Skip to main content
SpringerLink
Log in

Convergence of solutions of the weighted Allen–Cahn equations to Brakke type flow

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we study the parabolic Allen–Cahn equation, which has slow diffusion and fast reaction, with a potential K. In particular, the convergence of solutions to a generalized Brakke’s mean curvature flow is established in the limit of a small parameter \( \varepsilon \rightarrow 0\). 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 difference between the mean curvature vector and the normal part of \({\nabla K}/{2K}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95, 417–491 (1972)

    Article  MathSciNet  Google Scholar 

  2. Bethuel, F., Orlandi, G., Smets, D.: Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature. Ann. Math. (2) 163, 37–163 (2006)

    Article  MathSciNet  Google Scholar 

  3. Brakke, K.A.: The Motion of a Surface by its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)

    MATH  Google Scholar 

  4. Bronsard, L., Kohn, R.V.: Motion by mean curvature as the singular limit of Ginzburg–Landau dynamics. J. Differ. Equ. 90, 211–237 (1991)

    Article  MathSciNet  Google Scholar 

  5. Chen, X.: Generation and propagation of interfaces for reaction–diffusion equations. J. Differ. Equ. 96, 116–141 (1992)

    Article  MathSciNet  Google Scholar 

  6. De Lellis, C.: Rectifiable Sets, Density, and Tangent Measures, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich (2008)

    Book  Google Scholar 

  7. Deng, Y., Guo, Y., Lu, L.: On the collapse and concentration of Bose–Einstein condensates with inhomogeneous attratctive interactions. Calc. Var. Partial Differ. Equ. 54, 99–118 (2015)

    Article  Google Scholar 

  8. Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, 1097–1123 (1992)

    Article  MathSciNet  Google Scholar 

  9. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)

    Article  MathSciNet  Google Scholar 

  10. Ilmanen, T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)

    Article  MathSciNet  Google Scholar 

  11. Kagaya, T.: Convergence of the Allen–Cahn equation with Neumann boundary condition on non-convex domain, arXiv:1710.00526

  12. Kasai, K., Tonegawa, Y.: A general regularity theory for weak mean curvature flow. Calc. Var. Partial Differ. Equ. 50, 1–68 (2014)

    Article  MathSciNet  Google Scholar 

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Lefter, C., Radulescu, V.D.: On the Ginzburg–Landau energy with weight. Ann. Inst. H. Poincare Anal. Non Lineaire 13, 171–184 (1996)

    Article  MathSciNet  Google Scholar 

  15. Lin, F.-H.: Some dynamical properties of Ginzburg–Landau vortices. Commun. Pure Appl. Math. 49, 323–359 (1996)

    Article  MathSciNet  Google Scholar 

  16. Mizuno, M., Tonegawa, Y.: Convergence of the Allen–Cahn equation with Neumann boundary conditions. SIAM J. Math. Anal. 47(3), 1906–1932 (2015)

    Article  MathSciNet  Google Scholar 

  17. 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)

    Article  MathSciNet  Google Scholar 

  18. Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University Centre for Mathematical Analysis, Canberra (1983)

  19. Soner, H.M.: Convergence of the phase-field equations to the Mullins–Sekerka problem with kinetic undercooling. Arch. Ration. Mech. Anal. 131, 139–197 (1995)

    Article  MathSciNet  Google Scholar 

  20. Soner, H.M.: Ginzburg–Landau equation and motion by mean curvature. I. Convergence. J. Geom. Anal. 7, 437–475 (1997)

    Article  MathSciNet  Google Scholar 

  21. Soner, H.M.: Ginzburg–Landau equation and motion by mean curvature. II. Development of the initial interface. J. Geom. Anal. 7, 477–491 (1997)

    Article  MathSciNet  Google Scholar 

  22. Stahl, A.: Convergence of solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differ. Equ. 4, 421–441 (1996)

    Article  MathSciNet  Google Scholar 

  23. Takasao, K., Tonegawa, Y.: Existence and regularity of mean curvature flow with transport term in higher dimensions. Math. Ann. 364, 857–935 (2016)

    Article  MathSciNet  Google Scholar 

  24. Tonegawa, Y.: Integrality of varifolds in the singular limit of reaction–diffusion equations. Hiroshima Math. J. 33, 323–341 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Tonegawa, Y.: A second derivative Hölder estimate for weak mean curvature flow. Adv. Calc. Var. 7, 91–138 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32861, USA

    Yuanwei Qi

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Gao-Feng Zheng

Authors
  1. Yuanwei Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gao-Feng Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gao-Feng Zheng.

Additional information

Communicated by F.H. Lin.

Qi was partially supported by a Simons Collaboration grant, Grant No. 525042, and Zheng by NSFC Grants 11171126, 11571131. The authors thank Pengfei Guan for stimulating discussion.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, Y., Zheng, GF. Convergence of solutions of the weighted Allen–Cahn equations to Brakke type flow. Calc. Var. 57, 133 (2018). https://doi.org/10.1007/s00526-018-1409-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-018-1409-8

Keywords

Mathematics Subject Classification

Search

Navigation