Appendix: Skew-product flows and the uniform attractor

Abstract

In this appendix we discuss another approach to the asymptotic dynamics of non-autonomous equations, the uniform attractor, which was developed by Chepyzhov and Vishik (2002) [see also the appendix in the book by Vishik (1992)]. Reinterpreted in the language of processes, the uniform attractor is the minimal fixed (time-independent) compact subset ${\mathcal{A}}_{\Sigma }$ of the phase space that attracts all trajectories uniformly for bounded sets B of initial conditions and uniformly in the initial time:

$$ \begin{array}{rcl} \lim \limits_{t\rightarrow +\infty }\left [\sup \limits_{s\in \mathbb{R}}\,\mathrm{dist}(S(t + s,s)B,{\mathcal{A}}_{\Sigma })\right ] = 0.& & \\ \end{array}$$

Note that while this uniform attractor is a fixed subset of the phase space and is ‘attracting’, one cannot speak of the ‘dynamics on the uniform attractor’. The property of invariance of the global or non-autonomous attractor has been replaced by minimality (Definition 16.8).