale Supérieure, Tome 45 (2012), p. 637-680 / Harvested from Numdam

Résumé

Nous obtenons une estimation de la différence entre deux solutions de l’équation des Schrödinger maps de $T^1$ dans $S^2.$ Cette estimation fournit une propriété de continuité du flot associé pour la topologie de $L^2(T^1,S^2)$, quotientée par l’action continue du groupe $T^1$ via les translations. Nous démontrons également que sans cette prise de quotient, quel que soit $t>0$ l’application flot au temps $t$ est discontinue de $\mathcal {C}^\infty (T^1,S^2)$, équipé de la topologie faible de $H^{1/2},$ vers l’espace des distributions périodiques $(\mathcal {C}^\infty (T^1,\mathbb {R}^3))^*.$ L’argument repose de manière essentielle sur le lien étroit entre l’équation des Schrödinger maps et celle du flot par courbure binormale pour une courbe dans l’espace, et sur une nouvelle estimation concernant ce dernier.

Abstract

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t>0$ the flow map at time $t$ is discontinuous as a map from $\mathcal {C}^\infty (T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions $(\mathcal {C}^\infty (T^1,\mathbb {R}^3))^*.$ The argument relies in an essential way on the link between the Schrödinger map equation and the binormal curvature flow for curves in the euclidean space, and on a new estimate for the latter.

Détail

Publié le : 2012-01-01

MR 3059243

DOI : https://doi.org/10.24033/asens.2175
Classification:  35Q55,  53A04
Mots clés: Équation des schrödinger maps, flot par courbure binormale

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