Existence of weak solutions and regular solutions to the incompressible Schr\"odinger flow

TL;DR

This paper proves the existence of short-time regular solutions and global weak solutions for the incompressible Schrödinger flow in bounded domains, using novel methods and Sobolev space frameworks.

Contribution

It introduces a new approach for short-time regular solutions and employs the complex structure approximation to establish global weak solutions for the flow.

Findings

Existence of short-time regular solutions in Sobolev spaces for dimensions up to 3.

Global existence of weak solutions in any dimension.

Application of novel methods to analyze Schrödinger flow boundary value problems.

Abstract

In this paper, we are concerned with the initial-Neumann boundary value problem of the Schr\"{o}dinger flow for maps from a smooth bounded domain in an Euclidean space into $\mathbb{S}^2$. By adopting a novel method due to B. Chen and Y.D. Wang, we prove the existence of short-time regular solutions to this flow within the framework of Sobolev spaces when the underlying space is a smooth bounded domain in $\mathbb{R}^m$ with $m\leq 3$. Moreover, we also utilize the ``complex structure approximation method" to establish the global existence of weak solutions to the incompressible Schr\"{o}dinger flow in a smooth bounded domain of $\mathbb{R}^m$ (where $m\geq 1$).