Approximation theories for inertial manifolds

Luskin, Mitchell; Sell, George R.

ESAIM: Modélisation mathématique et analyse numérique, Attractors, Inertial Manifolds and their Approximation. Proceedings of the Marseille-Luminy... 1987, Tome 23 (1989) no. 3, pp. 445-461.

J. E. Billotti, J. P. Lasalle (1971), Dissipative periodic processes, Bull. Amer.Math. Soc, 77, pp. 1082-1088. | MR | Zbl

S.-N. Chow, K. Lu and G. R. Sell (1988), Smoothness of inertial manifolds, Preprint. | MR | Zbl

P. Constantin (1988), A construction of inertial manifolds, Preprint. | MR | Zbl

P. Constantin, C. Foias, B. Nicolaenko, R. Temam (1989), Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Mathematical Sciences, No 70, Springer-Verlag. | MR | Zbl

P. Constantin, C. Foias, B. Nicolaenko, R. Temam (1989), Spectral barriers and inertial manifolds for dissipative partial differential equations, J Dynamics and Differential Equations, 1 (to appear). | MR | Zbl

P. Constantin, C. Foias, R. Temam (1985), Attractors representing turbulent flows, Memoirs Amer. Math. Soc., 314. | MR | Zbl

C. R. Doering, J. D. Gibbon, D. D. Holm and B. Nicolaenko (1988), Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity (to appear). | MR | Zbl

E. Fabes, M. Luskin and G. R. Sell (1988), Construction of inertial manifolds by elliptic regularization, J. Differential Equations, to appear. | MR | Zbl

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi (1988), On the computation of inertial manifolds, Physics Letters A, Vol. 131, No 7, 8, pp. 433-436. | MR

C. Foias, B. Nicolaenko, G. R. Sell, R. Temam (1988), Inertial manifolds for the Kuramoto Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl., 67, pp. 197-226. | MR | Zbl

C. Foias, G. R. Sell and R. Temam (1986), Inertial manifolds for nonlinear evolutionary equations, IMA Preprint No 234, March, 1986 Also in, J. Differential Equations, 73 (1988), pp. 309-353. | MR | Zbl

C. Foias, G. R. Sell and E. S. Titi (1988), Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynamics and Differential Equations, to appear. | MR | Zbl

C. Foias and R. Temam (1979), Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl., 58, pp. 339-368. | MR | Zbl

J.-M. Ghidaglia, Discrétisation en temps et variétés inertielles pour des équations d'évolution aux dérivées partielles non linéaires, Preprint. | Zbl

J. K. Hale (1988), Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence. | MR | Zbl

J. K. Hale and G. R. Sell (1988), Inertial manifolds for gradient Systems.

D. Henry (1981), Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, No 840, Springer-Verlag, New York. | MR | Zbl

M. S. Jolly (1988), Explicit construction of an inertial manifold for a reaction diffusion equation, J. Differential Equations (to appear). | MR | Zbl

D. A. Kamaev (1981), Hopf's conjecture for a class of chemical kinetics equations, J. Soviet Math., 25, pp. 836-849. | Zbl

M. Luskin and G. R. Sell (1988), Parabolic regularization and the construction of inertial manifolds, Preprint.

J. Mallet-Paret (1976), Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22, pp. 331-348. | MR | Zbl

J. Mallet-Paret and G. R. Sell (1987), Inertial manifolds for reaction diffusion equations in higher space dimensions, IMA Preprint No. 331, June 1987, Also in, J. Amer. Math. Soc, 1, No. 4 (1988), pp. 805-866. | MR | Zbl

R. Mané (1977), Reduction of semilinear parabolic equations to finite dimensional C1 flows, Geometry and Topology, Lecture Notes in Math., vol. 597, Springer-Verlag, New York, pp.361-378. | MR | Zbl

R. Mané, (1981) On the dimension of the compact invariant sets of certain nonlinear maps, Lecture Notes in Math,, vol. 898, Springer-Verlag, New York, pp. 230-242. | MR | Zbl

M. Marion (1988), Inertial manifolds associated to partly dissipative reaction diffusion equations, J. Math. Anal. Appl. (to appear). | Zbl

X. Mora (1983), Finite dimensional attracting manifolds in reaction diffusion equations, Contemporary Math., 17, pp. 353-360. | MR | Zbl

X. Mora and J. Solà-Morales (1987), Existence and non-existence of finite dimensional globally attracting invariant manifolds in semilinear damped wave equations, Dynamics of Infinite Dimensional Systems, Springer-Verlag, New York, pp. 187-210. | MR | Zbl

X. Mora and J. Solà-Morales (1988), The singular limit dynamics of semilinear damped wave equations, Preprint, Univ. Autònoma Barcelona. | MR | Zbl

B. Nicolaenko, B. Scheurer and R. Temam, (1987), Some global dynamical properties of a class of pattern formation equations, IMA Preprint No. 381. | MR | Zbl

A. Pazy (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol.44, Springer-Verlag, New York. | MR | Zbl

R. J. Sacker (1964), On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, NYU Preprint No. 333, October 1964.

R. J. Sacker (1965), A new approach to the perturbation theory of invariant surfaces, Comm. Pure Appl. Math., 18, pp.717-732. | MR | Zbl

R. J. Sacker (1969), A perturbation theorem for invariant manifolds and Hölder continuity, J. Math. Mech., 18, pp.705-762. | MR | Zbl

G. R. Sell and Y. You (1988), Inertial manifolds : The non self adjoint case, Preprint. | Zbl

M. Taboada (1988), Finite dimensional asymptotic behavior for the Swift-Hohenberg model of convection, Nonlinear Analysis, TMA, to appear. | MR | Zbl

R. Temam (1988), Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York. | MR | Zbl