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 lambda is an element of Lambda, where Lambda is a complete metric space, such that for each lambda is an element of Lambda there exists a unique pullback attractor A(lambda)(t). Using the theory of Baire category we show under natural conditions that there exists a residual set Lambda(*) subset of Lambda such that for every t is an element of R the function lambda bar right arrow A(lambda)(t) is continuous at each lambda is an element of Lambda(*) with respect to the Hausdorff metric. Similarly, given a family of uniform attractors Lambda(lambda), there is a residual set at which the map lambda bar right arrow Lambda(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. (C) 2017 Elsevier Inc. All rights reserved.