Convergence Of Solutions Of The Weighted Allen-Cahn Equation To Brakke Type Flow With Transport Term

We establish the convergence of solutions to the parabolic Allen-Cahn equation with potential K and a transport term u to a generalized Brakke's mean curvature flow.More precisely,we show that a sequence of Radon measures,associated to energy density of 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 summation of the mean curvature vector and the normal part of u—?K/2K.