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In this paper, we study slow manifolds for infinite-dimensional evolutionequations. We compare two approaches: an abstract evolution equation frameworkand a finite-dimensional spectral Galerkin approximation. We prove that theslow manifolds constructed within each approach are asymptotically close undersuitable conditions. The proof is based upon Lyapunov-Perron methods and acomparison of the local graphs for the slow manifolds in scales of Banachspaces. In summary, our main result allows us to change between differentcharacterizations of slow invariant manifolds, depending upon the technicalchallenges posed by particular fast-slow systems.

Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions