Fast Reactions and Slow Manifolds

TL;DR

This paper extends Fenichel theory to infinite-dimensional fast-reaction systems, demonstrating that solutions can be approximated by slow flows and constructing families of slow manifolds with proven closeness to critical manifolds.

Contribution

It generalizes invariant manifold theory for fast-slow PDEs to include fast reaction terms and introduces a parameter-based family of slow manifolds.

Findings

Slow manifolds are close to the critical manifold.

Semi-flow on slow manifold converges to that on the critical manifold.

Theoretical results are verified through an example.

Abstract

In this paper we generalize the Fenichel theory for attracting critical/slow manifolds to fast-reaction systems in infinite dimensions. In particular, we generalize the theory of invariant manifolds for fast-slow partial differential equations in standard form to the case of fast reaction terms. We show that the solution of the fast-reaction system can be approximated by the corresponding slow flow of the limit system. Introducing an additional parameter that stems from a splitting in the slow variable space, we construct a family of slow manifolds and we prove that the slow manifolds are close to the critical manifold. Moreover, the semi-flow on the slow manifold converges to the semi-flow on the critical manifold. Finally, we apply these results to an example and show that the underlying assumptions can be verified in a straightforward way.