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

Maximilian Engel,Felix Hummel,Christian Kuehn

doi:10.1090/bproc/92

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.

Highlights

The perturbation theory of normally hyperbolic invariant manifolds introduced by Fenichel [4, 9] has proved to be a useful tool in the theory of dynamical systems
If we want to use a Galerkin approximation in both the slow and the fast variable, it is useful to impose similar conditions on X, i.e. that there is a splitting X = XFζ ⊕ XSζ such that the conditions (i)–(v) in Assumption (Bn) hold with Y and B being replaced by X and A, respectively
A truncation at |k| ≤ k0 = NΔ,ζ, for ε and ζ given as before, yields the described correspondence to system (2.6). Such a Galerkin approach is very insightful in situations of dynamical interest such as dynamic bifurcations in reaction-diffusion systems where geometric techniques can be applied to the finite-dimensional approximation and be extended to the infinite-dimensional limit

Summary

Introduction

The perturbation theory of normally hyperbolic invariant manifolds introduced by Fenichel [4, 9] has proved to be a useful tool in the theory of dynamical systems. 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. The advantage of (1.3) is that the fast variable is uniquely determined by the slow variable, i.e., the dimension of the dynamical problem (1.1) has been reduced It has been an open problem for a few decades, how to generalize Fenichel theory to the infinite-dimensional setting, with fast-slow systems of partial differential equations as an important application. The main ingredient of the latter procedure is a splitting of the slow variable space Y = YFζ ⊕YSζ into a quickly decaying part and a part on which the linear dynamics are invertible

The two approaches

The main result

Case study of an explicit reaction-diffusion problem