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

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


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.


Integral varifolds Mean curvature vector Parabolic Allen-Cahn equations Potential 

Mathematics Subject Classification

35K58 53C44 28A75 



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


  1. 1.
    Bethuel, F., Orlandi, G., Smets, D.: Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature. Ann. of Math. (2) 163, 37–163 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978) zbMATHGoogle Scholar
  3. 3.
    Bronsard, L., Kohn, R.V.: Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J. Differential Equations 90, 211–237 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Differential Equations 96, 116–141 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45, 1097–1123 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ilmanen, T.: Convergence of the Allen-Cahn equation to brakke’s motion by mean curvature. J. Differential Geom. 38, 417–461 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Katsoulakis, M., Kossioris, G.T., Reitich, F.: Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions. J. Geom. Anal. 5, 255–279 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mizuno, M., Tonegawa, Y.: Convergence of the Allen-Cahn equation with Neumann boundary conditions. SIAM J. Math. Anal. 47(3), 1906–1932 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Qi, Y., Zheng, G.-F.: Convergence of solutions of the weighted Allen-Cahn equations to Brakke type flow. Calc. Var. Partial Differ. Equ. 55 133 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sato, N.: A simple proof of convergence of the Allen-Cahn equation to brakke’s motion by mean curvature. Indiana Univ. Math. J. 57, 1743–1751 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Soner, H.M.: Convergence of the phase-field equations to the mullins-sekerka problem with kinetic undercooling. Arch. Rational Mech. Anal. 131, 139–197 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Soner, H.M.: Ginzburg-Landau equation and motion by mean curvature. I. Convergence. J. Geom. Anal. 7, 437–475 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Soner, H.M.: Ginzburg-Landau equation and motion by mean curvature. II. Development of the initial interface. J. Geom. Anal. 7, 477–491 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Stahl, A.: Convergence of solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differential Equations 4, 421–441 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Takasao, K.: Convergence of the allen-cahn equation with neumann boundary condition on non-convex domains (2017). arXiv:1710.00526
  16. 16.
    Takasao, K., Tonegawa, Y.: Existence and regularity of mean curvature flow with transport term in higher dimensions. Math. Ann. 364, 857–935 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tonegawa, Y.: Integrality of varifolds in the singular limit of reaction-diffusion equations. Hiroshima Math. J. 33, 323–341 (2003) MathSciNetCrossRefzbMATHGoogle Scholar

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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