We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from to This estimate yields some continuity properties of the flow map for the topology of , provided one takes its quotient by the continuous group action of given by translations. We also prove that without taking this quotient, for any the flow map at time is discontinuous as a map from , equipped with the weak topology of to the space of distributions 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.
Nous obtenons une estimation de la différence entre deux solutions de l’équation des Schrödinger maps de dans Cette estimation fournit une propriété de continuité du flot associé pour la topologie de , quotientée par l’action continue du groupe via les translations. Nous démontrons également que sans cette prise de quotient, quel que soit l’application flot au temps est discontinue de , équipé de la topologie faible de vers l’espace des distributions périodiques 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.
Keywords: schrödinger maps, binormal curvature flow
Mot clés : Équation des schrödinger maps, flot par courbure binormale
@article{ASENS_2012_4_45_4_637_0, author = {Jerrard, Robert L. and Smets, Didier}, title = {On {Schr\"odinger} maps from $T^1$ to~$S^2$}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {637--680}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 45}, number = {4}, year = {2012}, doi = {10.24033/asens.2175}, mrnumber = {3059243}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2175/} }
TY - JOUR AU - Jerrard, Robert L. AU - Smets, Didier TI - On Schrödinger maps from $T^1$ to $S^2$ JO - Annales scientifiques de l'École Normale Supérieure PY - 2012 SP - 637 EP - 680 VL - 45 IS - 4 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2175/ DO - 10.24033/asens.2175 LA - en ID - ASENS_2012_4_45_4_637_0 ER -
%0 Journal Article %A Jerrard, Robert L. %A Smets, Didier %T On Schrödinger maps from $T^1$ to $S^2$ %J Annales scientifiques de l'École Normale Supérieure %D 2012 %P 637-680 %V 45 %N 4 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2175/ %R 10.24033/asens.2175 %G en %F ASENS_2012_4_45_4_637_0
Jerrard, Robert L.; Smets, Didier. On Schrödinger maps from $T^1$ to $S^2$. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 4, pp. 637-680. doi : 10.24033/asens.2175. http://www.numdam.org/articles/10.24033/asens.2175/
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