Infinite dimensional Slow Manifolds for a Linear Fast-Reaction System

TL;DR

This paper reviews the theory of invariant slow manifolds in fast-slow systems and introduces new results on the convergence and construction of slow manifolds for linear fast-reaction PDEs, extending classical theorems.

Contribution

It provides a convergence analysis and explicit construction of slow manifolds for linear fast-reaction PDEs, generalizing Fenichel-Tikhonov theory to infinite-dimensional systems.

Findings

Solutions converge to the limit system as ε→0

Explicit slow manifold construction from solutions

Generalized Fenichel-Tikhonov theorem for linear systems

Abstract

The aim of this expository paper is twofold. We start with a concise overview of the theory of invariant slow manifolds for fast-slow dynamical systems starting with the work by Tikhonov and Fenichel to the most recent works on infinite-dimensional fast-slow systems. The main part focuses on a class of linear fast-reaction PDE, which are particular forms of fast-reaction systems. The first result shows the convergence of solutions of the linear system to the limit system as the time-scale parameter $\varepsilon$ goes to zero. Moreover, from the explicit solutions the slow manifold is constructed and the convergence to the critical manifold is proven. The subsequent result, then, states a generalized version of the Fenichel-Tikhonov theorem for linear fast-reaction systems.