Skip to main content
SpringerLink
Log in

On the Caginalp system with dynamic boundary conditions and singular potentials

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H 2, the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.

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. H. Abels, M. Wilke: Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67 (2007), 3176–3193.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Aizicovici, E. Feireisl: Long-time stabilization of solutions to a phase-field model with memory. J. Evol. Equ. 1 (2001), 69–84.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Aizicovici, E. Feireisl, F. Issard-Roch: Long-time convergence of solutions to a phase-field system. Math. Methods Appl. Sci. 24 (2001), 277–287.

    Article  MATH  MathSciNet  Google Scholar 

  4. P.W. Bates, S. Zheng: Inertial manifolds and inertial sets for phase-field equations. J. Dyn. Diff. Equations 4 (1992), 375–398.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Brochet, X. Chen, D. Hilhorst: Finite dimensional exponential attractors for the phase-field model. Appl. Anal. 49 (1993), 197–212.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Brokate, J. Sprekels: Hysteresis and phase transitions. Springer, New York, 1996.

    MATH  Google Scholar 

  7. G. Caginalp: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92 (1986), 205–245.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Cherfils, A. Miranville: Some results on the asymptotic behavior of the Caginalp system with singular potentials. Adv. Math. Sci. Appl. 17 (2007), 107–129.

    MATH  MathSciNet  Google Scholar 

  9. R. Chill, E. Fašangovά, J. Prüss: Convergence to steady states of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions. Math. Nachr. 279 (2006), 1448–1462.

    Article  MATH  MathSciNet  Google Scholar 

  10. H. P. Fischer, P. Maass, W. Dieterich: Novel surface modes in spinodal decomposition. Phys. Rev. Letters 79 (1997), 893–896.

    Article  Google Scholar 

  11. H. P. Fischer, P. Maass, W. Dieterich: Diverging time and length scales of spinodal decomposition modes in thin flows. Europhys. Letters 62 (1998), 49–54.

    Article  Google Scholar 

  12. C. G. Gal: A Cahn-Hilliard model in bounded domains with permeable walls. Math. Methods Appl. Sci. 29 (2006), 2009–2036.

    Article  MATH  MathSciNet  Google Scholar 

  13. C. G. Gal, M. Grasselli: The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 22 (2008), 1009–1040.

    MATH  MathSciNet  Google Scholar 

  14. S. Gatti, A. Miranville: Asymptotic behavior of a phase-field system with dynamic boundary conditions. Differential Equations: Inverse and Direct Problems (Proceedings of the workshop “Evolution Equations: Inverse and Direct Problems”, Cortona, June 21–25, 2004). A series of Lecture Notes in Pure and Applied Mathematics, Vol. 251 (A. Favini and A. Lorenzi, eds.). CRC Press, Boca Raton, 2006, pp. 149–170.

    Google Scholar 

  15. C. Giorgi, M. Grasselli, V. Pata: Uniform attractors for a phase-field model with memory and quadratic nonlinearity. Indiana Univ. Math. J. 48 (1999), 1395–1445.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Grasselli, A. Miranville, V. Pata, S. Zelik: Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials. Math. Nachr. 280 (2007), 1475–1509.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Grasselli, H. Petzeltovά, G. Schimperna: Long time behavior of solutions to the Caginalp system with singular potential. Z. Anal. Anwend. 25 (2006), 51–72.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Grasselli, H. Petzeltovά, G. Schimperna: Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Commun. Pure Appl. Anal. 5 (2006), 827–838.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Grasselli, H. Petzeltovά, G. Schimperna: A nonlocal phase-field system with inertial term. Q. Appl. Math. 65 (2007), 451–46.

    MATH  Google Scholar 

  20. M. A. Jendoubi: A simple unified approach to some convergence theorems of L. Simon. J. Funct. Anal. 153 (1998), 187–202.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl, W. Dieterich: Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions. Comput. Phys. Comm. 133 (2001), 139–157.

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Łojasiewicz: Ensembles semi-analytiques. IHES, Bures-sur-Yvette, 1965. (In French.)

    Google Scholar 

  23. A. Miranville, A. Rougirel: Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations. Z. Angew. Math. Phys. 57 (2006), 244–268.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Miranville, S. Zelik: Robust exponential attractors for singularly perturbed phase-field type equations. Electron. J. Differ. Equ. (2002), 1–28.

  25. A. Miranville, S. Zelik: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci. 28 (2005), 709–735.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Prüss, R. Racke, S. Zheng: Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions. Ann. Mat. Pura Appl. 185 (2006), 627–648.

    Article  MathSciNet  Google Scholar 

  27. J. Prüss, M. Wilke: Maximal L p -regularity and long-time behaviour of the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions. Operator Theory: Advances and Applications, Vol. 168. Birkhäuser, Basel, 2006, pp. 209–236.

    Google Scholar 

  28. R. Racke, S. Zheng: The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Diff. Equ. 8 (2003), 83–110.

    MATH  MathSciNet  Google Scholar 

  29. P. Rybka, K.-H. Hoffmann: Convergence of solutions to Cahn-Hilliard equation. Commun. Partial Differ. Equations 24 (1999), 1055–1077.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Simon: Asymptotics for a class of non-linear evolution equations, with applications to gemetric problems. Ann. Math. 118 (1983), 525–571.

    Article  Google Scholar 

  31. R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Springer, New York, 1997.

    MATH  Google Scholar 

  32. H. Wu, S. Zheng: Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. J. Differ. Equations 204 (2004), 511–531.

    Article  MATH  MathSciNet  Google Scholar 

  33. Z. Zhang: Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions. Commun. Pure Appl. Anal. 4 (2005), 683–693.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de La Rochelle, LMA, Avenue Michel Crépeau, 17042, La Rochelle Cedex, France

    L. Cherfils

  2. Université de Poitiers, Mathématiques, SP2MI, Téléport 2, Avenue Marie et Pierre Curie, BP 30179, 86962, Futuroscope Chasseneuil Cedex, France

    A. Miranville

Authors
  1. L. Cherfils
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Miranville
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Cherfils.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cherfils, L., Miranville, A. On the Caginalp system with dynamic boundary conditions and singular potentials. Appl Math 54, 89–115 (2009). https://doi.org/10.1007/s10492-009-0008-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10492-009-0008-6

Keywords

MSC 2000

Search

Navigation