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

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.