Existence and uniqueness of local regular solution to the Schr\"{o}dinger flow from a bounded domain in $\mathbb{R}^3$ into $\mathbb{S}^2$

TL;DR

This paper proves the local existence and uniqueness of regular solutions for the Schrödinger flow with Neumann boundary conditions from a bounded domain in three-dimensional space into the 2-sphere, using geometric and perturbation methods.

Contribution

It extends existing techniques to establish well-posedness of Schrödinger flows with boundary conditions in a bounded domain, combining geometric energy methods and perturbation approaches.

Findings

Established local existence of solutions

Proved uniqueness of solutions

Developed new boundary behavior observations

Abstract

In this paper, we show the existence and uniqueness of local regular solutions to the initial-Neumann boundary value problem of Schr\"{o}dinger flow from a smooth bounded domain $\Omega\subset\mathbb{R}^3$ into $\mathbb{S}^2$(namely Landau-Lifshitz equation without dissipation). The proof is built on a parabolic perturbation method, an intrinsic geometric energy argument and some observations on the behaviors of some geometric quantities on the boundary of the domain manifold. It is based on methods from Ding and Wang (one of the authors of this paper) for the Schr\"odinger flows of maps from a closed Riemannian manifold into a K\"ahler manifold as well as on methods by Carbou and Jizzini for solutions of the Landau-Lifshitz equation.