Continuity of pullback and uniform attractors

Luan T. Hoang, Eric J. Olson, James C. Robinson

Research output: Contribution to journalArticle

8 Scopus citations

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 Aλ(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↦Aλ(t) is continuous at each λ∈Λ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors Aλ, there is a residual set at which the map λ↦Aλ 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.

Original languageEnglish
Pages (from-to)4067-4093
Number of pages27
JournalJournal of Differential Equations
Volume264
Issue number6
DOIs
StatePublished - Mar 15 2018

Keywords

  • Pullback attractor
  • Uniform attractor

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