zbMATH — the first resource for mathematics

Some remarks on the smoothness of inertial manifolds. (English) Zbl 0723.58033
Nonlinear evolutionary partial differential equations give rise to infinite dimensional dynamical systems. It is believed that dissipative effects may lead to the fact that in the long run only a finite number of degrees of freedom is relevant. This may be captured by the notion of the inertial manifold, producing a splitting of the Banach space with a subspace which is finite dimensional, invariant and exponentially attracting the solutions.
In the paper conditions for the existence of such inertial manifolds are given and regularity properties are shown.
Reviewer: C.H.Cap (Zürich)

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
Full Text: DOI
[1] {\scConstantin} P., {\scFoias} C., {\scNicolaenko} B. & {\scTemam} R., Spectral barriers and inertial manifolds for dissipative partial differential equations, Dynam. diff. Eqns (to appear).
[2] Constantin, P.; Foias, C.; Nicolaenko, B.; Temam, R., Integral manifolds and inertial manifolds for dissipative partial differential equations, (1988), Springer New York · Zbl 0683.58002
[3] Chow, S.N.; Lu, K., Invariant manifolds for flows in Banach spaces, J. diff. eqns, 74, 285-317, (1988) · Zbl 0691.58034
[4] Demengel, F.; Ghidaglia, J.M., Inertial manifolds for partial differential evolution equations under time-discretization: existence, convergence and applications, C.R. acad. sci. Paris, 307, 453-458, (1988), série I · Zbl 0666.35049
[5] {\scDemengel} F. & {\scGhidaglia} J.M., Construction of inertial manifolds via the Lyapunov-Perron method, (to appear).
[6] Foias, C.; Sell, G.; Temam, R., Inertial manifolds for nonlinear evolutionary equations, J. diff. eqns, 73, 309-353, (1988) · Zbl 0643.58004
[7] Hirsch, M.W.; Pugh, C.C., Stable manifolds and hyperbolic sets, Proc. symp. pure math., 14, 133-163, (1970) · Zbl 0215.53001
[8] {\scLuskin} M. & {\scSell} G., Approximation theories for inertial manifolds, Proc. Luminy Conference (Edited by J.C. {\scSaut} and R. {\scTemam}) (to appear). · Zbl 0688.58035
[9] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, (1988), Springer New York · Zbl 0662.35001
[10] Foias, C.; Sell, G.; Titi, E., Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. dynam. diff. eqns, 1, 194-224, (1989)
[11] Mallet-Paret, J.; Sell, G., Inertial manifolds for reaction diffusion equations in higher space dimensions, J. am. math. soc., 1, 805-866, (1988) · Zbl 0674.35049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.