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 parameterised by a complete metric space $Λ$ such that for each $λ\inΛ$ there exists a unique pullback attractor $\mathcal A_λ(t)$. Using the theory of Baire category we show under natural conditions that there exists a residual set $Λ_*\subseteqΛ$ such that for every $t\in\mathbb R$ the function $λ\mapsto\mathcal A_λ(t)$ is continuous at each $λ\inΛ_*$ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors $\mathbb A_λ$, there is a residual set at which the map $λ\mapsto\mathbb A_λ$ 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.