Advertisement

Milan Journal of Mathematics

, Volume 79, Issue 2, pp 561–596 | Cite as

The Cahn-Hilliard Equation with Logarithmic Potentials

  • Laurence Cherfils
  • Alain MiranvilleEmail author
  • Sergey Zelik
Article

Abstract

Our aim in this article is to discuss recent issues related with the Cahn-Hilliard equation in phase separation with the thermodynamically relevant logarithmic potentials; in particular, we are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Neumann boundary conditions and then dynamic boundary conditions which account for the interactions with the walls in confined systems and have recently been proposed by physicists. We also present, in the case of dynamic boundary conditions, some numerical results.

Mathematics Subject Classification (2010)

35K55 35J60 80A22 

Keywords

Cahn-Hilliard equation logarithmic potential Neumann boundary conditions dynamic boundary conditions well-posedness asymptotic behavior 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abels H.: On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal. 194, 463–506 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abels H., Feireisl E.: On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J. 57, 659–698 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Abels H., Wilke M.: Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67, 3176–3193 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Alt H.W., Pawłow I.: A mathematical model of dynamics of non-isothermal phase separation. Physica D 59, 389–416 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Alt H.W., Pawłow I.: Existence of solutions for non-isothermal phase separation. Adv. Math. Sci. Appl. 1, 319–409 (1992)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Babin A.V., Vishik M.I.: Attractors of evolution equations. North-Holland, Amsterdam (1992)zbMATHGoogle Scholar
  8. 8.
    Bai F., Elliott C.M., Gardiner A.: A. Spence and A.M. Stuart, The viscous Cahn-Hilliard equation. Part I:. computations, Nonlinearity 8, 131–160 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Baňas L., Nürnberg R.: A multigrid method for the Cahn-Hilliard equation with obstacle potential. Appl. Math. Comput. 213, 290–303 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Barrett J.W., Blowey J.F.: An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. Numer. Math. 72, 257–287 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Barrett J.W., Blowey J.F.: Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility. Math. Comp. 68, 487–517 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Barrett J.W., Blowey J.F., Garcke H.: Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37, 286–318 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bartels S., Müller R.: A posteriori error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations. Interfaces Free Bound. 12, 45–73 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    S. Bartels and R. Müller, Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential, submitted.Google Scholar
  15. 15.
    Bates P.W., Han J.: The Neumann boundary problem for a nonlocal Cahn-Hilliard equation. J. Diff. Eqns. 212, 235–277 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Binder K., Frisch H.L.: Dynamics of surface enrichment: A theory based on the Kawasaki spin-exchange model in the presence of a wall. Z. Phys. B 84, 403–418 (1991)CrossRefGoogle Scholar
  17. 17.
    Blömker D., Gawron B., Wanner T.: Nucleation in the one-dimensional stochastic Cahn-Hilliard model. Discrete Cont. Dyn. Systems 27, 25–52 (2010)zbMATHCrossRefGoogle Scholar
  18. 18.
    Blömker D., Maier-Paape S., Wanner T.: Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation. Trans. Amer. Math. Soc. 360, 449–489 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Blowey J.F., Elliott C.M.: The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I. Mathematical analysis. Eur. J. Appl. Math. 2, 233–280 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Bo L., Shi K., Wang Y.: Support theorem for a stochastic Cahn-Hilliard equation. Electron. J. Prob. 15, 484–525 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Bo L., Wang Y.: Stochastic Cahn-Hilliard partial differential equations with Lévy spacetime white noise. Stochastics and Dynamics 6, 229–244 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Bonetti E., Colli P., Dreyer W., Gilardi G., Schimperna G., Sprekels J.: On a model for phase separation in binary alloys driven by mechanical effects. Physica D 165, 48–65 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Bonetti E., Dreyer W., Schimperna G.: Global solution to a viscous Cahn-Hilliard equation for tin-lead alloys with mechanical stresses. Adv. Diff. Eqns. 2, 231–256 (2003)MathSciNetGoogle Scholar
  24. 24.
    Bonfoh A., Grasselli M., Miranville A.: Long time behavior of a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Math. Methods Appl. Sci. 31, 695–734 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Bonfoh A., Grasselli M., Miranville A.: Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation. Topol. Methods Nonlinear Anal. 35, 155–185 (2010)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Bonfoh A., Grasselli M., Miranville A.: Singularly perturbed 1D Cahn-Hilliard equation revisited. Nonlinear Diff. Eqns. Appl. (NoDEA) 17, 663–695 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Bonfoh A., Miranville A.: On Cahn-Hilliard-Gurtin equations. Nonlinear Anal. 47, 3455–3466 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Boyer F.: Nonhomogeneous Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 225–259 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Boyer F.: A theoretical and numerical model for the study of incompressible mixture flows. Computers and Fluids 31, 41–68 (2002)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Boyer F., Chupin L., Fabrie P.: Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model. Eur. J. Mech. B Fluids 23, 759–780 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Boyer F., Lapuerta C.: Study of a three component Cahn-Hilliard flow model. M2AN Math. Model. Numer. Anal. 40, 653–687 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Boyer F., Lapuerta C., Minjeaud S., Piar B.: A local adaptive refinement method with multigrid preconditionning illustrated by multiphase flows simulations. ESAIM Proc. 27, 15–53 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Boyer F., Lapuerta C., Minjeaud S., Piar B., Quintard M.: Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows. Transp. Porous Media 82, 463–483 (2010)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Boyer F., Minjeaud S.: Numerical schemes for a three component Cahn-Hilliard model, M2AN. Math. Model. Numer. Anal. 45, 697–738 (2011)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Caffarelli L.A., Muler N.E.: An L bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 133, 129–144 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Caginalp G.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems, in On the evolution of phase boundaries, IMA Vol. Math. Appl. 43, M. Gurtin ed., Springer, New York, 1–27, 1992.Google Scholar
  38. 38.
    Caginalp G., Chen X.: Convergence of the phase field model to its sharp interface limits. European J. Appl. Math. 9, 417–445 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Cahn J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)CrossRefGoogle Scholar
  40. 40.
    Cahn J.W., Elliott C.M., Novick-Cohen A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Eur. J. Appl. Math. 7, 287–301 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Cahn J.W., Hilliard J.E.: Free energy of a nonuniform system I Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)CrossRefGoogle Scholar
  42. 42.
    Cahn J.W., Hilliard J.E.: Free energy of a nonuniform system III. Nucleation in a two component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959)CrossRefGoogle Scholar
  43. 43.
    Cardon-Weber C.: Cahn-Hilliard stochastic equation: existence of the solution and of its density. Bernouilli 7, 777–816 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Carrive M., Miranville A., Piétrus A.: The Cahn-Hilliard equation for deformable continua. Adv. Math. Sci. Appl. 10, 539–569 (2000)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Carvalho A.N., Dlotko T.: Dynamics of the viscous Cahn-Hilliard equation. J. Math. Anal. Appl. 344, 703–725 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    V. Chalupeckí, Numerical studies of Cahn-Hilliard equations and applications in image processing, in Proceedings of Czech-Japanese Seminar in Applied Mathematics 2004 (August 4-7, 2004), Czech Technical University in Prague.Google Scholar
  47. 47.
    Cherfils L., Miranville A.: Generalized Cahn-Hilliard equations with a logarithmic free energy. Rev. Real Acad. Sci. 94, 19–32 (2000)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Cherfils L., Miranville A.: On the Caginalp system with dynamic boundary conditions and singular potentials. Appl. Math. 54, 89–115 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    L. Cherfils, S. Gatti and A. Miranville, Corrigendum to “Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials” [J. Math. Anal. Appl. 343 (2008), 557–566], J. Math. Anal. Appl. 348 (2008), 1029–1030.Google Scholar
  50. 50.
    Cherfils L., Petcu M., Pierre M.: A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete Cont. Dyn. Systems 27, 1511–1533 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Chill R., Fašangová E., Prüss J.: Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions. Math. Nachr. 279, 1448–1462 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Cholewa J.W., Dlotko T.: Global attractor for the Cahn-Hilliard system. Bull. Austral. Math. Soc. 49, 277–293 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Cholewa J.W., Dlotko T.: Global attractors in abstract parabolic problems, London Mathematical Society Lecture Notes Series, Vol. 278,. Cambridge University Press, Cambridge (2000)Google Scholar
  54. 54.
    Choo S.M., Chung S.K.: Asymtotic behaviour of the viscous Cahn-Hilliard equation. J. Appl. Math. Comput. 11, 143–154 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Chupin L.: An existence result for a mixture of non-newtonian fluids with stressdiffusion using the Cahn-Hilliard formulation. Discrete Cont. Dyn. Systems B 3, 45–68 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Cohen D., Murray J.M.: A generalized diffusion model for growth and dispersion in a population. J. Math. Biol. 12, 237–248 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Conti M., Coti Zelati M.: Attractors for the Cahn-Hilliard equation with memory in 2D. Nonlinear Anal. 72, 1668–1682 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Conti M., Mola G.: 3-D viscous Cahn-Hilliard equation with memory. Math. Methods Appl. Sci. 32, 1370–1395 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Cook H.: Brownian motion in spinodal decomposition. Acta Metall. 18, 297–306 (1970)CrossRefGoogle Scholar
  60. 60.
    Copetti M.I.M., Elliott C.M.: Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63, 39–65 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Da Prato G.: A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal. 26, 241–263 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Dal Passo R., Giacomelli L.: A. Novick-Cohen, Existence for an Allen-Cahn/Cahn- Hilliard system with degenerate mobility. Interfaces Free Bound. 1, 199–226 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Debussche A., Dettori L.: On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 24, 1491–1514 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Debussche A., Zambotti L.: Conservative stochastic Cahn-Hilliard equation with reflection. Ann. Probab. 35, 1706–1739 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Dlotko T.: Global attractor for the Cahn-Hilliard equation in H2 and H3. J. Diff. Eqns. 113, 381–393 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Dolcetta I.C., Vita S.F.: Area-preserving curve-shortening flows: from phase separation to image processing. Interfaces Free Bound. 4, 325–343 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Eden A., Foias C., Nicolaenko B., Temam R.: Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, Vol. 37,. John-Wiley, New York (1994)Google Scholar
  68. 68.
    Eden A., Kalantarov V.K.: The convective Cahn-Hilliard equation. Appl. Math. Lett. 20, 455–461 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Eden A., Kalantarov V.K.: 3D convective Cahn-Hilliard equation. Commun. Pure Appl. Anal. 6, 1075–1086 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    A. Eden, V. Kalantarov and S.V. Zelik, Infinite energy solutions for the Cahn-Hilliard equation in cylindrical domains, submitted.Google Scholar
  71. 71.
    Efendiev M., Gajewski H., Zelik S.: The finite dimensional attractor for a 4th order system of the Cahn-Hilliard type with a supercritical nonlinearity. Adv. Diff. Eqns. 7, 1073–1100 (2002)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Efendiev M., Miranville A.: New models of Cahn-Hilliard-Gurtin equations. Contin. Mech. Thermodyn. 16, 441–451 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Efendiev M., Miranville A., Zelik S.: Exponential attractors for a nonlinear reactiondiffusion system in \({\mathbb {R}^3}\) . C.R. Acad. Sci. Paris Série I Math. 330, 713–718 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Efendiev M., Miranville A., Zelik S.: Exponential attractors for a singularly perturbed Cahn-Hilliard system. Math. Nach. 272, 11–31 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Elezovic N., Mikelic A.: On the stochastic Cahn-Hilliard equation. Nonlinear Anal. 16, 1169–1200 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical models for phase change problems, J.F. Rodrigues ed., International Series of Numerical Mathematics, Vol. 88, Birkhäuser, Basel, 1989.Google Scholar
  77. 77.
    Elliott C.M., French D.A.: Numerical studies of the Cahn-Hilliard equation for phase separation. IMA J. Appl. Math. 38, 97–128 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Elliott C.M., French D.A.: A non-conforming finite element method for the twodimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26, 884–903 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Elliott C.M., French D.A., Milner F.A.: A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54, 575–590 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Elliott C.M., Garcke H.: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Elliott C.M., Garcke H.: Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Physica D 109, 242–256 (1997)MathSciNetCrossRefGoogle Scholar
  82. 82.
    C.M. Elliott and S. Luckhaus, A generalized diffusion equation for phase separation of a multi-component mixture with interfacial energy, SFB 256 Preprint No. 195, University of Bonn, 1991.Google Scholar
  83. 83.
    Elliott C.M., Stuart A.M.: Viscous Cahn-Hilliard equation II. Analysis. J. Diff. Eqns. 128, 387–414 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Elliott C.M., Zheng S.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96, 339–357 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Eyre J.D.: Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53, 1686–1712 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    J.D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, in Computational and mathematical models of microstructural evolution, J.W. Bullard, R. Kalia, M. Stoneham and L.Q. Chen eds., The Materials Research Society, 1998.Google Scholar
  87. 87.
    Feng X., Prohl A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99, 47–84 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Feng X., Prohl A.: Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Interfaces Free bound. 7, 1–28 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Feng W.M., Yu P., Hu S.Y., Liu Z.K., Du Q., Chen L.Q.: A Fourier spectral moving mesh method for the Cahn-Hilliard equation with elasticity. Commun. Comput. Phys. 5, 582–599 (2009)MathSciNetGoogle Scholar
  90. 90.
    Fischer H.P., Maass P., Dieterich W.: Novel surface modes in spinodal decomposition. Phys. Rev. Lett. 79, 893–896 (1997)CrossRefGoogle Scholar
  91. 91.
    Fischer H.P., Maass P., Dieterich W.: Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett 42, 49–54 (1998)Google Scholar
  92. 92.
    Fischer H.P., Reinhard J., Dieterich W., Gouyet J.-F., Maass P., Majhofer A., Reinel D.: Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. Chem. Phys 108, 3028–3037 (1998)Google Scholar
  93. 93.
    FreeFem++ is freely available at http://www.freefem.org/ff++.
  94. 94.
    Furihata D.: A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math 87, 675–699 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Gajewski H. Zacharias K.: On a nonlocal phase separation model, J. Math. Anal. Appl 286, 11–31 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Gal C.G.: A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci 29, 2009–2036 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Gal C.G.: Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls, Electron. J. Diff. Eqns 2006, 1–23 (2006)Google Scholar
  98. 98.
    Gal C.G., Grasselli M.: Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 401–436 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Gal C.G. Grasselli M.: Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Cont. Dyn. Systems 28, 1–39 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Gal C.G., Miranville A.: Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl 10, 1738–1766 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Gal C.G., Miranville A.: Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions, Discrete Cont. Dyn. Systems S 2, 113–147 (2009)MathSciNetzbMATHGoogle Scholar
  102. 102.
    GalC.G. Wu H.: Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Cont. Dyn. Systems 22, 1041–1063 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Galenko P., Jou D: Kinetic contribution to the fast spinodal decomposition controlled by diffusion. Physica A 388, 3113–3123 (2009)MathSciNetCrossRefGoogle Scholar
  104. 104.
    Galenko P., Lebedev V.: Analysis of the dispersion relation in spinodal decomposition of a binary system, Philos. Mag. Lett 87, 821–827 (2007)CrossRefGoogle Scholar
  105. 105.
    Galenko P. Lebedev V: Local nonequilibrium effect on spinodal decomposition in a binary system, Int. J. Thermodyn 11, 21–28 (2008)Google Scholar
  106. 106.
    Galenko P. Lebedev V.: Nonequilibrium effects in spinodal decomposition of a binary system, Phys. Lett. A 372, 985–989 (2008)zbMATHCrossRefGoogle Scholar
  107. 107.
    GaoW. Yin J.: System of Cahn-Hilliard equations with nonconstant interaction matrix, Chinese Ann. Math., Ser. A 20, 169–176 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    Garcke H.: On Cahn-Hilliard systems with elasticity, Proc. Roy. Soc. Edinburgh A 133, 307–331 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    Garcke H.: On a Cahn-Hilliard model for phase separation with elastic misfit, Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 165–185 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    H. Garcke, Mechanical effects in the Cahn-Hilliard model: A review on mathematical results, in Mathematical methods and models in phase transitions, A. Miranville ed., Nova Sci. Publ., New York, 43–77, 2005.Google Scholar
  111. 111.
    Garcke H. Weikard U.: Numerical approximation of the Cahn-Larché equation, Numer. Math 100, 639–662 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Gatti S., Grasselli M., Miranville A., Pata V.: On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation, J. Math. Anal. Appl 312, 230–247 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Gatti S., Grasselli M., Miranville A., Pata V.: Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D, Math. Models Methods Appl. Sci 15, 165–198 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Gatti S., Grasselli M., Miranville A., Pata V.: Memory relaxation of first order evolution equations. Nonlinearity 18, 1859–1883 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of the onedimensional Cahn-Hilliard equation, in Dissipative phase transitions, Ser. Adv. Math. Appl. Sci., Vol. 71, World Sci. Publ., Hackensack, NJ, 101–114, 2006.Google Scholar
  116. 116.
    Giacomin G. Lebowitz J.L.: Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits, J. Statist. Phys 87, 37–61 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  118. 118.
    Gilardi G., Miranville A., Schimperna G.: On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal. 8, 881–912 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Gilardi G., Miranville A., Schimperna G.: Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B 31, 679–712 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Goldstein G.R., Miranville A., Schimperna G.: A Cahn-Hilliard model in a domain with non-permeable walls. Physica D 240, 754–766 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 121.
    Golovin A.A., Nepomnyashchy A.A., Davis S.H., Zaks M.A.: Convective Cahn-Hilliard models: from coarsening to roughening. Phys. Rev. Lett 86, 1550–1553 (2001)CrossRefGoogle Scholar
  122. 122.
    Gomez H., Calo V.M., Basilevs Y., Hughes T.J.R.: Isogeometric analysis of Cahn-Hilliard phase field model. Comput. Methods Appl. Mech. Engrg 197, 4333–4352 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    Goudenège L.: Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection. Stochastic Process. Appl 119, 3516–3548 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    L. Goudenège, D. Martin and G. Vial, High order finite element calculations for the deterministic Cahn-Hilliard equation, submitted.Google Scholar
  125. 125.
    Grasselli M., Miranville A., Rossi R., Schimperna G.: Analysis of the Cahn-Hilliard equation with a chemical potential dependent mobility, Commun. Partial Diff. Eqns 36, 1193–1238 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    Grasselli M. Pierre M.: A splitting method for the Cahn-Hilliard equation with inertial term, Math. Models Methods Appl. Sci 20, 1–28 (2010)MathSciNetCrossRefGoogle Scholar
  127. 127.
    Grasselli M., Schimperna G., Miranville A.: The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Cont. Dyn. Systems 28, 67–98 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    Grasselli M., Schimperna G., Segatti A., Zelik S.: On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Eqns 9, 371–404 (2009)MathSciNetCrossRefGoogle Scholar
  129. 129.
    Grasselli M., Schimperna G., Zelik S.: On the 2D Cahn-Hilliard equation with inertial term, Commun. Partial Diff. Eqns 34, 137–170 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  130. 130.
    Grasselli M., Schimperna G., Zelik S.: Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term. Nonlinearity 23, 707–737 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  131. 131.
    Guo B., Wang G., Wang S.: Well posedness for the stochastic Cahn-Hilliard equation driven by Lévy space-time white noise, Diff. Int. Eqns 22, 543–560 (2009)zbMATHGoogle Scholar
  132. 132.
    Gurtin M.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178–192 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Gurtin M., Polignone D., Vinals J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci 6, 815–831 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  134. 134.
    He L., Liu Y.: A class of stable spectral methods for the Cahn-Hilliard equation. J. Comput. Phys 228, 5101–5110 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    Injrou S., Pierre M.: Stable discretizations of the Cahn-Hilliard-Gurtin equations. Discrete Cont. Dyn. Systems 22, 1065–1080 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  136. 136.
    Injrou S., Pierre M.: Error estimates for a finite element discretization of the Cahn-Hilliard-Gurtin equations. Diff. Int. Eqns 15, 1161–1192 (2010)MathSciNetzbMATHGoogle Scholar
  137. 137.
    Jacquemin D.: Calculation of two phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys 155, 96–127 (1999)MathSciNetCrossRefGoogle Scholar
  138. 138.
    Kay D., Styles V., Süli E.: Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM J. Numer. Anal 47, 2660–2685 (2009)zbMATHCrossRefGoogle Scholar
  139. 139.
    Kay D., Styles V., Welford R.: Finite element approximation of a Cahn-Hilliard-Navier-Stokes system. Interfaces Free Bound 10, 15–43 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  140. 140.
    Kenmochi N., Niezgódka M., Pawłow I.: Subdifferential operator approach to the Cahn-Hilliard equation with constraint. J. Diff. Eqns 117, 320–356 (1995)zbMATHCrossRefGoogle Scholar
  141. 141.
    Kenzler R., Eurich F., Maass P., Rinn B., Schropp J., Bohl E., Dieterich W.: Phase separation in confined geometries: solving the Cahn-Hilliard equation with generic boundary conditions. Comput. Phys. Commun 133, 139–157 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  142. 142.
    Kim J.: Phase field computations for ternary fluid flows. Comput. Methods Appl. Mech. Engrg 196, 4779–4788 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  143. 143.
    Kim J.: Three-dimensional numerical simulations of a phase-field model for anisotropic interfacial energy. Commun. Korean Math. Soc 22, 453–464 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  144. 144.
    Kim J.: A numerical method for the Cahn-Hilliard equation with a variable mobility. Commun. Nonlinear Sci. Numer. Simul 12, 1560–1571 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  145. 145.
    Kim J.: A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows. Comput. Methods Appl. Mech. Engrg 198, 37–40 (2009)Google Scholar
  146. 146.
    Kim J., Kang K.: A numerical method for the ternary Cahn-Hilliard system with a degenerate mobility. Appl. Numer. Math 59, 1029–1042 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  147. 147.
    Kim J., Lowengrub J.: Phase field modeling and simulation of three-phase flows. Interfaces Free Bound 7, 435–466 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  148. 148.
    Klapper I., Dockery J.: Role of cohesion in the material description of biofilms. Phys. Rev. E 74, 0319021–0319028 (2006)MathSciNetCrossRefGoogle Scholar
  149. 149.
    Kohn R.V., Otto F.: Upper bounds for coarsening rates. Commun. Math. Phys 229, 375–395 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  150. 150.
    Langer J.S.: Theory of spinodal decomposition in alloys, Ann. Phys 65, 53–86 (1975)Google Scholar
  151. 151.
    Lecoq N., Zapolsky H., Galenko P.: Evolution of the structure factor in a hyperbolic model of spinodal decomposition, Eur. Phys. J. Special Topics 177, 165–175 (2009)CrossRefGoogle Scholar
  152. 152.
    Lee H.G., Kim J.: A second-order accurate non-linear difference scheme for the N-component Cahn-Hilliard system. Physica A 387, 19–20 (2008)Google Scholar
  153. 153.
    Li D., Zhong C.: Global attractor for the Cahn-Hilliard system with fast growing nonlinearity. J. Diff. Eqns 149, 191–210 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  154. 154.
    Liu C.: Convective Cahn-Hilliard equation with degenerate mobility. Dyn. Cont. Discrete Impuls. Systems Ser. A Math. Anal 16, 15–25 (2009)zbMATHGoogle Scholar
  155. 155.
    Liu C., Shen J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179, 211–228 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  156. 156.
    Lowengrub J., Truskinovski L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. Royal Soc. London Ser. A 454, 2617–2654 (1998)zbMATHCrossRefGoogle Scholar
  157. 157.
    Ma T., Wang S.: Cahn-Hilliard equations and phase transition dynamics for binary systems, Discrete Cont. Dyn. Systems 11, 741–784 (2009)zbMATHGoogle Scholar
  158. 158.
    Maier-Paape S., Mischaikow K., Wanner T.: Rigorous numerics for the Cahn-Hilliard equation on the unit square. Rev. Mat. Complutense 21, 351–426 (2008)MathSciNetzbMATHGoogle Scholar
  159. 159.
    Maier-Paape S., Wanner T.: Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate, Commun. Math. Phys 195, 435–464 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  160. 160.
    Maier-Paape S., Wanner T.: Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics. Arch. Ration. Mech. Anal 151, 187–219 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  161. 161.
    Miranville A.: Some generalizations of the Cahn-Hilliard equation. Asymptotic Anal 22, 235–259 (2000)MathSciNetzbMATHGoogle Scholar
  162. 162.
    Miranville A.: Long-time behavior of some models of Cahn-Hilliard equations in deformable continua. Nonlinear Anal. Real World Appl 2, 273–304 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  163. 163.
    Miranville A.: Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions. Physica D 158, 233–257 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  164. 164.
    MiranvilleA. Piétrus A.: A new formulation of the Cahn-Hilliard equation. Nonlinear Anal. Real World Appl 7, 285–307 (2006)MathSciNetCrossRefGoogle Scholar
  165. 165.
    Miranville A., Rougirel A.: Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations. Z. Angew. Math. Phys 57, 244–268 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  166. 166.
    MiranvilleA. Schimperna G.: Nonisothermal phase separation based on a microforce balance. Discrete Cont. Dyn. Systems B 5, 753–768 (2005)CrossRefGoogle Scholar
  167. 167.
    Miranville A., Schimperna G.: Generalized Cahn-Hilliard equations for multicomponent alloys. Adv. Math. Sci. Appl 19, 131–154 (2009)MathSciNetzbMATHGoogle Scholar
  168. 168.
    Miranville A., Schimperna G.: On a doubly nonlinear Cahn-Hilliard-Gurtin system. Discrete Cont. Dyn. Systems B 14, 675–697 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  169. 169.
    MiranvilleA. Zelik S.: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci 27, 545–582 (2004)MathSciNetCrossRefGoogle Scholar
  170. 170.
    Miranville A., Zelik S.: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci 28, 709–735 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  171. 171.
    A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations, Vol. 4, C.M. Dafermos and M. Pokorny eds., Elsevier, Amsterdam, 103–200, 2008.Google Scholar
  172. 172.
    Miranville A., Zelik S.: Doubly nonlinear Cahn-Hilliard-Gurtin equations. Hokkaido Math. J 38, 315–360 (2009)MathSciNetzbMATHGoogle Scholar
  173. 173.
    Miranville A., Zelik S.: The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete Cont. Dyn. Systems 28, 275–310 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  174. 174.
    B. Nicolaenko and B. Scheurer, Low dimensional behaviour of the pattern formation equations, in Trends and practice of nonlinear analysis, V. Lakshmikantham ed., North-Holland, 1985.Google Scholar
  175. 175.
    Nicolaenko B., Scheurer B., Temam R.: Some global dynamical properties of a class of pattern formation equations. Commun. Partial Diff. Eqns 14, 245–297 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  176. 176.
    Novick-Cohen A.: Energy methods for the Cahn-Hilliard equation. Quart. Appl. Math 46, 681–690 (1988)MathSciNetzbMATHGoogle Scholar
  177. 177.
    A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material instabilities in continuum and related problems, J.M. Ball ed., Oxford University Press, Oxford, 329–342, 1988.Google Scholar
  178. 178.
    Novick-Cohen A.: The Cahn-Hilliard equation: Mathematical and modeling perspectives. Adv. Math. Sci. Appl 8, 965–985 (1998)MathSciNetzbMATHGoogle Scholar
  179. 179.
    Novick-Cohen A.: Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system. Physica D 137, 1–24 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  180. 180.
    A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations, Evolutionary Partial Differential Equations, Vol. 4, C.M. Dafermos and M. Pokorny eds., Elsevier, Amsterdam, 201–228, 2008.Google Scholar
  181. 181.
    Oron A., Davis S.H., Bankoff S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys 69, 931–980 (1997)CrossRefGoogle Scholar
  182. 182.
    Pawłow I., Zajaczkowski W.M.: Strong solvability of 3-D Cahn-Hilliard system in elastic solids. Math. Methods Appl. Sci 32, 879–914 (2008)Google Scholar
  183. 183.
    Pawłow I. Zajaczkowski W.M.: Weak solutions to 3-D Cahn-Hilliard system in elastic solids. Topol. Methods Nonlinear Anal 32, 347–377 (2008)MathSciNetzbMATHGoogle Scholar
  184. 184.
    Pawłow I., Zajaczkowski W.M.: Global regular solutions to Cahn-Hilliard system coupled with viscoelasticity. Math. Methods Appl. Sci 32, 2197–2242 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  185. 185.
    Pawłow I., Zajaczkowski W.M.: Long time behaviour of a Cahn-Hilliard system coupled with viscoelasticity. Ann. Polon. Math 98, 1–21 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  186. 186.
    M. Pierre, Personal communication.Google Scholar
  187. 187.
    Prüss J., Racke R., Zheng S.: Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions. Ann. Mat. Pura Appl 4(185), 627–648 (2006)CrossRefGoogle Scholar
  188. 188.
    Prüss J., Vergara V., Zacher R.: Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete Cont. Dyn. Systems 26, 625–647 (2010)zbMATHGoogle Scholar
  189. 189.
    Qian T., Wang X.-P., Sheng P.: A variational approach to moving contact line hydrodynamics. J. Fluid Mech 564, 333–360 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  190. 190.
    Racke R., Zheng S.: The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Diff. Eqns 8, 83–110 (2003)MathSciNetzbMATHGoogle Scholar
  191. 191.
    Rajagopal A., Fischer P., Kuhl E., Steinmann P.: Natural element analysis of the Cahn-Hilliard phase-field model. Comput. Mech. 46, 471–493 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  192. 192.
    Rougirel A.: Convergence to steady state and attractors for doubly nonlinear equations. J. Math. Anal. Appl 339, 281–294 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  193. 193.
    Rossi R.: On two classes of generalized viscous Cahn-Hilliard equations. Commun. Pure Appl. Anal 4, 405–430 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  194. 194.
    R. Rossi, Global attractor for the weak solutions of a class of viscous Cahn-Hilliard equations, in Dissipative phase transitions, Ser. Adv. Math. Appl. Sci., Vol. 71, World Sci. Publ., Hackensack, NJ, 247–268, 2006.Google Scholar
  195. 195.
    Rybka P., Hoffmann K.-H.: Convergence of solutions to Cahn-Hilliard equation. Commun. Partial Diff. Eqns 24, 1055–1077 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  196. 196.
    Savaré G., Visintin A.: Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl 8, 49–89 (1997)MathSciNetzbMATHGoogle Scholar
  197. 197.
    Schimperna G.: Weak solution to a phase-field transmission problem in a concentrated capacity. Math. Methods Appl. Sci 22, 1235–1254 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  198. 198.
    Schimperna G.: Global attractor for Cahn-Hilliard equations with nonconstant mobility. Nonlinearity 20, 2365–2387 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  199. 199.
    G. Schimperna and S. Zelik, Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations, submitted.Google Scholar
  200. 200.
    Shen J., Yang X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Cont. Dyn. Systems 28, 1669–1691 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  201. 201.
    Shen J., Yang X.: Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows. Chinese Ann. Math., Ser. B 31, 743–758 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  202. 202.
    Shen W., Zheng S.: On the coupled Cahn-Hilliard equations. Commun. Partial Diff. Eqns 18, 701–727 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  203. 203.
    Stogner R.H., Carey G.F., Murray B.T.: Approximation of Cahn-Hilliard diffuse interface models using parallel adaptative mesh refinement and coarsening with C1 elements. Int. J. Numer. Methods Engrg 76, 636–661 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  204. 204.
    Temam R.: Infinite-dimensional dynamical systems in mechanics and physics, Second edition, Applied Mathematical Sciences, Vol 68. Springer-Verlag, New York (1997)Google Scholar
  205. 205.
    Thiele U., Knobloch E.: Thin liquid films on a slightly inclined heated plate. Physica D 190, 213–248 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  206. 206.
    Tremaine S.: On the origin of irregular structure in Saturn’s rings. Astron. J. 125, 894–901 (2003)CrossRefGoogle Scholar
  207. 207.
    Watson S.J., Otto F., Rubinstein B., Davis S.H.: Coarsening dynamics of the convective Cahn-Hilliard equation. Physica D 178, 127–148 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  208. 208.
    Wells G.N., KuhlE. Garikipati K.: A discontinuous Galerkin method for Cahn-Hilliard equation. J. Comput. Phys 218, 860–877 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  209. 209.
    Wise S., Kim J. Lowengrub J.: Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys 226, 414–446 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  210. 210.
    Wu H., Zheng S.: Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. J. Diff. Eqns 204, 511–531 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  211. 211.
    Xia Y., Xu Y., Shu C.-W.: Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. Commun. Comput. Phys 5, 821–835 (2009)MathSciNetGoogle Scholar
  212. 212.
    J. Yin and C. Liu, Cahn-Hilliard type equations with concentration dependent mobility, in Mathematical methods and models in phase transitions, A. Miranville ed., Nova Sci.Publ., New York, 79–93, 2005Google Scholar
  213. 213.
    Yue P., Feng J.J., Liu C., Shen J.: A diffuse-interface method for simulating twophase flows of complex fluids. J. Fluid Mech 515, 293–317 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  214. 214.
    Zhang T., Wang Q.: Cahn-Hilliard vs singular Cahn-Hilliard equations in phase field modeling. Commun. Comput. Phys. 7, 362–382 (2010)MathSciNetGoogle Scholar
  215. 215.
    Zhao L.-Y., Wu H., Huang H.-Y.: Convergence to equilibrium for a phase-field model for the mixture of two incompressible fluids. Commun. Math. Sci 7, 939–962 (2009)MathSciNetzbMATHGoogle Scholar
  216. 216.
    Zheng S.: Asymptotic behavior of solution to the Cahn-Hilliard equation. Appl. Anal 23, 165–184 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  217. 217.
    Zhou J.X., Li M.E.: Solving phase field equations with a meshless method. Commun. Numer. Methods Engrg 22, 1109–1115 (2006)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Laurence Cherfils
    • 1
  • Alain Miranville
    • 2
    Email author
  • Sergey Zelik
    • 3
  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 6086 - SP2MIChasseneuil Futuroscope CedexFrance
  3. 3.Department of MathematicsUniversity of SurreyGuildfordUnited Kingdom

Personalised recommendations