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Proceedings of the American Mathematical Society Series B

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


Authors: Maximilian Engel, Felix Hummel and Christian Kuehn
Journal: Proc. Amer. Math. Soc. Ser. B 8 (2021), 252-266
MSC (2020): Primary 37L15, 37L25, 37L65; Secondary 34E15, 35K57
DOI: https://doi.org/10.1090/bproc/92
Published electronically: August 19, 2021
MathSciNet review: 4302153
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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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Additional Information

Maximilian Engel
Affiliation: Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
MR Author ID: 1217727
ORCID: 0000-0002-1406-8052
Email: maximilian.engel@fu-berlin.de

Felix Hummel
Affiliation: Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748 Garching bei München, Germany
MR Author ID: 1319692
ORCID: 0000-0002-2374-7030
Email: hummel@ma.tum.de

Christian Kuehn
Affiliation: Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748 Garching bei München, Germany
MR Author ID: 875693
ORCID: 0000-0002-7063-6173
Email: ckuehn@ma.tum.de

Keywords: Fast-slow systems, reaction-diffusion equations, Galerkin discretization, infinite-dimensional dynamics
Received by editor(s): March 1, 2021
Published electronically: August 19, 2021
Additional Notes: The first author was supported by Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).
The second author acknowledges partial support via the SFB/TR109 “Discretization in Geometry and Dynamics” as well as partial support of the EU within the TiPES project funded the European Unions Horizon 2020 research and innovation programme under grant agreement No. 820970.
The third author acknowledges support via a Lichtenberg Professorship as well as support via the SFB/TR109 “Discretization in Geometry and Dynamics” as well as partial support of the EU within the TiPES project funded the European Unions Horizon 2020 research and innovation programme un der grant agreement No. 820970.
This project is TiPES contribution #80: This project had received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 820970
The authors gratefully acknowledge the financial support of the TUM publishing fund for an open access publication of this article
Communicated by: Wenxian Shen
Article copyright: © Copyright 2021 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)