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

Anna  Kostianko; Sergey Zelik

doi:10.3934/cpaa.2015.14.2069

Article Contents

2015, Volume 14, Issue 5: 2069-2094. Doi: 10.3934/cpaa.2015.14.2069

This issue Previous Article A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains Next Article A nonlocal diffusion population model with age structure and Dirichlet boundary condition

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

Received: September 30, 2014

Revised: January 31, 2015

Published: August 2015

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 35B40, 35B42.

Citation:

Related Papers

Cited by

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, Indiana Univ. Math. J., 21 (1971/1972), 193-226.
[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.