Jerrard, Robert L.; Smets, Didier 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 Ann. Sci. Éc. Norm. Supér. (4) 45, No. 4, 637-680 (2012). 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. Cited in 10 Documents 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 PDFBibTeX XMLCite \textit{R. L. Jerrard} and \textit{D. Smets}, Ann. Sci. Éc. Norm. Supér. (4) 45, No. 4, 637--680 (2012; Zbl 1308.58023) Full Text: arXiv