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

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.
Citation: Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure and 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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[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.

[23]

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

[24]

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

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[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.

[23]

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

[24]

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

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