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

TL;DR

This paper establishes local existence and uniqueness of smooth solutions to the Schrödinger flow with Neumann boundary conditions for maps into the 2-sphere, and proves global extension in one dimension.

Contribution

It provides the first rigorous proof of local well-posedness and boundary compatibility conditions, and extends 1D solutions globally.

Findings

Local existence and uniqueness of smooth solutions in Sobolev spaces

Boundary compatibility conditions for initial data

Global extension of solutions in 1D case

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 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.