On the phase-field-crystal model with logarithmic nonlinear terms

Original Paper
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Abstract

Our aim in this paper is to study the well-posedness for the phase-field-crystal model with logarithmic nonlinear terms. More precisely, we prove the existence and uniqueness of variational solutions, based on a variational inequality.

Keywords

Phase-field-crystal model Logarithmic nonlinear terms  Variational solutions Well-posedness 

Mathematics Subject Classification

35B45 35K55 
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Notes

Acknowledgments

The author wishes to thank an anonymous referee for her/his careful reading of the paper and useful comments.

References

  1. 1.
    Berry, J., Elder, K.R., Grant, M.: Simulation of an atomistic dynamic field theory for monatomic liquids: freezing and glass formation. Phys. Rev. E 77, 061506 (2008)CrossRefGoogle Scholar
  2. 2.
    Berry, J., Grant, M., Elder, K.R.: Diffusive atomistic dynamics of edge dislocations in two dimensions. Phys. Rev. E 73, 031609 (2006)CrossRefGoogle Scholar
  3. 3.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2, 258–267 (1958)CrossRefGoogle Scholar
  4. 4.
    Chen, F., Shen, J.: Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn–Hilliard systems. Commun. Comp. Phys. 13, 1189–1208 (2013)MathSciNetGoogle Scholar
  5. 5.
    Cherfils, L., Gatti, S., Miranville, A.: A variational approach to a Cahn–Hilliard model in a domain with nonpermeable walls. J. Math. Sci. 189, 604–636 (2013)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Cherfils, L., Miranville, A., Zelik, S.: The Cahn–Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596 (2011)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Conti, M., Gatti, S., Miranville, A.: Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions. Anal. Appl. 11, 1350024 (2013)CrossRefMathSciNetGoogle Scholar
  8. 8.
    de Gennes, P.G.: Dynamics of fluctuations and spinodal decomposition in polymer blends. J. Chem. Phys. 72, 4756–4763 (1980)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Elder, K.R., Grant, M.: Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. 70, 051605 (2004)Google Scholar
  10. 10.
    Emmerich, H., Löwen, H., Wittkowski, R., Gruhn, T., Tóth, G.I., Tegze, G., Gránásy, L.: Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview. Adv. Phys. 61, 665–743 (2012)CrossRefGoogle Scholar
  11. 11.
    Frigeri, S., Grasselli, M.: Nonlocal Cahn–Hilliard–Navier–Stokes systems with singular potentials. Dyn. PDE 9, 273–304 (2012)Google Scholar
  12. 12.
    Galenko, P., Danilov, D., Lebedev, V.: Phase-field-crystal and Swift–Hohenberg equations with fast dynamics. Phys. Rev. E 79, 051110 (2009)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Grasselli, M., Miranville, A., Schimperna, G.: The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discret. Cont. Dyn. Syst. 28, 67–98 (2010)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Gompper, G., Kraus, M.: Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations. Phys. Rev. E 47, 4289–4300 (1993)CrossRefGoogle Scholar
  15. 15.
    Gompper, G., Kraus, M.: Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations. Phys. Rev. E 47, 4301–4312 (1993)CrossRefGoogle Scholar
  16. 16.
    Grasselli, M., Wu, H.: Well-posedness and longtime behavior for the modified phase-field crystal equation. Math. Models Methods Appl. Sci. 24, 2743–2783 (2014)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Grasselli, M., Wu, H.: Robust exponential attractors for the modified phase-field crystal equation. Discret. Cont. Dyn. Syst. 35, 2539–2564 (2015)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Hu, Z., Wise, S.M., Wang, C., Lowengrub, J.S.: Stable finite difference, nonlinear multigrid simulation of the phase field crystal equation. J. Comput. Phys. 228, 5323–5339 (2009)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Korzec, M., Nayar, P., Rybka, P.: Global weak solutions to a sixth order Cahn–Hilliard type equation. SIAM J. Math. Anal. 44, 3369–3387 (2012)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Korzec, M., Rybka, P.: On a higher order convective Cahn–Hilliard type equation. SIAM J. Appl. Math. 72, 1343–1360 (2012)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Miranville, A.: Asymptotic behavior of a sixth-order Cahn–Hilliard system. Central Eur. J. Math. 12, 141–154 (2014)MathSciNetMATHGoogle Scholar
  22. 22.
    Miranville, A.: Sixth-order Cahn–Hilliard systems with dynamic boundary conditions. Math. Methods Appl. Sci. 38, 1127–1145 (2015)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Miranville, A.: Sixth-order Cahn–Hilliard equations with singular nonlinear terms. Appl. Anal. Int. J. (2014). doi: 10.1080/00036811.2014.972384
  24. 24.
    Miranville, A., Zelik, S.: The Cahn–Hilliard equation with singular potentials and dynamic boundary conditions. Discret. Cont. Dyn. Syst. 28, 275–310 (2010)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Pawlow, I., Schimperna, G.: On a Cahn–Hilliard model with nonlinear diffusion. SIAM J. Math. Anal. 45, 31–63 (2013)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Pawlow, I., Schimperna, G.: A Cahn–Hilliard equation with singular diffusion. J. Differ. Equ. 254, 779–803 (2013)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Pawlow, I., Zajaczkowski, W.: A sixth order Cahn–Hilliard type equation arising in oil–water–surfactant mixtures. Commun. Pure Appl. Anal. 10, 1823–1847 (2011)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Pawlow, I., Zajaczkowski, W.: On a class of sixth order viscous Cahn–Hilliard type equations. Discret. Cont. Dyn. Syst. S 6, 517–546 (2013)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Savina, T.V., Golovin, A.A., Davis, S.H., Nepomnyashchy, A.A., Voorhees, P.W.: Faceting of a growing crystal surface by surface diffusion. Phys. Rev. E 67, 021606 (2003)CrossRefGoogle Scholar
  30. 30.
    Torabi, S., Lowengrub, J., Voigt, A., Wise, S.: A new phase-field model for strongly anisotropic systems. Proc. R. Soc. A 465, 1337–1359 (2009)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Wang, C., Wise, S.M.: Global smooth solutions of the modified phase field crystal equation. Methods Appl. Anal. 17, 191–212 (2010)MathSciNetMATHGoogle Scholar
  32. 32.
    Wang, C., Wise, S.M.: An energy stable and convergent finite difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49, 945–969 (2011)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Wise, S.M., Wang, C., Lowengrub, J.S.: An Energy stable and convergent finite difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47, 2269–2288 (2009)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Applications, UMR CNRS 7348-SP2MIUniversité de PoitiersChasseneuil Futuroscope CedexFrance

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