Smooth solutions to the Schrdinger flow for maps from smooth bounded domains in Euclidean spaces into mathbbS²

The results of this paper are twofold. First, we establish the local existence and uniqueness of very regular or smooth solutions to the initial-Neumann boundary value problem of the Schrödinger flow for maps from a smooth bounded domain Rᵐ with m=1, 2, 3 into S² in the scale of Sobolev spaces. In this part, we also provide a precise description of the compatibility conditions required at the boundary for the initial data. Second, we further prove that the local smooth solution obtained for the initial-Neumann boundary value problem of the 1-dimensional Schrödinger flow can be extended to a global smooth one.