Attractors. Then and now

Abstract

This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini-courses taught by the author at the University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs, including attractors for autonomous and non-autonomous equations, dynamical systems in general topological spaces, various types of trajectory, pullback and random attractors, exponential attractors, determining functionals and inertial manifolds, as well as the dimension theory for the classes of attractors mentioned above. The theoretical results are illustrated by a number of clarifying examples and counterexamples. Bibliography: 248 titles.

The non-autonomous Navier–Stokes–Brinkman–Forchheimer equation with Dirichlet boundary conditions: dissipativity, regularity, and attractors

“…The proof of this proposition in a more general setting can be found in [8], see also [42]. We mention here that since Φ is a reflexive Banach space, bounded sets in it are precompact in a weak topology, so if we are given a bounded absorbing/attracting set, its closed convex hull will be a compact absorbing/attracting set, so the standard asymptotic compactness condition is satisfied.…”

“…Proof. Indeed, since ∂ t u n are uniformly bounded in the space L 2 (0, 1; L 2 (Ω)), due to Cauchy-Schwarz inequality, we have see [41,42] for the details. This finishes the proof of the lemma.…”

“…(3) A w un is a minimal (by inclusion) set which satisfies properties ( 1) and (2), see [8,42] for the proof of the equivalence of these definitions. However, if we want to present the attractor as a union of complete bounded trajectories, we need to introduce the hull H(g) and the cocycle S ξ (t).…”

“…for some new monotone function Q and this, in turn, allows us to study the longtime behaviour of solutions of (1.1) in terms of uniform attractors for the cocycle generated by equation (1.1), see [3,7,8,18,42] and references therein. Note that, in contrast to the standard situation, it looks difficult here to establish the existence of a compact uniformly absorbing set for the corresponding cocycle based only on the compactness of Sobolev embeddings, so we utilize the energy method (see [4]), the central part of which is to establish the energy identity (1.9) for any weak solution of (1.1).…”

Hausdorff Dimension Estimates for Interconnected Systems With Variable Metrics

“…Also, periodic oscillations are in general difficult to analyze in spite of their importance. Furthermore, complex attractors may not be manifolds, and thus, dimension with non-integer values becomes important in the analysis [7]. In this paper, for these reasons, we are concerned with finding estimates of Hausdorff dimensions of invariant sets.…”