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

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 $m\geq 1$).