On Schrödinger maps from T 1 to  S 2

Robert L. Jerrard; Didier Smets

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 4, page 637-680
  • ISSN: 0012-9593

Abstract

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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 𝒞 ∞ ( T 1 , S 2 ) , equipped with the weak topology of  H 1 / 2 , to the space of distributions ( 𝒞 ∞ ( T 1 , 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.

How to cite

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Jerrard, Robert L., and Smets, Didier. "On Schrödinger maps from $T^1$ to $S^2$." Annales scientifiques de l'École Normale Supérieure 45.4 (2012): 637-680. <http://eudml.org/doc/272227>.

@article{Jerrard2012,
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 $t&gt;0$ 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.},
author = {Jerrard, Robert L., Smets, Didier},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {schrödinger maps; binormal curvature flow},
language = {eng},
number = {4},
pages = {637-680},
publisher = {Société mathématique de France},
title = {On Schrödinger maps from $T^1$ to $S^2$},
url = {http://eudml.org/doc/272227},
volume = {45},
year = {2012},
}

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
PB - Société mathématique de France
VL - 45
IS - 4
SP - 637
EP - 680
AB - 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 $t&gt;0$ 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.
LA - eng
KW - schrödinger maps; binormal curvature flow
UR - http://eudml.org/doc/272227
ER -

References

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