The Landau-Lifshitz equation of the ferromagnetic spin chain and Oseen-Frank flow

TL;DR

This paper studies the Landau-Lifshitz equation for ferromagnetic spin chains under Oseen-Frank energy, proving global existence, uniqueness of weak solutions, and finiteness of singular points.

Contribution

It establishes the first comprehensive results on global solutions and singularity structure for this class of ferromagnetic equations.

Findings

Global existence and uniqueness of weak solutions

Finite number of singular points

Applicability to large energy data

Abstract

In this paper, we consider the Landau-Lifshitz equation of the ferromagnetic spin chain from $\R^2$ to the unit sphere $S^2$ under the general Oseen-Frank energy. We obtain global existence and uniqueness of weak solutions for large energy data; moreover, the number of singular points is finite.