Existence of weak solutions and regular solutions to the incompressible Schrödinger flow

In this paper, we are concerned with the initial-Neumann boundary value problem of the Schrödinger 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 $m\le 3$. Moreover, we also utilize the ``complex structure approximation method" to establish the global existence of weak solutions to the incompressible Schrödinger flow in a smooth bounded domain of ${\mathbb{R}}^m$ (where $m\ge 1$).