Nous construisons une variété inertielle avec retard (IMD) pour les systèmes dissipatifs du second ordre en temps. Ce résultat est appliqué à l'étude des propriétés asymptotiques des solutions. En utilisant cette IMD, nous construisons les variétés inertielles approchées contenant toutes les solutions stationnaires et donnons une nouvelle caractérisation de la variété K-invariante.
Inertial manifold with delay (IMD) for dissipative systems of second order in time is constructed. This result is applied to the study of different asymptotic properties of solutions. Using IMD, we construct approximate inertial manifolds containing all the stationary solutions and give a new characterization of the K-invariant manifold.
@article{AIF_2004__54_5_1547_0,
author = {Rezounenko, Alexander V.},
title = {Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay},
journal = {Annales de l'Institut Fourier},
volume = {54},
year = {2004},
pages = {1547-1564},
doi = {10.5802/aif.2058},
mrnumber = {2127857},
zbl = {1080.35168},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2004__54_5_1547_0}
}
Rezounenko, Alexander V. Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay. Annales de l'Institut Fourier, Tome 54 (2004) pp. 1547-1564. doi : 10.5802/aif.2058. http://gdmltest.u-ga.fr/item/AIF_2004__54_5_1547_0/
[1] , & , Variétés Inertielles des équations différentielles dissipatives, C.R. Acad. Sci. Paris, Série I 301 (1985) p. 139-141 | MR 801946 | Zbl 0591.35062
[2] & , Invariant manifolds for flows in Banach spaces, J. Diff. Eqns 74 (1988) p. 285-317 | MR 952900 | Zbl 0691.58034
[3] , & , Sur l'interaction des petits et grands tourbillons dans les écoulements turbulents, C.R. Acad. Sci. Paris, Série I 305 (1987) p. 497-500 | MR 916319 | Zbl 0624.76072
[4] , Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967) p. 1-34 | Numdam | MR 223716 | Zbl 0176.54103
[5] , & , Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dyn. Diff. Eqns 1 (1989) p. 199-224 | MR 1010966 | Zbl 0692.35053
[6] , Introduction to the Theory of Infinite-Dimensional Dissipative Systems (in Russian), 2002 | Zbl 1100.37046
[7] , Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, J. Dyn. Diff. Eqns 7 (1995) no.4 p. 549-566 | MR 1362670 | Zbl 0840.60054
[8] & , Inertial manifolds and their dimension, ed. S.I. Andersson, A.E. Anderson and O. Ottson, Dynamical Systems, Theory and Applications, World Scientific, 1993 | MR 1386913
[9] , , & , Integral manifolds and inertial manifolds for dissipative partial differential equations, Springer, 1989 | MR 966192 | Zbl 0683.58002
[10] , Infinite dimensional dynamical systems in mechanics and physics, Springer, 1988 | MR 953967 | Zbl 0662.35001
[11] , & , Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis 34 (1998) p. 907-925 | MR 1636608 | Zbl 0954.34064
[12] , Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Res. Notes in Math 155 (1987) p. 172-183 | MR 907731 | Zbl 0642.35061
[13] & , Some second-order vibrating systems cannot tolerate small delays in their damping, J. Optim. Theory. Appl 70 (1991) no.3 p. 521-537 | MR 1124776 | Zbl 0791.34045
[14] & , Some new generalizations of inertial manifolds, Discr. Contin. Dynamical Systems 2 (1996) p. 543-558 | MR 1414085 | Zbl 0948.35018
[15] , Inertial manifolds with and without delay, Discr. Contin. Dynamical Systems 5 (1999) p. 813-824 | MR 1722373 | Zbl 0954.37037
[16] & , Invariant manifolds for retarded semilinear wave equations, J. Diff. Eqns 114 (1994) p. 337-369 | MR 1303032 | Zbl 0815.34067
[17] , Theory of functional differential equations, Springer, 1977 | MR 508721 | Zbl 0352.34001
[18] , Asymptotic behavior of dissipative systems, Amer. Math. Soc., 1988 | MR 941371 | Zbl 0642.58013
[19] & , Existence and stability for partial functional differential equations, Transactions of AMS 200 (1974) p. 395-418 | MR 382808 | Zbl 0299.35085
[20] , , & , Delay equations. Functional, complex, and nonlinear analysis, Applied Mathematical Sciences 110, Springer-Verlag, 1995 | MR 1345150 | Zbl 0826.34002
[21] , Theory and applications of partial functional-differential equation, Applied Mathematical Sciences 119, Springer-Verlag, 1996 | Zbl 0870.35116
[22] , On a certain system of equations with delay, occurring in aeroelasticity, J. of Soviet Mathematics 58 (1992) p. 385-390 | Zbl 0783.73046
[23] & , Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, série I 321 (1995) p. 607-612 | MR 1356562 | Zbl 0845.35129
[24] , & , Long-time behaviour of strong solutions for a class of retarded nonlinear P.D.E.s, Commun. in Partial Differential Equations 22 (1997) no.9-10 p. 1453-1474 | MR 1469578 | Zbl 0891.35159
[25] , Inertial manifolds with delay for retarded semilinear parabolic equations, Discr. Contin. Dynamical Systems 6 (2000) p. 829-840 | MR 1788255 | Zbl 1011.37046
[26] , Inertial manifolds for retarded second order in time evolution equations, Nonlinear Analysis 51 (2002) p. 1045-1054 | MR 1926084 | Zbl 1023.35087
[27] , Steady approximate inertial manifolds of exponential order for semilinear parabolic equations, Differential and Integral Equations 15 (2002) no.11 p. 1345-1356 | MR 1920691 | Zbl 01839894
[28] , Approximate inertial manifolds for retarded semilinear parabolic equations, J. Math. Anal. Appl 282 (2003) no.2 p. 614-628 | MR 1989676 | Zbl 1039.35133
[29] , & , Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D 44 (1990) p. 38-60 | MR 1069671 | Zbl 0704.58030