Convergence of Solutions of Some Allen-Cahn Equations to Brakke’s Mean Curvature Flow

Abstract

The convergence of solutions of the parabolic Allen-Cahn equation with potential $K$ and a transport term $u$ to a generalized Brakke’s mean curvature flow is established. More precisely, we show that a sequence of Radon measures, associated to the solutions to the parabolic Allen-Cahn equation, converges to a weight measure of an integral varifold. Moreover, the limiting varifold evolves by a vector which is the sum of the mean curvature vector and the normal part of $u-{\nabla K}/{2K}$ in weak sense.