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Higher-Order Allen–Cahn Models with Logarithmic Nonlinear Terms

  • Laurence Cherfils
  • Alain Miranville
  • Shuiran Peng
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 69)

Abstract

Our aim in this chapter was to study higher-order (in space) Allen–Cahn models with logarithmic nonlinear terms. In particular, we obtain well-posedness results, as well as the existence of the global attractor.

Keywords

Variational Inequality Nonlinear Term Global Attractor Variational Solution Interpolation Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27,1085–1095 (1979)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Berry, J., Grant, M., Elder, K.R.: Diffusive atomistic dynamics of edge dislocations in two dimensions. Phys. Rev. E 73, 031609 (2006)Google Scholar
  4. 4.
    Caginalp, G., Esenturk, E.: Anisotropic phase field equations of arbitrary order. Discret. Contin. Dyn. Syst. S 4, 311–350 (2011)Google Scholar
  5. 5.
    Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)Google Scholar
  6. 6.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2, 258–267 (1958)Google Scholar
  7. 7.
    Chen, F., Shen, J.: Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems. Commun. Comput. Phys. 13, 1189–1208 (2013)Google Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    Cherfils, L., Miranville, A., Peng, S.: Higher-order models in phase separation. J. Appl. Anal. Comput. (to appear)Google Scholar
  10. 10.
    Cherfils, L., Miranville, A., Zelik, S.: The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596 (2011)Google Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    de Gennes, P.G.: Dynamics of fluctuations and spinodal decomposition in polymer blends. J. Chem. Phys. 72, 4756–4763 (1980)Google Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Frigeri, S., Grasselli, M.: Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials. Dyn. PDE 9, 273–304 (2012)Google Scholar
  15. 15.
    Galenko, P., Danilov, D., Lebedev, V.: Phase-field-crystal and Swift-Hohenberg equations with fast dynamics. Phys. Rev. E 79, 051110 (2009)Google Scholar
  16. 16.
    Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits. J. Stat. Phys. 87, 37–61 (1997)Google Scholar
  17. 17.
    Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interaction II. Interface motion. SIAM J. Appl. Math. 58, 1707–1729 (1998)Google Scholar
  18. 18.
    Gompper, G., Kraus, M.: Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations. Phys. Rev. E 47, 4289–4300 (1993)Google Scholar
  19. 19.
    Gompper, G., Kraus, M.: Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations. Phys. Rev. E 47 4301–4312 (1993)Google Scholar
  20. 20.
    Grasselli, M.: Finite-dimensional global attractor for a nonlocal phase-field system. Istit. Lombardo Accad. Sci. Lett. Rend. A 146, 3–22 (2012)Google Scholar
  21. 21.
    Grasselli, M., Miranville, A., Schimperna, G.: The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discret. Contin. Dyn. Syst. 28, 67–98 (2010)Google Scholar
  22. 22.
    Grasselli, M., Schimperna, G.: Nonlocal phase-field systems with general potentials. Discret. Contin. Dyn. Syst. A 33, 5089–5106 (2013)Google Scholar
  23. 23.
    Grasselli, M., Wu, H.: Well-posedness and longtime behavior for the modified phase-field crystal equation. Math. Model. Methods Appl. Sci. 24, 2743–2783 (2014)Google Scholar
  24. 24.
    Grasselli, M., Wu, H.: Robust exponential attractors for the modified phase-field crystal equation. Discret. Contin. Dyn. Syst. 35, 2539–2564 (2015)Google Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    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)Google Scholar
  27. 27.
    Korzec, M., Rybka, P.: On a higher order convective Cahn-Hilliard type equation. SIAM J. Appl. Math. 72, 1343–1360 (2012)Google Scholar
  28. 28.
    Miranville, A.: Some mathematical models in phase transition. Discret. Contin. Dyn. Syst. S 7, 271–306 (2014)Google Scholar
  29. 29.
    Miranville, A.: Asymptotic behavior of a sixth-order Cahn-Hilliard system. Central Eur. J. Math. 12, 141–154 (2014)Google Scholar
  30. 30.
    Miranville, A.: Sixth-order Cahn-Hilliard equations with logarithmic nonlinear terms. Appl. Anal. 94, 2133–2146 (2015)Google Scholar
  31. 31.
    Miranville, A.: Sixth-order Cahn-Hilliard systems with dynamic boundary conditions. Math. Methods Appl. Sci. 38, 1127–1145 (2015)Google Scholar
  32. 32.
    Miranville, A.: On the phase-field-crystal model with logarithmic nonlinear terms. RACSAM (to appear)Google Scholar
  33. 33.
    Miranville, A., Zelik, S.: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci. 27, 545–582 (2004)Google Scholar
  34. 34.
    Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Dafermos, C.M., Pokorny, M. (eds.) Handbook of Differential Equations, Evolutionary Partial Differential Equations, vol. 4, pp. 103–200. Elsevier, Amsterdam (2008)Google Scholar
  35. 35.
    Miranville, A., Zelik, S.: The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discret. Cont. Dyn. Syst. 28, 275–310 (2010)Google Scholar
  36. 36.
    Novick-Cohen, A.: The Cahn-Hilliard equation. In: Dafermos, C.M., Pokorny, M. (eds.) Handbook of Differential Equations, Evolutionary Partial Differential Equations, pp. 201–228. Elsevier, Amsterdam (2008)Google Scholar
  37. 37.
    Pawlow, I., Schimperna, G.: On a Cahn-Hilliard model with nonlinear diffusion. SIAM J. Math. Anal. 45, 31–63 (2013)Google Scholar
  38. 38.
    Pawlow, I., Schimperna, G.: A Cahn-Hilliard equation with singular diffusion. J. Diff. Equ. 254, 779–803 (2013)Google Scholar
  39. 39.
    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)Google Scholar
  40. 40.
    Pawlow, I., Zajaczkowski, W.: On a class of sixth order viscous Cahn-Hilliard type equations. Discret. Contin. Dyn. Syst. S 6, 517–546 (2013)Google Scholar
  41. 41.
    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)Google Scholar
  42. 42.
    Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)Google Scholar
  43. 43.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Wang, C., Wise, S.M.: Global smooth solutions of the modified phase field crystal equation. Methods Appl. Anal. 17, 191–212 (2010)Google Scholar
  45. 45.
    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)Google Scholar
  46. 46.
    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)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Laurence Cherfils
    • 1
  • Alain Miranville
    • 2
  • Shuiran Peng
    • 2
  1. 1.Université de La Rochelle, Laboratoire Mathématiques, Image et ApplicationsLa Rochelle CedexFrance
  2. 2.Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MIChasseneuil Futuroscope CedexFrance

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