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

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

2011-05-13

arxiv.org/abs/1105.2736v1

Mathematics

Analysis of PDEs

44 pages, no figures

Scientific paper

We prove an estimate for the difference of two solutions of the Schr\"odinger 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 $(\mathcal{C}^\infty(T^1,\R^3))^*.$

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