Attractors for dissipative partial differential equations in bounded and unbounded domains. (English) Zbl 1221.37158

Miranville, A.; Zelik, S.

Dafermos, C.M.(ed.) et al., Handbook of differential equations: Evolutionary equations. Vol. IV. Amsterdam: Elsevier/North-Holland (ISBN 978-0-444-53034-9/hbk). Handbook of Differential Equations, 103-200 (2008).

This survey concerns attractors of infinite-dimensional dissipative dynamical systems.
After an introduction, in Section 2 the main definitions and results about global attractors, their dimension and their robustness are presented. Section 3 deals with exponential attractors and inertial manifolds. Section 4 concerns uniform attractors and pullback attractors and pullback attractors of nonautonomous systems, and finite-dimensional reductions of nonautonomous systems are described.
The main section is Section 5 about dissipative PDEs on unbounded domains. The dynamics of those PDEs is purely infinite-dimensional, in general, and, hence, does not possess any finite-dimensional reduction. Moreover, interactions between spatial and temporal chaotic modes can lead to so-called space-time chaos with infinite Lyapunov dimension and infinite topological entropy.
Finally, generalizations to ill-posed problems (by means of trajectory attractors) are given in Section 6.
For the entire collection see [Zbl 1173.35002].

Reviewer: Lutz Recke (Berlin)

MSC:

37L30	Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41	Attractors
35B42	Inertial manifolds

Keywords:

dimension; robustness; exponential attractors; uniform attractors; pullback attractors; nonautonomous systems; space-time chaos; trajectory attractors

Citations:

Zbl 1173.35002