Dynamics of Evolutionary Equations pp 569-592 | Cite as

# Inertial Manifolds: The Reduction Principle

Chapter

## Abstract

In the previous chapters, we have seen several illustrations of finite dimensional structures within the infinite dimensional dynamical systems. For example, many dissipative systems have global attractors, and oftentimes, the attractor A has finite Hausdorff and fractal dimensions. During the last few years it has been shown that some infinite dimensional nonlinear dissipative evolutionary equations have inertial manifolds. We will give the definition shortly.

## Keywords

Strong Solution Invariant Manifold Mild Solution Global Attractor Unique Fixed Point
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## Additional Readings

- D B Henry (1981),
*Geometric Theory of Semilinear Parabolic Equations*, Lecture Notes in Mathematics, No 840, Springer Verlag, New York.Google Scholar - P Constantin, C Foias, B Nicolaenko, and R Temam (1988),
*Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations*, Appl Math Sciences, No 70, Springer Verlag, New York.Google Scholar - J Mallet-Paret and G R Sell (1988),
*Inertial manifolds for reaction diffusion equations in higher space dimensions*, J Am Math Soc**1**, 805–866.MathSciNetCrossRefGoogle Scholar - R J Sacker (1965),
*A new approach to the perturbation theory of invariant surfaces*, Comm Pure Appl Math**18**, 717–732.MathSciNetzbMATHCrossRefGoogle Scholar - E Fabes, M Luskin and G R Sell (1991),
*Construction of inertial manifolds by elliptic regularization*, J Dynamics Differential Equations**89**, 355–387.MathSciNetzbMATHCrossRefGoogle Scholar - R J Sacker (1969),
*A perturbation theorem for invariant manifolds and Hölder continuity*, J Math Mech**18**, 705–762.MathSciNetzbMATHGoogle Scholar - R Maíié (1977),
*Reduction of semilinear parabolic equations to finite dimensional C1 flows*, Lecture Notes in Math, vol 597, Springer Verlag, New York, pp. 361–378.Google Scholar - C Foias, G R Sell and E S Titi (1989),
*Exponential tracking and approximation of inertial manifolds for dissipative equations*, J Dynamics Differential Equations**1**, 199–244.MathSciNetzbMATHCrossRefGoogle Scholar - A Eden, C Foias, and B Nicolaenko (1994),
*Exponential attractors of optimal Lyapunov dimension for the Navier-Stokes equations*, J Dynamics Differential Equations**6**, 301–323.MathSciNetzbMATHCrossRefGoogle Scholar - A Eden, C Foias, B Nicolaenko, and R Temam (1994),
*Exponential Attractors for Dissipative Evolution Equations*, Research in Applied Mathematics Series, Masson, Paris.Google Scholar - X Mora and J Solà-Morales (1989),
*Inertial manifolds of damped semilinear wave equations, Attractors*, Inertial Manifolds and their Approximation (Marseille-Luminy, 1987), RAIRO Model Math Anal Numer, vol. 23, pp. 489–505.MathSciNetzbMATHGoogle Scholar - M Taboada and Y You (1992),
*Global attactor, inertial manifolds and stabilizaton of nonlinear damped beam equations*, IMA Preprint, No 851.Google Scholar - Y You (1993a),
*Spillover problem and global dynamics of nonlinear beam equations*, in: Differential Equations, Dynamical Systems and Control Science, K D Elworthy, W N Everitt, and E B Lee (eds), Marcel Dekker, New York, pp. 891–912.Google Scholar - Y You (1996a),
*Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping*, Abstr Appl Anal**1**, 83–102.MathSciNetzbMATHCrossRefGoogle Scholar - Y You (1996b),
*Nonlinear wave equations with asymptotically monotone damping*, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, ( AG Kartsatos, ed.), Marcel Dekker, New York, pp. 299–311.Google Scholar - G Bianchi and A Marzocchi (1998),
*Inertial manifold for the motion of strongly damped nonlinear elastic beams*, NoDEA Nonlinear Differential Equations**5**, 181–192.MathSciNetzbMATHCrossRefGoogle Scholar

## Additional Readings

- S-N Chow, K Lu, and G R Sell (1992),
*Smoothness of inertial manifolds*, J Math Anal Appl**169**, 283–312.MathSciNetzbMATHCrossRefGoogle Scholar - P Constantin, C Foias, B Nicolaenko, and R Temam (1988),
*Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations*, Appl Math Sciences, No 70, Springer Verlag, New York.Google Scholar - E Fabes, M Luskin and G R Sell (1991),
*Construction of inertial manifolds by elliptic regularization*, J Dynamics Differential Equations**89**, 355–387.MathSciNetzbMATHCrossRefGoogle Scholar - C Foias, B Nicolaenko, and R Temam (1989),
*Spectral barriers and inertial manifolds for dissipative partial differential equations*, J Dynamics Differential Equations**1**, 45–73.MathSciNetzbMATHCrossRefGoogle Scholar - C Foias, G R Sell and R Temam (1988),
*Inertial manifolds for nonlinear evolutionary equations*, J Differential Equations**73**, 309–353.MathSciNetzbMATHCrossRefGoogle Scholar - C Foias, G R Sell and E S Titi (1989),
*Exponential tracking and approximation of inertial manifolds for dissipative equations*, J Dynamics Differential Equations**1**, 199–244.MathSciNetzbMATHCrossRefGoogle Scholar - M S Jolly (1989),
*Explicit construction of an inertial manifold for a reaction diffusion equation*, J Differential Equations**78**, 220–261.MathSciNetzbMATHCrossRefGoogle Scholar - J Mallet-Paret (1988),
*Morse decompositions for delay differential equations*, J Differential Equations**72**, 270–315.MathSciNetzbMATHCrossRefGoogle Scholar - J Mallet-Paret, G R Sell, and Z Shao (1993),
*Obstructions for the existence of normally hyperbolic inertial manifolds*, Indiana J Math**42**, 1027–1055.MathSciNetzbMATHCrossRefGoogle Scholar - M Marion (1989a),
*Inertial manifolds associated to partly dissipative reaction diffusion equations*, J Math Anal Appl**143**, 295–326.MathSciNetzbMATHCrossRefGoogle Scholar - R Rosa and R Temam (1996),
*Inertial manifolds and normal hyperbolicity*, Acta Appl Math**45**, 1–50.MathSciNetzbMATHCrossRefGoogle Scholar - G R Sell and M Taboada (1992),
*Local dassipitavity and attractors for the KuramotoSivashinsky equation in thin 2D domains*, Nonlinear Anal**18**, 671–687.MathSciNetzbMATHCrossRefGoogle Scholar - G R Sell and Y You (1992),
*Inertial manifolds: The non-self adjoint case*, J Differential Equations**96**, 203–255.MathSciNetzbMATHCrossRefGoogle Scholar - Z Shao (1998),
*Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions*, J Differential Equations**144**, 1–43.MathSciNetzbMATHCrossRefGoogle Scholar - M Taboada and Y You (1992),
*Global attactor, inertial manifolds and stabilizaton of nonlinear damped beam equations*, IMA Preprint, No 851.Google Scholar - R Temam (1988),
*Infinite Dimensional Dynamical Systems in Mechanics and Physics*, Applied Mathematical Sciences, Vol 68, Springer Verlag, New York.CrossRefGoogle Scholar - Y You (1993b),
*Inertial manifolds and stabilization of nonlinear elastic systems with structural damping*, in: Differential Equations with Applications to Mathematical Physics, W F Ames, E Harrell, and J V Herod (eds), Academic Press, New York, pp. 335–346.CrossRefGoogle Scholar - Y You (1996a),
*Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping*, Abstr Appl Anal**1**, 83–102.MathSciNetzbMATHCrossRefGoogle Scholar - Y You (1996b),
*Nonlinear wave equations with asymptotically monotone damping*, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, ( AG Kartsatos, ed.), Marcel Dekker, New York, pp. 299–311.Google Scholar - G R Sell (2001), References on dynamical systems, http://www.math.umn.edu/sell/.

## Copyright information

© Springer Science+Business Media New York 2002