Annali di Matematica Pura ed Applicata (1923 -)

, Volume 194, Issue 4, pp 1071–1106 | Cite as

Cahn–Hilliard equation with nonlocal singular free energies

  • Helmut Abels
  • Stefano Bosia
  • Maurizio GrasselliEmail author


We consider a Cahn–Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is characterized by a Helmholtz free energy density, which can be of logarithmic type. Moreover, the spatial interactions between the different phases are modeled by a singular kernel. As a consequence, the chemical potential \(\mu \) contains an integral operator acting on the concentration difference \(c\), instead of the usual Laplace operator. We analyze the equation on a bounded domain subject to no-flux boundary condition for \(\mu \) and by assuming constant mobility. We first establish the existence and uniqueness of a weak solution and some regularity properties. These results allow us to define a dissipative dynamical system on a suitable phase–space, and we prove that such a system has a (connected) global attractor. Finally, we show that a Neumann-like boundary condition can be recovered for \(c\), provided that it is supposed to be regular enough.


Cahn–Hilliard equation Nonlocal free energy Regional fractional Laplacian Logarithmic potential Monotone operators Global attractors 

Mathematical Subject Classification (2010)

35B41 37L99 45K05 47H05 47J35 80A22 


  1. 1.
    Abels, H., Kassmann, M.: An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control. J. Differ. Eqs. 236, 29–56 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Abels, H., Wilke, M.: Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67, 3176–3193 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  4. 4.
    Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations. J. Nonlinear Sci. 7(5), 475–502 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ball, J.M.: Erratum: “Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations”. J. Nonlinear Sci. 8(2), 233 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bates, P.W., Han, J.: The Dirichlet boundary problem for a nonlocal Cahn–Hilliard equation. J. Math. Anal. Appl. 311, 289–312 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bates, P.W., Han, J.: The Neumann boundary problem for a nonlocal Cahn–Hilliard equation. J. Differ. Eqs. 212, 235–277 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Probab. Theory Rel. Fields 127, 89–152 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50)Google Scholar
  11. 11.
    Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)CrossRefGoogle Scholar
  12. 12.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial Energy. J. Chem. Phys. 28, 258–267 (1958)Google Scholar
  13. 13.
    Cherfils, L., Miranville, A., Zelik, S.: The Cahn–Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Colli, P., Krejčí, P., Rocca, E., Sprekels, J.: Nonlinear evolution inclusions arising from phase change models. Czechoslov. Math. J. 57, 1067–1098 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Debussche, A., Dettori, L.: On the Cahn–Hilliard equation with a logarithmic free energy. Nonlinear Anal. 24(10), 1491–1514 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Elliott, C.M., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Elliott, C.M., Zheng, S.: On the Cahn–Hilliard equation. Arch. Rational Mech. Anal. 96, 339–357 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fife, P.C.: Models for phase separation and their mathematics. Electron. J. Differ. Eqs. pages No. 48, 26 pp. (electronic) (2000)Google Scholar
  19. 19.
    Folland, G.B.: Real Analysis. Pure and Applied Mathematics (New York), 2nd edition. Wiley, New York (1999). Modern Techniques and Their Applications, A Wiley-Interscience PublicationGoogle Scholar
  20. 20.
    Gajewski, H., Zacharias, K.: On a nonlocal phase separation model. J. Math. Anal. Appl. 286, 11–31 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gal, C.G., Grasselli, M.: Longtime behavior of nonlocal Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. Ser. A 34, 145–179 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. Limits. J. Stat. Phys. 87, 37–61 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. II. Interface motion. SIAM J. Appl. Math. 58, 1707–1729 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Londen, S.-O., Petzeltová, H.: Convergence of solutions of a non-local phase–field system. Discret. Contin. Dyn. Syst. Ser. S 4, 653–670 (2011)zbMATHGoogle Scholar
  25. 25.
    Londen, S.-O., Petzeltová, H.: Regularity and separation from potential barriers for a non-local phase–field system. J. Math. Anal. Appl. 379, 724–735 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Nec, A.A., Nepomnyashchy, Y., Golovin, A.A.: Front-type solutions of fractional Allen–Cahn equation. Phys. D 237, 3237–3251 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of a class of pattern formation equations. Comm. Part. Differ. Eqs. 14, 245–297 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Novick-Cohen, A.: The Cahn–Hilliard equation: mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8, 965–985 (1998)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Novick-Cohen, A.: The Cahn–Hilliard equation. In: Handbook of Differential Equations: Evolutionary Equations. Vol. IV. Handb. Differ. Equ., pp. 201–228. Elsevier/North-Holland, Amsterdam (2008)Google Scholar
  30. 30.
    Pata, V., Zelik, S.: A result on the existence of global attractors for semigroups of closed operators. Commun. Pure Appl. Anal. 6, 481–486 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Volume 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1997)Google Scholar
  32. 32.
    Simon, J.: Sobolev, Besov and Nikol’skiĭ fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 4(157), 117–148 (1990)CrossRefGoogle Scholar
  33. 33.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A. Springer, New York (1990)zbMATHCrossRefGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Helmut Abels
    • 1
  • Stefano Bosia
    • 2
  • Maurizio Grasselli
    • 2
    Email author
  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Politecnico di MilanoDipartimento di MatematicaMilanItaly

Personalised recommendations