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On Schrödinger maps from $$T^1$$ to $$S^2$$. (À propos des Schrödinger maps de $$T^1$$ dans $$S^2$$.) (English. French summary) Zbl 1308.58023
Summary: 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 $$\mathcal C(T^1,S^2)$$, equipped with the weak topology of $$H^{1/2}$$ to the space of distributions $$(\mathcal C(T^1,S^2))^*$$. 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.

##### MSC:
 58J99 Partial differential equations on manifolds; differential operators 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
##### Keywords:
Schrödinger maps; binormal curvature flow
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