Advertisement

Applications of Mathematics

, Volume 54, Issue 2, pp 89–115 | Cite as

On the Caginalp system with dynamic boundary conditions and singular potentials

  • L. CherfilsEmail author
  • A. Miranville
Article

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.

Keywords

Caginalp phase field system singular potential dynamic boundary conditions global existence global attractor Łojasiewicz-Simon inequality convergence to a steady state 

MSC 2000

35B40 35B41 80A22 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Abels, M. Wilke: Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67 (2007), 3176–3193.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S. Aizicovici, E. Feireisl: Long-time stabilization of solutions to a phase-field model with memory. J. Evol. Equ. 1 (2001), 69–84.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    P.W. Bates, S. Zheng: Inertial manifolds and inertial sets for phase-field equations. J. Dyn. Diff. Equations 4 (1992), 375–398.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    D. Brochet, X. Chen, D. Hilhorst: Finite dimensional exponential attractors for the phase-field model. Appl. Anal. 49 (1993), 197–212.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Brokate, J. Sprekels: Hysteresis and phase transitions. Springer, New York, 1996.zbMATHGoogle Scholar
  7. [7]
    G. Caginalp: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92 (1986), 205–245.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [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.zbMATHMathSciNetGoogle Scholar
  9. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    H. P. Fischer, P. Maass, W. Dieterich: Novel surface modes in spinodal decomposition. Phys. Rev. Letters 79 (1997), 893–896.CrossRefGoogle Scholar
  11. [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.CrossRefGoogle Scholar
  12. [12]
    C. G. Gal: A Cahn-Hilliard model in bounded domains with permeable walls. Math. Methods Appl. Sci. 29 (2006), 2009–2036.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    C. G. Gal, M. Grasselli: The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 22 (2008), 1009–1040.zbMATHMathSciNetGoogle Scholar
  14. [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. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    M. Grasselli, H. Petzeltovά, G. Schimperna: A nonlocal phase-field system with inertial term. Q. Appl. Math. 65 (2007), 451–46.zbMATHGoogle Scholar
  20. [20]
    M. A. Jendoubi: A simple unified approach to some convergence theorems of L. Simon. J. Funct. Anal. 153 (1998), 187–202.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    S. Łojasiewicz: Ensembles semi-analytiques. IHES, Bures-sur-Yvette, 1965. (In French.)Google Scholar
  23. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    A. Miranville, S. Zelik: Robust exponential attractors for singularly perturbed phase-field type equations. Electron. J. Differ. Equ. (2002), 1–28.Google Scholar
  25. [25]
    A. Miranville, S. Zelik: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci. 28 (2005), 709–735.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [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.CrossRefMathSciNetGoogle Scholar
  27. [27]
    J. Prüss, M. Wilke: Maximal Lp-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. [28]
    R. Racke, S. Zheng: The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Diff. Equ. 8 (2003), 83–110.zbMATHMathSciNetGoogle Scholar
  29. [29]
    P. Rybka, K.-H. Hoffmann: Convergence of solutions to Cahn-Hilliard equation. Commun. Partial Differ. Equations 24 (1999), 1055–1077.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    L. Simon: Asymptotics for a class of non-linear evolution equations, with applications to gemetric problems. Ann. Math. 118 (1983), 525–571.CrossRefGoogle Scholar
  31. [31]
    R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Springer, New York, 1997.zbMATHGoogle Scholar
  32. [32]
    H. Wu, S. Zheng: Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. J. Differ. Equations 204 (2004), 511–531.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    Z. Zhang: Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions. Commun. Pure Appl. Anal. 4 (2005), 683–693.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  1. 1.Université de La Rochelle, LMALa Rochelle CedexFrance
  2. 2.Université de PoitiersFuturoscope Chasseneuil CedexFrance

Personalised recommendations