Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

Abstract

Abstract. We suggest in this article a new explicit algorithm allowing to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to nonautonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting finite-dimensional time-dependent set in the phase space. In particular, this result shows that, for a wide class of nonautonomous equations of mathematical physics, the limit dynamics remains finite-dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this article on the model example of a nonautonomous reaction-diffusion system in a bounded domain.Introduction.

Singularly perturbed 1D Cahn–Hilliard equation revisited

“…In the present paper, as in [24], we construct for S (t) a robust family of exponential attractors M ,δ on B 5,δ H 0 with respect to the metric induced by the H 0 -norm, whose common basin of attraction is the whole phase-space K δ , with a uniform upper bound on the fractal dimension as well as a uniform rate of attraction of trajectories. Here, inspired by [13], we show that the map → M ,δ is Hölder continuous in the following sense…”

“…also [9]). In particular, the new strategy devised in [11] allows to construct a robust (under perturbations) family of exponential attractors (see also [12][13][14]24,34] and references therein). This family is characterized by an explicit estimate on the symmetric distance between exponential attractors of the unperturbed and perturbed problems, and the continuity with respect to the perturbations does not involve time shifts as in the previous results.…”

“…Inspired by the recent contributions [13] and [34], the main goal of this paper is to construct a family of exponential attractors which is continuous with respect to by using a metric which does not depend on as goes to 0. More precisely, in Sects.…”

Lower semicontinuity of attractors for non-autonomous dynamical systems

“…Firstly, we can obtain the existence of the unstable manifold under the assumption that (15) holds for all z = (z + , z − ) ∈ Z with some suitably small ρ > 0. Also, supposing that (15) holds for all z = (z + , z − ) ∈ Z, we get continuity of the unstable and stable manifolds.…”

“…(b) On the other hand, observe that nowhere do we need the pullback attraction property of the family {A η (t)} t∈R ; so our results are also applicable to any kind of nonautonomous family of attractors {A η (t)} t∈R , described as in the above theorem. Currently, there is much active research on the relation between pullback, forward and uniform attraction for non-autonomous dynamical systems (see, for example, Cheban et al [13], Efendiev et al [15], Rodríguez-Bernal and Vidal-López [28], Carvalho et al [11] or Langa et al [24]). Our result would cover all of these different kinds of attraction.…”