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 $C(I;\dot{H}^1(\mathbf{R}^2)\cap \dot{H}^2(\mathbf{R}^2))$ 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.
Citation
Ikkei Shimizu. "Remarks on local theory for Schrödinger maps near harmonic maps." Kodai Math. J. 43 (2) 278 - 324, June 2020. https://doi.org/10.2996/kmj/1594313555