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Fast Reactions and Slow Manifolds

Christian Kuehn, Jan-Eric Sulzbach

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.

Fast Reactions and Slow Manifolds

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.
Paper Structure (14 sections, 18 theorems, 136 equations)

This paper contains 14 sections, 18 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Fast Reaction Systems
  3. Function spaces and assumptions
  4. Existence of solutions
  5. Approximation by slow flow
  6. Slow Manifolds
  7. Assumptions
  8. Existence of slow manifolds
  9. Distance to critical manifold
  10. Attraction of trajectories and approximation by the slow flow
  11. Example
  12. Remarks
  13. Tools from Functional Analysis
  14. A different approach in controlling the fast-reaction term

Key Result

Proposition 2.5

Let $\varepsilon>0$ be small. Then, there exists a unique strict solution $(u^\varepsilon,v^\varepsilon)$ of the system full eq, i.e the solution satisfies $(u^\varepsilon,v^\varepsilon)\in C^1([0,\infty);X\times Y) \cap C([0,\infty);X_1\times Y_1)$.

Theorems & Definitions (54)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 44 more