Comptes Rendus
no. 11
A sufficient condition for the existence of approximate inertial manifolds containing the global attractor
[Une condition suffisante pour l'existence de variétés inertielles approchées contenant l'attracteur global]
Alexander Rezounenko1
1 Department of Mechanics and Mathematics, Kharkov University, 4 Svobody square, Kharkov 61077, Ukraine
Comptes Rendus. Mathématique, Volume 334 (2002) no. 11, pp. 1015-1020.

Utilisant le récent concept de variété inertielle avec retard, nous présentons une nouvelle méthode de construction de variétés inertielles approchées. Dans le cas où l'attracteur global peut être plongé dans une variété C2 de dimension finie, nous construisons une variété inertielle approchée de la même dimension contenant l'attracteur.

Using the recently introduced concept of inertial manifold with delay we present a new method of construction of approximate inertial manifolds (AIMs). In the case when the global attractor can be embedded into a finite-dimensional C2-manifold we construct AIMs of the same dimension which contain the attractor.

Reçu le :
Révisé le :
Publié le :
DOI : 10.1016/S1631-073X(02)02385-3
Alexander Rezounenko 1

1 Department of Mechanics and Mathematics, Kharkov University, 4 Svobody square, Kharkov 61077, Ukraine 
@article{CRMATH_2002__334_11_1015_0,
     author = {Alexander Rezounenko},
     title = {A sufficient condition for the existence of approximate inertial manifolds containing the global attractor},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1015--1020},
     publisher = {Elsevier},
     volume = {334},
     number = {11},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02385-3},
     language = {en},
}
TY  - JOUR
AU  - Alexander Rezounenko
TI  - A sufficient condition for the existence of approximate inertial manifolds containing the global attractor
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 1015
EP  - 1020
VL  - 334
IS  - 11
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02385-3
LA  - en
ID  - CRMATH_2002__334_11_1015_0
ER  - 
%0 Journal Article
%A Alexander Rezounenko
%T A sufficient condition for the existence of approximate inertial manifolds containing the global attractor
%J Comptes Rendus. Mathématique
%D 2002
%P 1015-1020
%V 334
%N 11
%I Elsevier
%R 10.1016/S1631-073X(02)02385-3
%G en
%F CRMATH_2002__334_11_1015_0
Alexander Rezounenko. A sufficient condition for the existence of approximate inertial manifolds containing the global attractor. Comptes Rendus. Mathématique, Volume 334 (2002) no. 11, pp. 1015-1020. doi : 10.1016/S1631-073X(02)02385-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02385-3/

[1] A.V. Babin; M.I. Vishik Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992

[2] S.-N. Chow; K. Lu Invariant manifolds for flows in Banach spaces, J. Differential Equations, Volume 74 (1988), pp. 285-317

[3] I.D. Chueshov Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999 (in Russian)

[4] I.D. Chueshov; A.V. Rezounenko Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Série I., Volume 321 (1995), pp. 607-612

[5] I.D. Chueshov Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, J. Dynamic Differential Equations, Volume 7 (1995) no. 4, pp. 549-566

[6] I.D. Chueshov Global attractors for nonlinear problems of mathematical physics, Russian Math. Surveys, Volume 48 (1993) no. 3, pp. 133-161

[7] P. Constantin; C. Foias; B. Nicolaenko; R. Temam Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer, Berlin, 1989

[8] A. Debussche; R. Temam Convergent families of approximate inertial manifolds, J. Math. Pure Appl., Volume 73 (1994), pp. 489-522

[9] A. Debussche; R. Temam Some new generalizations of inertial manifolds, Discr. Contin. Dynamical Systems, Volume 2 (1996), pp. 543-558

[10] A. Eden; C. Foias; B. Nicolaenko; R. Temam Exponential Attractors for Dissipative Evolution Equations, Collection Recherches au Mathematiques Appliquees, Masson, Paris, 1994

[11] C. Foias; O. Manley; R. Temam Sur l'interaction des petits et grands tourbillons dans les ecoulements turbulents, C. R. Acad. Sci. Paris, Série I, Volume 305 (1987), pp. 497-500

[12] C. Foias; E. Olson Finite fractal dimension and Hölder–Lipschtz parametrization, Indiana Univ. Math. J., Volume 45 (1996) no. 3, pp. 603-616

[13] C. Foias; G. Sell; R. Temam Variétés Inertielles des équations différentielles dissipatives, C. R. Acad. Sci. Paris, Série I, Volume 301 (1985), pp. 139-142

[14] C. Foias; G. Sell; E. Titi Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dynamics Differential Equations, Volume 1 (1989), pp. 199-224

[15] J.K. Hale Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988

[16] D. Henry Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981

[17] A.V. Rezounenko, Steady approximate inertial manifolds of exponential order for semilinear parabolic equations, Differential and Integral Equations (accepted)

[18] A.V. Rezounenko Inertial manifolds with delay for retarded semilinear parabolic equations, Discr. Contin. Dynamical Systems, Volume 6 (2000), pp. 829-840

[19] J. Robinson Inertial manifolds with and without delay, Discr. Contin. Dynamical Systems, Volume 5 (1999), pp. 813-824

[20] A.V. Romanov Finite-dimensional limiting dynamics of dissipative parabolic equations, Sb. Math., Volume 191 (2000) no. 3, pp. 415-429 Translation from Mat. Sb. 191 (3) (2000) 99–112

[21] E.M. Stein Singular Integrals and Differentiality Properties of Functions, Princeton University Press, Princeton, 1970

[22] R. Temam Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988

Cité par Sources :

☆ For my parents Vyacheslav and Larisa.

Commentaires - Politique

Ces articles pourraient vous intéresser

Non-local PDEs with a state-dependent delay term presented by Stieltjes integral

Alexander V. Rezounenko

C. R. Math (2011)

A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case

Ciprian Foias; Ricardo M.S. Rosa; Roger Temam

C. R. Math (2010)

Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains

Tomás Caraballo; Grzegorz Łukaszewicz; José Real

C. R. Math (2006)