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*et al*and are obtained from the Navier- Stokes Equations by a global modification of the nonlinear advection term. The existence of the attractor is obtained via the flattening property, which is verified.

*Discrete and Continuous Dynamical Systems - Series B*on the asymptotic dynamics of non-autonomous systems.

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If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.

In this paper, for non-autonomous RDS we study cocycle attractors with autonomous attraction universes, i.e. pullback attracting some autonomous random sets, instead of non-autonomous ones. We first compare cocycle attractors with autonomous and non-autonomous attraction universes, and then for autonomous ones we establish some existence criteria and characterization. We also study for cocycle attractors the continuity of sections indexed by non-autonomous symbols to find that the upper semi-continuity is equivalent to uniform compactness of the attractor, while the lower semi-continuity is equivalent to an equi-attracting property under some conditions. Finally, we apply these theoretical results to 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing.

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