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, 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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