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

Engel, Maximilian; Hummel, Felix; Kuehn, Christian

doi:10.1090/bproc/92

Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions
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by Maximilian Engel, Felix Hummel and Christian Kuehn;

Proc. Amer. Math. Soc. Ser. B 8 (2021), 252-266

DOI: https://doi.org/10.1090/bproc/92

Published electronically: August 19, 2021

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Abstract:

In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.

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