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

  • Gui-Chun Jiang
  • Chang-Jian Wang
  • Gao-Feng ZhengEmail author
Article

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.

Keywords

Integral varifolds Mean curvature vector Parabolic Allen-Cahn equations Potential 

Mathematics Subject Classification

35K58 53C44 28A75 

Notes

Acknowledgements

The authors are grateful to the referees for their helpful comments and suggestions that improve the presentation of this paper.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Gui-Chun Jiang
    • 1
  • Chang-Jian Wang
    • 1
  • Gao-Feng Zheng
    • 1
    Email author
  1. 1.School of Mathematics and StatisticsCentral China Normal UniversityWuhanP.R. China

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