Smooth solutions to the Schrödinger flow for maps from smooth bounded domains in Euclidean spaces into ${\mathbb{S}}^2$

Abstract

The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schrödinger flow for maps from a smooth bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^m$ with $m=1,2,3$ into ${\mathbb{S}}^2$ in the scale of Sobolev spaces. In this part, we provide a precise description of the compatibility conditions at the boundary for the initial data. The other is that we further prove that the locally smooth solution to the initial-Neumann boundary value problem of the 1-dimensional Schrödinger flow can be extended to a global smooth one.

The final version, to appear in Comm. Anal. Geom