Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay

Alexander V. Rezounenko[1]

  • [1] Kharkov University, Department of Mechanics and Mathematics, 4 Svobody sqr., Kharkov 61077 (Ukraine)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1547-1564
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

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 54.5 (2004): 1547-1564. <http://eudml.org/doc/116151>.

@article{Rezounenko2004,
abstract = {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.},
affiliation = {Kharkov University, Department of Mechanics and Mathematics, 4 Svobody sqr., Kharkov 61077 (Ukraine)},
author = {Rezounenko, Alexander V.},
journal = {Annales de l’institut Fourier},
keywords = {approximate inertial manifolds; inertial manifolds with delay; retarded non-linear partial differential equations; retarded nonlinear partial differential equations},
language = {eng},
number = {5},
pages = {1547-1564},
publisher = {Association des Annales de l'Institut Fourier},
title = {Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay},
url = {http://eudml.org/doc/116151},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Rezounenko, Alexander V.
TI - Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1547
EP - 1564
AB - 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.
LA - eng
KW - approximate inertial manifolds; inertial manifolds with delay; retarded non-linear partial differential equations; retarded nonlinear partial differential equations
UR - http://eudml.org/doc/116151
ER -

References

top
  1. C. Foias, G. Sell, R. Temam, Variétés Inertielles des équations différentielles dissipatives, C.R. Acad. Sci. Paris, Série I 301 (1985), 139-141 Zbl0591.35062MR801946
  2. S.-N. Chow, K. Lu, Invariant manifolds for flows in Banach spaces, J. Diff. Eqns 74 (1988), 285-317 Zbl0691.58034MR952900
  3. C. Foias, O. Manley, R. Temam, Sur l'interaction des petits et grands tourbillons dans les écoulements turbulents, C.R. Acad. Sci. Paris, Série I 305 (1987), 497-500 Zbl0624.76072MR916319
  4. C. Foias G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967), 1-34 Zbl0176.54103MR223716
  5. C. Foias, G. Sell, E. Titi, Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dyn. Diff. Eqns 1 (1989), 199-224 Zbl0692.35053MR1010966
  6. I.D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems (in Russian), Acta, Kharkov (1999) (2002), http://www.emis.de/monographs/Chueshov/ Zbl1100.37046
  7. I.D. Chueshov, Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, J. Dyn. Diff. Eqns 7 (1995), 549-566 Zbl0840.60054MR1362670
  8. A. Debussche, R. Temam, Inertial manifolds and their dimension, (1993), World Scientific Zbl0769.34047MR1386913
  9. P. Constantin, C. Foias, B. Nicolaenko, R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, (1989), Springer, Berlin-Heidelberg-New York Zbl0683.58002MR966192
  10. R. Temam, Infinite dimensional dynamical systems in mechanics and physics, (1988), Springer, Berlin-Heidelberg-New York Zbl0662.35001MR953967
  11. L. Boutet de Monvel, I.D. Chueshov, A.V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis 34 (1998), 907-925 Zbl0954.34064MR1636608
  12. X. Mora, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Res. Notes in Math 155 (1987), 172-183 Zbl0642.35061MR907731
  13. R. Datko, Y.C. You, Some second-order vibrating systems cannot tolerate small delays in their damping, J. Optim. Theory. Appl 70 (1991), 521-537 Zbl0791.34045MR1124776
  14. A. Debussche, R. Temam, Some new generalizations of inertial manifolds, Discr. Contin. Dynamical Systems 2 (1996), 543-558 Zbl0948.35018MR1414085
  15. J. Robinson, Inertial manifolds with and without delay, Discr. Contin. Dynamical Systems 5 (1999), 813-824 Zbl0954.37037MR1722373
  16. M. Taboada, Y.C. You, Invariant manifolds for retarded semilinear wave equations, J. Diff. Eqns 114 (1994), 337-369 Zbl0815.34067MR1303032
  17. J.K. Hale, Theory of functional differential equations, (1977), Springer, Berlin - Heidelberg - New York Zbl0352.34001MR508721
  18. J.K. Hale, Asymptotic behavior of dissipative systems, (1988), Amer. Math. Soc., Providence, Rhode Island Zbl0642.58013MR941371
  19. C.C. Travis, D.F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS 200 (1974), 395-418 Zbl0299.35085MR382808
  20. O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay equations. Functional, complex, and nonlinear analysis, 110 (1995), Springer-Verlag, New York Zbl0826.34002MR1345150
  21. J. Wu, Theory and applications of partial functional-differential equation, 119 (1996), Springer-Verlag, New York Zbl0870.35116
  22. I.D. Chueshov, On a certain system of equations with delay, occurring in aeroelasticity, J. of Soviet Mathematics 58 (1992), 385-390 Zbl0783.73046
  23. I.D. Chueshov, A.V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, série I 321 (1995), 607-612 Zbl0845.35129MR1356562
  24. L. Boutet de Monvel, I.D. Chueshov, and A.V. Rezounenko, Long-time behaviour of strong solutions for a class of retarded nonlinear P.D.E.s, Commun. in Partial Differential Equations 22 (1997), 1453-1474 Zbl0891.35159MR1469578
  25. A.V. Rezounenko, Inertial manifolds with delay for retarded semilinear parabolic equations, Discr. Contin. Dynamical Systems 6 (2000), 829-840 Zbl1011.37046MR1788255
  26. A.V. Rezounenko, Inertial manifolds for retarded second order in time evolution equations, Nonlinear Analysis 51 (2002), 1045-1054 Zbl1023.35087MR1926084
  27. A.V. Rezounenko, Steady approximate inertial manifolds of exponential order for semilinear parabolic equations, Differential and Integral Equations 15 (2002), 1345-1356 Zbl1161.35427MR1920691
  28. A.V. Rezounenko, Approximate inertial manifolds for retarded semilinear parabolic equations, J. Math. Anal. Appl 282 (2003), 614-628 Zbl1039.35133MR1989676
  29. M.S. Jolly, I.G. Kevrekidis, E.S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D 44 (1990), 38-60 Zbl0704.58030MR1069671

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.