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Archive for Rational Mechanics and Analysis

, Volume 233, Issue 1, pp 167–247 | Cite as

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

  • Chun Liu
  • Hao WuEmail author
Article

Abstract

The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn–Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager’s principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints: conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of the asymptotic limit as \({t\to+\infty}\) and characterize the stability of local energy minimizers for the system.

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Notes

Acknowledgements

The authorswould like to thank the anonymous referees for their careful reading of an initial version of this paper and for several helpful comments that allowed us to improve the presentation. The authors also want to thank Professors P. Colli, T. Fukao, C. Gal, H. Garcke, T.-Z. Qian and U. Stefanelli for helpful discussions.

Compliance with Ethical Standards

Funding

C. Liu is partially supported byNSF grantsDMS-1714401, DMS-1412005. H. Wu is partially supported by NNSFC grant No. 11631011 and the Shanghai Center for Mathematical Sciences at Fudan University.

Conflict of Interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Abels, H., Wilke, M.: Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67(11), 3176–3193 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akagi, G.: Stability of non-isolated asymptotic profiles for fast diffusion. Commun. Math. Phys. 345(1), 77–100 (2016)ADSMathSciNetzbMATHGoogle Scholar
  3. 3.
    Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30(1), 139–165 (1997)ADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bai, F., Elliott, C.M., Gardiner, A., Spence, A., Stuart, A.M.: The viscous Cahn-Hilliard equation. I. Computations. Nonlinearity8, 131–160 (1995)ADSMathSciNetzbMATHGoogle Scholar
  5. 5.
    Bates, P., Fife, P.: The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53(4), 990–1008 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brezzi, F., Gilardi, G.: Part I. FEM mathematics.Finite Element Handbook, (Ed. Kardestuncer H.) McGraw-Hill Book Co., New York, 1987Google Scholar
  7. 7.
    Caffarelli, L.A., Muller, N.E.: An \(L^\infty \) bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 133, 129–144 (1995)MathSciNetGoogle Scholar
  8. 8.
    Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)Google Scholar
  9. 9.
    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
  10. 10.
    Cavaterra, C., Gal, C.G., Grasselli, M.: Cahn-Hilliard equations with memory and dynamic boundary conditions. Asymptot. Anal. 71, 123–162 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cavaterra, C., Grasselli, M., Wu, H.: Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Commun. Pure Appl. Anal. 13(5), 1855–1890 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cazenave, T., Haraux, A.:An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications, Vol. 13. Oxford University Press, New York, 1998Google Scholar
  13. 13.
    Chen, X.F.: Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44, 262–311 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chen, X.F., Wang, X.P., Xu, X.M.: Analysis of the Cahn-Hilliard equation with a relaxation boundary condition modeling the contact angle dynamics. Arch. Ration. Mech. Anal. 213, 1–24 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cherfils, L., Gatti, S., Miranville, A.: A variational approach to a Cahn–Hilliard model in a domain with nonpermeable walls.J. Math. Sci. (N.Y.) 189, 604–636, 2013Google Scholar
  16. 16.
    Cherfils, L., Miranville, A., Zelik, S.: The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chill, R.: On the Łojasiewicz-Simon gradient inequality. J. Funct. Anal. 201(2), 572–601 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Chill, R., Fasangová, E., Prüss, J.: Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions. Math. Nachr. 279(13–14), 1448–1462 (2006)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Colli, P., Fukao, T.: Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary. J. Math. Anal. Appl. 429(2), 1190–1213 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Colli, P., Fukao, T.: Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials. Nonlinear Anal. 127, 413–433 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Colli, P., Gilardi, G., Sprekels, J.: On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential. J. Math. Anal. Appl. 419(2), 972–994 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Colli, P., Gilardi, G., Sprekels, J.: On a Cahn–Hilliard system with convection and dynamic boundary conditions.Ann. Mat. Pura Appl. (4) 197(5), 1445–1475, 2018Google Scholar
  23. 23.
    Denk, R., Prüss, J., Zacher, R.: Maximal \(L^p\)-regularity of parabolic problems with boundary dynamics of relaxation type. J. Funct. Anal. 255, 3149–3187 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Du, Q., Liu, C., Ryham, R., Wang, X.Q.: Energetic variational approaches in modeling vesicle and fluid interactions. Physica D238, 923–930 (2009)ADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Dziuk, G., Elliott, C.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Eisenberg, B., Hyon, Y., Liu, C.: Energy variational analysis of ions in water and channels: field theory for primitive models of complex ionic fluids. J. Chem. Phys. 133(10), 104104 (2010)ADSGoogle Scholar
  27. 27.
    Elliott, C.M., Stuart, A.M.: Viscous Cahn-Hilliard equation. II. Analysis. J. Differ. Equ. 128, 387–414 (1996)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Elliott, C.M., Zheng, S.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96, 339–357 (1986)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Feireisl, E., Simondon, F.: Convergence for semilinear degenerate parabolic equations in several space dimensions. J. Dyn. Differ. Equ. 12(3), 647–673 (2000)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Fischer, H.P., Maass, P., Dieterich, W.: Novel surface modes in spinodal decomposition. Phys. Rev. Lett. 79, 893–896 (1997)ADSGoogle Scholar
  31. 31.
    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)ADSGoogle Scholar
  32. 32.
    Forster, J.: Mathematical Modeling of Complex Fluids, Master's Thesis, University of Würzburg, 2013Google Scholar
  33. 33.
    Fried, E., Gurtin, M.E.: Continuum theory of thermally induced phase transitions based on an order parameter. Physica D68, 326–343 (1993)ADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Gal, C.G.: A Cahn-Hilliard model in bounded domains with permeable walls. Math. Methods Appl. Sci. 29, 2009–2036 (2006)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Gal, C.G.: Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 12, 1241–1274 (2007)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gal, C.G.: Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard equation with dynamic boundary conditions. Dyn. Partial Differ. Equ. 5, 39–67 (2008)MathSciNetzbMATHGoogle Scholar
  37. 37.
    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)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Gal, C.G., Miranville, A.: Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete Contin. Dyn. Syst. Ser. S2, 113–147 (2009)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Gal, C.G., Wu, H.: Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete Contin. Dyn. Syst. 22, 1041–1063 (2008)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Garcke, H., Knopf, P.: Weak solutions of the Cahn–Hilliard system with dynamic boundary conditions: a gradient flow approach, preprint, 2018. arXiv:1810.09817
  41. 41.
    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)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Gilardi, G., Miranville, A., Schimperna, G.: Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Chin. Ann. Math. Ser. B31, 679–712 (2010)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Giorgini, A., Grasselli, M., Miranville, A.: The Cahn-Hiliard-Oono equation with singular potential. Math. Models Methods Appl. Sci. 27(13), 2485–2510 (2017)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Giorgini, A., Grasselli, M., Wu, H.: On the Cahn–Hilliard–Hele–Shaw system with singular potential.Ann. Inst. H. Poincaré Anal. Non Lineaire 35(4), 1079–1118, 2018Google Scholar
  45. 45.
    Goldstein, G., Miranville, A., Schimperna, G.: A Cahn-Hilliard model in a domain with non-permeable walls. Physica D240(8), 754–766 (2011)ADSMathSciNetzbMATHGoogle Scholar
  46. 46.
    Grinfeld, M., Novick-Cohen, A.: Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments. Proc. R. Soc. Edinb. Sect. A125, 351–370 (1995)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Gurtin, M.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D92, 178–192 (1996)ADSMathSciNetzbMATHGoogle Scholar
  48. 48.
    Haraux, A., Jendoubi, M.A.: Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity. Asymptot. Anal. 26(1), 21–36 (2001)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Heida, M.: On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system. Int. J. Eng. Sci. 62, 126–156 (2013)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Heida, M.: Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation. Appl. Math. 60(1), 51–90 (2015)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Henry, D.:Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics, Vol. 840. Springer, Berlin, 1981Google Scholar
  52. 52.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)Google Scholar
  53. 53.
    Huang, S.Z., Takáč, P.: Convergence in gradient-like systems which are asymptotically autonomous and analytic. Nonlinear Anal. 46, 675–698 (2001)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Hyon, Y., Kwak, D.Y., Liu, C.: Energetic variational approach in complex fluids: maximum dissipation principle. Discrete Contin. Dyn. Syst. 26(4), 1291–1304 (2010)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Kajiwara, N.: Global well-posedness for a Cahn–Hilliard equation on bounded domains with permeable and non-permeable walls in maximal regularity spaces. Adv. Math. Sci. Appl. 27(2), 277–298, 2018Google Scholar
  56. 56.
    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)ADSMathSciNetzbMATHGoogle Scholar
  57. 57.
    Khain, E., Sander, L.M.: Generalized Cahn-Hilliard equation for biological applications. Phys. Rev. E77, 051129 (2008)ADSGoogle Scholar
  58. 58.
    Koba, H., Liu, C., Giga, Y.: Energetic variational approaches for incompressible fluid systems on an evolving surface. Quart. Appl. Math. 75(2), 359–389 (2017)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Ladyzhenskaya, O., Solonnikov, V., Ural'ceva, N.:Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs. American Mathematical Society, Providence, 1968Google Scholar
  60. 60.
    Lions, J.L., Magenes, E.:Non-Homogeneous Boundary Value Problems and Applications Vol. 1, Die Grundlehren der mathematischen Wissenschaften, Vol. 181, Springer-Verlag Berlin Heidelberg, 1972Google Scholar
  61. 61.
    Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D179(3–4), 211–228 (2003)ADSMathSciNetzbMATHGoogle Scholar
  62. 62.
    Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions.R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978), 2617–2654, 1998Google Scholar
  63. 63.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  64. 64.
    Miranville, A.: The Cahn-Hilliard equation and some of its variants. AIMS Math. 2(3), 479–544 (2017)Google Scholar
  65. 65.
    Miranville, A., Zelik, S.: Exponential attractors for the Cahn-Hilliard equation with dynamical boundary conditions. Math. Methods Appl. Sci. 28, 709–735 (2005)ADSMathSciNetzbMATHGoogle Scholar
  66. 66.
    Miranville, A., Zelik, S.: The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete Contin. Dyn. Syst. 28, 275–310 (2010)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Nirenberg, L.: On elliptic partial differential equations. Annali della Scoula Norm. Sup. Pisa13, 115–162 (1959)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Novick-Cohen, A.: On the viscous Cahn–Hilliard equation.Material Instabilities in Continuum Mechanics (Edinburgh, 1985–1986), pp. 329–342. Oxford University Press, New York, 1988Google Scholar
  69. 69.
    Novick-Cohen, A.: The Cahn–Hilliard equation.Evolutionary Equations, Handbook of Differential Equations, Vol. 4, pp. 201–228 (Eds. Dafermos C.M. and Pokorný M.) Elsevier/North-Holland, Amsterdam, 2008Google Scholar
  70. 70.
    Oden, J.T., Prudencio, E.E., Hawkins-Daarud, A.: Selection and assessment of phenomenological models of tumor growth. Math. Models Methods Appl. Sci. 23, 1309–1338 (2013)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Onsager, L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426 (1931)ADSzbMATHGoogle Scholar
  72. 72.
    Onsager, L.: Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265–2279 (1931)ADSzbMATHGoogle Scholar
  73. 73.
    Pego, R.: Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. A422, 261–278 (1989)ADSMathSciNetzbMATHGoogle Scholar
  74. 74.
    Prüss, J., Racke, R., Zheng, S.: Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions. Annali di Matematica Pura ed Applicata185(4), 627–648 (2006)MathSciNetzbMATHGoogle Scholar
  75. 75.
    Qian, T.Z., Qiu, C.Y., Sheng, P.: A scaling approach to the derivation of hydrodynamic boundary conditions. J. Fluid Mech. 611, 333–364 (2008)ADSMathSciNetzbMATHGoogle Scholar
  76. 76.
    Qian, T.Z., Wang, X.P., Sheng, P.: A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360 (2006)ADSMathSciNetzbMATHGoogle Scholar
  77. 77.
    Racke, R., Zheng, S.: The Cahn-Hilliard equation with dynamical boundary conditions. Adv. Differ. Eqs. 8(1), 83–110 (2003)zbMATHGoogle Scholar
  78. 78.
    Rätz, A., Voigt, A.: PDE's on surfaces–a diffuse interface approach. Commun. Math. Sci. 4(3), 575–590 (2006)MathSciNetzbMATHGoogle Scholar
  79. 79.
    Rayleigh, L., Strutt, J.W.: Some general theorems relating to vibrations. Proc. Lond. Math. Soc. 4, 357–368 (1873)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Rybka, P., Hoffmann, K.H.: Convergence of solutions to Cahn-Hillard equation. Commun. Partial Differ. Equ. 24(5–6), 1055–1077 (1999)zbMATHGoogle Scholar
  81. 81.
    Simon, J.: Compact sets in the space \(L^p(0, T; B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Simon, L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math. 118(3), 525–571 (1983)MathSciNetzbMATHGoogle Scholar
  83. 83.
    Sonnet, A.M., Virga, E.G.: Dissipative Ordered Fluids Theories for Liquid Crystals. Springer, New York (2012)zbMATHGoogle Scholar
  84. 84.
    Thompson, P.A., Robbins, M.O.: Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63, 766–769 (1989)ADSGoogle Scholar
  85. 85.
    Wei, J., Winter, M.: Stationary solutions for the Cahn–Hilliard equation.Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 459–492, 1998Google Scholar
  86. 86.
    Wu, H.: Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition. Asymptot. Anal. 54, 71–92 (2007)MathSciNetzbMATHGoogle Scholar
  87. 87.
    Wu, H., Xu, X., Liu, C.: On the general Ericksen-Leslie system: Parodi's relation, well-posedness and stability. Arch. Ration. Mech. Anal. 208(1), 59–107 (2013)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Wu, H., Zheng, S.: Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary condition. J. Differ. Equ. 204, 511–531 (2004)ADSMathSciNetzbMATHGoogle Scholar
  89. 89.
    Xu, S.X., Sheng, P., Liu, C.: An energetic variational approach for ion transport. Commun. Math. Sci. 12(4), 779–789 (2014)MathSciNetzbMATHGoogle Scholar
  90. 90.
    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(4), 939–962 (2009)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Zheng, S.: Asymptotic behavior of solution to the Cahn-Hillard equation. Appl. Anal. 23(3), 165–184 (1986)MathSciNetzbMATHGoogle Scholar
  92. 92.
    Zheng, S.: Nonlinear Evolution Equations, Pitman Series Monographs and Survey in Pure and Applied Mathematics, vol. 133. Chapman & Hall/CRC, Boca Raton (2004)Google Scholar

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Authors and Affiliations

  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina
  3. 3.Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University)Ministry of EducationShanghaiChina
  4. 4.Shanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghaiChina

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