Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow

TL;DR

This paper proves that solutions of a Landau-Lifshitz system, modeled as multi-phase Allen-Cahn equations with constraints, converge to a Brakke flow representing multi-phase mean curvature evolution.

Contribution

It establishes the convergence of a Landau-Lifshitz based phase field model to Brakke's weak solution for multi-phase mean curvature flow.

Findings

Varifolds derived from solutions form a Brakke flow.

Energy limits match total variation in the singular limit.

Convergence holds under specific energy assumptions.

Abstract

We study the convergence of the system of the Allen-Cahn equations to the weak solution for the multi-phase mean curvature flow in the sense of Brakke. The Landau-Lifshitz equation in this paper can be regarded as a system of Allen-Cahn equations with the Lagrange multiplier, which is a phase field model of the multi-phase mean curvature flow. Under an assumption that the limit of the energies of the solutions for the equations matches with the total variation for the singular limit of the solutions, we show that the family of the varifolds derived from the energies is a Brakke flow.