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A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains
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 |
References:
[1] |
A. Babin and M. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (1992).
|
[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.
|
[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).
|
[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. |
[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.
|
[7] |
C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, \emph{Mathematical Models for Phase Change Problems}, (1989), 35.
|
[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,, \emph{J. Differential Equations}, 73 (1988), 309.
doi: 10.1016/0022-0396(88)90110-6. |
[10] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[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.
|
[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. |
[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. |
[15] |
M. Miklavcic, A sharp condition for existence of an inertial manifold,, \emph{J. Dynam. Differential Equations}, 3 (1991), 437.
doi: 10.1007/BF01049741. |
[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. |
[17] |
A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, \emph{Adv. Math. Sci. Appl.}, 8 (1998), 965.
|
[18] |
J. Robinson, Dimensions, Embeddings, and Attractors,, Cambridge Tracts in Mathematics, (2011).
|
[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. |
[20] |
A. Romanov, Finite-dimensionality of dynamics on an attractor for nonlinear parabolic equations,, \emph{Izv. Math.}, 65 (2001), 977.
doi: 10.1070/IM2001v065n05ABEH000359. |
[21] |
A. Romanov, Finite-dimensional limit dynamics of dissipative parabolic equations,, \emph{Sb. Math.}, 191 (2000), 415.
doi: 10.1070/SM2000v191n03ABEH000466. |
[22] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997).
doi: 10.1007/978-1-4612-0645-3. |
[23] |
S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs,, \emph{Proc. Royal Soc. Edinburgh}, 144A (2014), 1245.
doi: 10.1017/S0308210513000073. |
[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, (1992).
|
[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.
|
[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).
|
[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. |
[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.
|
[7] |
C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, \emph{Mathematical Models for Phase Change Problems}, (1989), 35.
|
[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,, \emph{J. Differential Equations}, 73 (1988), 309.
doi: 10.1016/0022-0396(88)90110-6. |
[10] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[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.
|
[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. |
[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. |
[15] |
M. Miklavcic, A sharp condition for existence of an inertial manifold,, \emph{J. Dynam. Differential Equations}, 3 (1991), 437.
doi: 10.1007/BF01049741. |
[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. |
[17] |
A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, \emph{Adv. Math. Sci. Appl.}, 8 (1998), 965.
|
[18] |
J. Robinson, Dimensions, Embeddings, and Attractors,, Cambridge Tracts in Mathematics, (2011).
|
[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. |
[20] |
A. Romanov, Finite-dimensionality of dynamics on an attractor for nonlinear parabolic equations,, \emph{Izv. Math.}, 65 (2001), 977.
doi: 10.1070/IM2001v065n05ABEH000359. |
[21] |
A. Romanov, Finite-dimensional limit dynamics of dissipative parabolic equations,, \emph{Sb. Math.}, 191 (2000), 415.
doi: 10.1070/SM2000v191n03ABEH000466. |
[22] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997).
doi: 10.1007/978-1-4612-0645-3. |
[23] |
S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs,, \emph{Proc. Royal Soc. Edinburgh}, 144A (2014), 1245.
doi: 10.1017/S0308210513000073. |
[24] |
S. Zelik, Inertial Manifolds for 1D convective reaction-diffusion equations,, submitted., (). Google Scholar |
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