Continuity of pullback and uniform attractors

Abstract

We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterized by λ ∈ Λ, where Λ is a complete metric space, such that for each λ ∈ Λ there exists a unique pullback attractor A λ (t). Using the theory of Baire category we show under natural conditions that there exists a residual set Λ * ⊆ Λ such that for every t ∈ R the function λ → A λ (t) is continuous at each λ ∈ Λ * with respect to the Hausdorff metric. Similarly, given a family of uniform attractors A λ , there is a residual set at which the map λ → A λ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when Λ is compact that the continuity of pullback attractors and uniform attractors with respect to λ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier-Stokes equations.

Pullback attractor of a non-autonomous order-2γ parabolic equation for an epitaxial thin film growth model

“…Similarly, multiplying (10) by e ξ t and integrating it over (t, τ ), using (13), it is easy to get t τ e ξ s ∇h(s)…”

“…In [12], the norm-to-weak continuous process has been proposed and the proof of the existence of the pullback D-attractor for non-autonomous equation in H 1 0 was showed by Li and Zhong. The continuity of pullback and uniform attractors was studied by Hoanga, Olsonb and Robinson [13]. And in [14], Cheskidov and Kavlie did many studies with pullback attractors.…”

Asymptotics of singularly perturbed damped wave equations with super-cubic exponent

“…Following the theory that constructs the fractional power spaces associated with A in [19], we denote for every α > 0…”

“…In Proposition 3.8, we showed that the dynamic process {S ε (t, τ )} t≥τ generated by (1.2) is dissipative in X 2 × X 1 and that its asymptotic compactness can be deduced in X 2 × X 1 by the decomposition method. According to the theory of infinite dynamical system ( [27,19]), the dynamical processes {S ε (t, τ )} t≥τ admits a compact uniform attractor A ε in X 2 × X 1 .…”

“…[ [19]]Let X and Λ be metric spaces and {A λ } λ∈Λ a family of subsets of X. We say that the family A λ is upper semicontinuous as λ…”