On Schr\"odinger maps from $T^1$ to $S^2$

We prove an estimate for the difference of two solutions of the Schr"odinger map equation for maps from $T^1$ to $S^2\mathrm{.}$ 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 ${\mathcal{C}}^{\mathrm{\infty }}(T^1,S^2)$, equipped with the weak topology of $H^{1\mathrm{/}2},$ to the space of distributions $({\mathcal{C}}^{\mathrm{\infty }}(T^1,{\mathbb{R}}^3){)}^{\ast }\mathrm{.}$