Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-12T22:23:03.082Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-local Cahn–Hilliard equations with fractional dynamic boundary conditions

Published online by Cambridge University Press:  01 December 2016

CIPRIAN G. GAL
CIPRIAN G. GAL*
Affiliation:
Department of Mathematics & Statistics, Florida International University, Miami, FL 33199, USA email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a non-local version of the Cahn–Hilliard equation characterized by the presence of a fractional diffusion operator, and which is subject to fractional dynamic boundary conditions. Our system generalizes the classical system in which the dynamic boundary condition was used to describe any relaxation dynamics of the order-parameter at the walls. The proposed fractional dynamic boundary condition appears to be more general in the sense that it incorporates non-local effects which were completely ignored in the classical approach. We aim to deduce well-posedness and regularity results as well as to establish the existence of finite-dimensional attractors for this system.

Keywords

Non-local Cahn–Hilliard equationfractional dynamic boundary conditionsregional fractional Laplacianfractional Wentzell Laplacianwell-posednessregularityglobal attractor
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 5 , October 2017 , pp. 736 - 788
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abels, H., Bosia, S. & Grasselli, M. (2015) Cahn-Hilliard equation with nonlocal singular free energies. Ann. Mat. Pura Appl. 194 (4), 1071–1106.CrossRefGoogle Scholar
[2] Bogdan, K., Burdzy, K. & Chen, Z. Q. (2003) Censored stable processes. Probab. Theory Rel. Fields 127, 89–152.CrossRefGoogle Scholar
[3] Bates, P. W. & Han, J. (2005) The Neumann boundary problem for a nonlocal Cahn-Hilliard equation. J. Differ. Equ. 212, 235–277.CrossRefGoogle Scholar
[4] Bates, P. W. & Han, J. (2005) The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation. J. Math. Anal. Appl. 311, 289–312.CrossRefGoogle Scholar
[5] Binder, K. & Frisch, H. L. (1991) 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.CrossRefGoogle Scholar
[6] Cahn, J. W. & Hilliard, J. E. (1958) Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
[7] Cherfils, L., Miranville, A. & Zelik, S. (2011) The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596.CrossRefGoogle Scholar
[8] Colli, P., Frigeri, S. & Grasselli, M. (2012) Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system. J. Math. Anal. Appl. 386, 428–444.CrossRefGoogle Scholar
[9] Deimling, K. (1984) Nonlinear Functional Analysis, Berlin, Heidelberg, New York, Tokyo, Springer-Verlag.Google Scholar
[10] Fischer, M. E. (1974) The renormalization group in the theory of critical behavior. Rev. Mod. Phys. 46, 597–616.CrossRefGoogle Scholar
[11] Frigeri, S. & Grasselli, M. (2012) Global and trajectory attractors for a nonlocal Cahn-Hilliard system. J. Dyn. Diff. Eqns. 24, 827–856.CrossRefGoogle Scholar
[12] Frigeri, S. & Grasselli, M. (2012) Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials. Dyn. Partial Differ. Equ. 9, 273–304.CrossRefGoogle Scholar
[13] Fischer, H. P., Reinhard, J., Dieterich, W., Gouyet, J.-F., Maass, P., Majhofer, A. & Reinel, D. (1998) Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall. J. Chem. Phys. 108, 3028–3037.CrossRefGoogle Scholar
[14] Gajewski, H. & Zacharias, K. (2003) On a nonlocal phase separation model. J. Math. Anal. Appl. 286, 11–31.CrossRefGoogle Scholar
[15] Gal, C. G. (2017) On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations. Discrete Continuous Dyn. Syst. A 37, 1–37.Google Scholar
[16] Gal, C. G. & Grasselli, M. (2014) Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 34 (1), 145–179.CrossRefGoogle Scholar
[17] Gal, C. G. & Grasselli, M. (2013) Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete Contin. Dyn. Syst. Ser. B 18 (6), 1581–1610.Google Scholar
[18] Gal, C. G. & Warma, M. (2016) Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete Contin. Dyn. Syst. 36, 1279–1319.CrossRefGoogle Scholar
[19] Gal, C. G. & Warma, M. (2016) Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evol. Equ. Control Theory 5, 61–103.Google Scholar
[20] Gal, C. G. & Warma, M. (2016) Bounded solutions for nonlocal boundary valued problems on Lipschitz manifolds with boundary. Adv. Nonlinear Stud. 16, 529–550.CrossRefGoogle Scholar
[21] Giacomin, G. & Lebowitz, J. L. (1997) Phase segregation dynamics in particle systems with long range interactions, I. Macroscopic limits. J. Statist. Phys. 87, 37–61.CrossRefGoogle Scholar
[22] Guan, Q. Y. (2006) Integration by parts formula for regional fractional Laplacian. Comm. Math. Phys. 266, 289–329.CrossRefGoogle Scholar
[23] Guan, Q. Y. & Ma, Z. M. (2006) Reflected symmetric α-stable processes and regional fractional Laplacian. Probab. Theory Rel. Fields 134, 649–694.CrossRefGoogle Scholar
[24] Haasen, P. (1978) Physical Metallurgy, Cambridge University Press, Cambridge.Google Scholar
[25] Khachaturyan, A. G. (1983) Theory of Structural Transformations in Solids, Wiley-Interscience Publications, New York.Google Scholar
[26] Londen, S.-O. & Petzeltová, H. (2011) Convergence of solutions of a non-local phase-field system. Discrete Contin. Dyn. Syst. Ser. S 4, 653–670.Google Scholar
[27] Londen, S.-O. & Petzeltová, H. (2011) Regularity and separation from potential barriers for a non-local phase-field system. J. Math. Anal. Appl. 379, 724–735.CrossRefGoogle Scholar
[28] Miranville, A. & Zelik, S. (2008) Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Dafermos, C. M. & Pokorný, M. (editors), Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Vol. 4, Elsevier/North-Holland, Amsterdam, pp. 103–200.CrossRefGoogle Scholar
[29] Novick-Cohen, A. (2008) The Cahn-Hilliard equation. In: Dafermos, C. M. & Pokorný, M. (editors) Evolutionary Equations, Handb. Differ. Equ., Vol. 4, Elsevier/North-Holland, Amsterdam, pp. 201–228.Google Scholar
[30] Di Nezza, E., Palatucci, G. & Valdinoci, E. (2012) Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573.CrossRefGoogle Scholar
[31] Warma, M. (2015) The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42, 499–547.CrossRefGoogle Scholar