American Institute of Mathematical Sciences

Advanced
September  2015, 14(5): 2069-2094. doi: 10.3934/cpaa.2015.14.2069

Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions

Anna Kostianko 1, and  Sergey Zelik 2,

1. 

University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom
2. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  October 2014 Revised  February 2015 Published  June 2015

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.
Keywords: inertial manifold., spatial averaging principle, Cahn-Hilliard equation.
Mathematics Subject Classification: 35B40, 35B4.
Citation: Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069
References:
[1]

A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

A. Bonfoh, M. Grasselli and A. Miranville, Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Topol. Methods Nonlinear Anal., 35 (2010), 155-185.  Google Scholar

[3]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. Google Scholar

[4]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[5]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4.  Google Scholar

[6]

A. Eden, V. Kalantarov and S. Zelik, Counterexamples to the Regularity of Mané Projections in the Attractors Theory, Russian Math. Surveys, 68 (2013), 199-226.  Google Scholar

[7]

C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, Mathematical Models for Phase Change Problems, 35-73, Internat. Ser. Numer. Math., 88, Birkhauser, Basel, 1989.  Google Scholar

[8]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, \emph{Indiana Univ. Math. J.}, 21 (): 193.   Google Scholar

[9]

C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[11]

N. Koksch, Almost sharp conditions for the existence of smooth inertial manifolds, in Equadiff 9: Conference on Differential Equations and their Applications: Proceedings (Z. Dosla, J. Kuben and J. Vosmansky eds.), Masaryk University, Brno, 1998, 139-166. Google Scholar

[12]

H. Kwean, An extension of the principle of spatial averaging for inertial manifolds, J. Austral. Math. Soc. (Series A), 66 (1999), 125-142.  Google Scholar

[13]

J. Mallet-Paret and G. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1 (1988), 805-866. doi: 10.2307/1990993.  Google Scholar

[14]

J. Mallet-Paret, G. Sell and Z. Shao, Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055. doi: 10.1512/iumj.1993.42.42048.  Google Scholar

[15]

M. Miklavcic, A sharp condition for existence of an inertial manifold, J. Dynam. Differential Equations, 3 (1991), 437-456. doi: 10.1007/BF01049741.  Google Scholar

[16]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In Handbook of Differential Equations: Evolutionary Equations, Vol. IV, 103-200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[17]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.  Google Scholar

[18]

J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.  Google Scholar

[19]

A. Romanov, Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations, Russian Acad. Sci. Izv. Math., 43 (1994), 31-47. doi: 10.1070/IM1994v043n01ABEH001557.  Google Scholar

[20]

A. Romanov, Finite-dimensionality of dynamics on an attractor for nonlinear parabolic equations, Izv. Math., 65 (2001), 977-1001. doi: 10.1070/IM2001v065n05ABEH000359.  Google Scholar

[21]

A. Romanov, Finite-dimensional limit dynamics of dissipative parabolic equations, Sb. Math., 191 (2000), 415-429. doi: 10.1070/SM2000v191n03ABEH000466.  Google Scholar

[22]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[23]

S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proc. Royal Soc. Edinburgh, 144A (2014), 1245-1327. doi: 10.1017/S0308210513000073.  Google Scholar

[24]

S. Zelik, Inertial Manifolds for 1D convective reaction-diffusion equations,, submitted., ().   Google Scholar

show all references

References:
[1]

A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

A. Bonfoh, M. Grasselli and A. Miranville, Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Topol. Methods Nonlinear Anal., 35 (2010), 155-185.  Google Scholar

[3]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. Google Scholar

[4]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[5]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4.  Google Scholar

[6]

A. Eden, V. Kalantarov and S. Zelik, Counterexamples to the Regularity of Mané Projections in the Attractors Theory, Russian Math. Surveys, 68 (2013), 199-226.  Google Scholar

[7]

C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, Mathematical Models for Phase Change Problems, 35-73, Internat. Ser. Numer. Math., 88, Birkhauser, Basel, 1989.  Google Scholar

[8]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, \emph{Indiana Univ. Math. J.}, 21 (): 193.   Google Scholar

[9]

C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[11]

N. Koksch, Almost sharp conditions for the existence of smooth inertial manifolds, in Equadiff 9: Conference on Differential Equations and their Applications: Proceedings (Z. Dosla, J. Kuben and J. Vosmansky eds.), Masaryk University, Brno, 1998, 139-166. Google Scholar

[12]

H. Kwean, An extension of the principle of spatial averaging for inertial manifolds, J. Austral. Math. Soc. (Series A), 66 (1999), 125-142.  Google Scholar

[13]

J. Mallet-Paret and G. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1 (1988), 805-866. doi: 10.2307/1990993.  Google Scholar

[14]

J. Mallet-Paret, G. Sell and Z. Shao, Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055. doi: 10.1512/iumj.1993.42.42048.  Google Scholar

[15]

M. Miklavcic, A sharp condition for existence of an inertial manifold, J. Dynam. Differential Equations, 3 (1991), 437-456. doi: 10.1007/BF01049741.  Google Scholar

[16]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In Handbook of Differential Equations: Evolutionary Equations, Vol. IV, 103-200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[17]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.  Google Scholar

[18]

J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.  Google Scholar

[19]

A. Romanov, Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations, Russian Acad. Sci. Izv. Math., 43 (1994), 31-47. doi: 10.1070/IM1994v043n01ABEH001557.  Google Scholar

[20]

A. Romanov, Finite-dimensionality of dynamics on an attractor for nonlinear parabolic equations, Izv. Math., 65 (2001), 977-1001. doi: 10.1070/IM2001v065n05ABEH000359.  Google Scholar

[21]

A. Romanov, Finite-dimensional limit dynamics of dissipative parabolic equations, Sb. Math., 191 (2000), 415-429. doi: 10.1070/SM2000v191n03ABEH000466.  Google Scholar

[22]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[23]

S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proc. Royal Soc. Edinburgh, 144A (2014), 1245-1327. doi: 10.1017/S0308210513000073.  Google Scholar

[24]

S. Zelik, Inertial Manifolds for 1D convective reaction-diffusion equations,, submitted., ().   Google Scholar

[1]

Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations & Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315
[2]

Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855
[3]

Maurizio Grasselli, Nicolas Lecoq, Morgan Pierre. A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term. Conference Publications, 2011, 2011 (Special) : 543-552. doi: 10.3934/proc.2011.2011.543
[4]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[5]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[6]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[7]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[8]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303
[9]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[10]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[11]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[12]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[13]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[14]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[15]

Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure & Applied Analysis, 2021, 20 (2) : 495-510. doi: 10.3934/cpaa.2020277
[16]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[17]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[18]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[19]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[20]

Nguyen Huy Tuan. Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021113

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences