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

TL;DR

This paper establishes continuity properties of the Schr"odinger map flow from the circle to the sphere in certain topologies, and demonstrates discontinuity without quotienting by translations, highlighting subtle regularity issues.

Contribution

It provides new estimates for solution differences and analyzes flow map continuity in quotient and non-quotient settings for Schr"odinger maps from $T^1$ to $S^2$.

Findings

Flow map is continuous in $L^2$ topology modulo translations.

Flow map is discontinuous in weak $H^{1/2}$ topology without quotient.

New estimate for the difference of two solutions.

Abstract

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