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 $S^2.$ 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 $(\mathcal{C}^\infty(T^1,\R^3))^*.$