# American Institute of Mathematical Sciences

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 & Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069
##### References:
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##### References:
 [1] A. Babin and M. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (1992). Google Scholar [2] A. Bonfoh, M. Grasselli and A. Miranville, Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation,, \emph{Topol. Methods Nonlinear Anal.}, 35 (2010), 155. Google Scholar [3] J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, \emph{J. Chem. Phys.}, 28 (1958), 258. Google Scholar [4] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar [5] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, \emph{Milan J. Math.}, 79 (2011), 561. 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,, \emph{Russian Math. Surveys}, 68 (2013), 199. Google Scholar [7] C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, \emph{Mathematical Models for Phase Change Problems}, (1989), 35. 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,, \emph{J. Differential Equations}, 73 (1988), 309. doi: 10.1016/0022-0396(88)90110-6. Google Scholar [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar [11] N. Koksch, Almost sharp conditions for the existence of smooth inertial manifolds,, in \emph{Equadiff 9: Conference on Differential Equations and their Applications: Proceedings (Z. Dosla, (1998), 139. Google Scholar [12] H. Kwean, An extension of the principle of spatial averaging for inertial manifolds,, \emph{J. Austral. Math. Soc. (Series A)}, 66 (1999), 125. Google Scholar [13] J. Mallet-Paret and G. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions,, \emph{J. Amer. Math. Soc.}, 1 (1988), 805. doi: 10.2307/1990993. Google Scholar [14] J. Mallet-Paret, G. Sell and Z. Shao, Obstructions to the existence of normally hyperbolic inertial manifolds,, \emph{Indiana Univ. Math. J.}, 42 (1993), 1027. doi: 10.1512/iumj.1993.42.42048. Google Scholar [15] M. Miklavcic, A sharp condition for existence of an inertial manifold,, \emph{J. Dynam. Differential Equations}, 3 (1991), 437. doi: 10.1007/BF01049741. Google Scholar [16] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, In \emph{Handbook of Differential Equations: Evolutionary Equations}, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [17] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, \emph{Adv. Math. Sci. Appl.}, 8 (1998), 965. Google Scholar [18] J. Robinson, Dimensions, Embeddings, and Attractors,, Cambridge Tracts in Mathematics, (2011). Google Scholar [19] A. Romanov, Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations,, \emph{Russian Acad. Sci. Izv. Math.}, 43 (1994), 31. doi: 10.1070/IM1994v043n01ABEH001557. Google Scholar [20] A. Romanov, Finite-dimensionality of dynamics on an attractor for nonlinear parabolic equations,, \emph{Izv. Math.}, 65 (2001), 977. doi: 10.1070/IM2001v065n05ABEH000359. Google Scholar [21] A. Romanov, Finite-dimensional limit dynamics of dissipative parabolic equations,, \emph{Sb. Math.}, 191 (2000), 415. doi: 10.1070/SM2000v191n03ABEH000466. Google Scholar [22] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [23] S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs,, \emph{Proc. Royal Soc. Edinburgh}, 144A (2014), 1245. doi: 10.1017/S0308210513000073. Google Scholar [24] S. Zelik, Inertial Manifolds for 1D convective reaction-diffusion equations,, submitted., (). Google Scholar
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