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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

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