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.
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This work was partially supported by NSFC Grants 11171126, 11571131.
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Jiang, GC., Wang, CJ. & Zheng, GF. Convergence of Solutions of Some Allen-Cahn Equations to Brakke’s Mean Curvature Flow. Acta Appl Math 167, 149–169 (2020). https://doi.org/10.1007/s10440-019-00272-2
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DOI: https://doi.org/10.1007/s10440-019-00272-2