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.

中文翻译：

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某些Allen-Cahn方程解对Brakeke平均曲率流的收敛性

建立了抛物型Allen-Cahn方程具有势\（K \）和输运项\（u \）的解与广义Brakke平均曲率流的收敛性。更准确地说，我们显示了与抛物线Allen-Cahn方程的解相关的一系列Radon测度收敛到积分变量的权重测度。此外，极限曲率由一个向量演化而成，该向量是平均曲率向量与\（u-{\ nabla K} / {2K} \）的法线部分的和，在弱意义上。