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We study the convergence of the system of the Allen-Cahn equations to theweak 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 ofAllen-Cahn equations with the Lagrange multiplier, which is a phase field modelof the multi-phase mean curvature flow. Under an assumption that the limit ofthe energies of the solutions for the equations matches with the totalvariation for the singular limit of the solutions, we show that the family ofthe varifolds derived from the energies is a Brakke flow.

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