Remarks on local theory for Schrödinger maps near harmonic maps

Abstract

We consider the initial-value problem for the equivariant Schrödinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J. 145(3) 537--583, 2008]. We also prove that the solution near harmonic maps is unique in $C(I;{\dot{H}}^1({\mathbb{R}}^2)\cap {\dot{H}}^2({\mathbb{R}}^2))$ for time interval $I$. In the proof, we give a justification of the derivation of the modified Schrödinger map equation in low regularity settings without smallness of energy.