Concentration sets of the Landau–Lifshitz system and quasi-mean curvature flows

Abstract

The aim of this work is to analyze the concentration set of the stationary weak solutions to the Landau-Lifshitz system of the ferromagnetic spin chain from Rm into the unit sphere S2 of R3. Suppose that uk → u weakly in W1,2(Rm × R+, S2) and that Σt is the concentration set for fixed t. In the present paper we first prove that Σt is a \(\mathcal{H}^{m-2}\)-rectifiable set for almost all t ∈ R+. And then we verify that Σt moves by the quasi-mean curvature under some assumptions, which is a new codimension 2 curvature flow. Finally we analyze the behavior of the solution at the singular point and get the blow up formulas. The main barrier to Landau–Lifshtiz system is that there is no energy monotonicity inequality. After the seminal work to on the study of the concentration set of minimizing energy harmonic maps by Leon Simon, there are several papers dealing with the stationary harmonic maps and its heat flows, and so on. This investigation is inspired by the study on the heat flow of harmonic maps and it largely depends on our result of the partial regularity.