Slow Manifolds for Infinite-Dimensional Evolution Equations

2020/08/24 by Felix Hummel, Christian Kuehn, Hummel, Felix +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #35A24 #35B25 #37D10 #37L25 #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2008.10700

openalex publication_date 2020/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend classical finite-dimensional Fenichel theory in two directions to infinite dimensions. Under comparably weak assumptions we show that the solution of an infinite-dimensional fast-slow system is approximated well by the corresponding slow flow. After that we construct a two-parameter family of slow manifolds S_ε,ζ under more restrictive assumptions on the linear part of the slow equation. The second parameter ζ does not appear in the finite-dimensional setting and describes a certain splitting of the slow variable space in a fast decaying part and its complement. The finite-dimensional setting is contained as a special case in which S_ε,ζ does not depend on ζ. Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.

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