## 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.

Original language | English |
---|---|

Pages (from-to) | 4067-4093 |

Number of pages | 27 |

Journal | Journal of Differential Equations |

Volume | 264 |

Issue number | 6 |

DOIs | |

State | Published - Mar 15 2018 |

## Keywords

- Pullback attractor
- Uniform attractor