Continuity of pullback and uniform attractors

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.