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

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 Schr\"{o}dinger flow for maps from a smooth bounded domain $\Omega\subset \mathbb{R}^m$ 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 Schr\"{o}dinger flow can be extended to a global smooth one.