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 $\Lambda$ such that for each $\lambda\in\Lambda$ there exists a unique pullback attractor $\mathcal A_\lambda(t)$. Using the theory of Baire category we show under natural conditions that there exists a residual set $\Lambda_*\subseteq\Lambda$ such that for every $t\in\mathbb R$ the function $\lambda\mapsto\mathcal A_\lambda(t)$ is continuous at each $\lambda\in\Lambda_*$ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors $\mathbb A_\lambda$, there is a residual set at which the map $\lambda\mapsto\mathbb A_\lambda$ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when $\Lambda$ is compact that the continuity of pullback attractors and uniform attractors with respect to $\lambda$ 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.