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Slow Manifolds for PDE with Fast Reactions and Small Cross Diffusion

Laurent Desvillettes, Christian Kuehn, Jan-Eric Sulzbach, Bao Quoc Tang, Bao-Ngoc Tran

TL;DR

The paper develops a rigorous slow-manifold theory for fast-slow PDEs with small cross diffusion by extending Fenichel-type ideas to infinite dimensions. It combines geometric and analytic methods, proving the existence of attracting slow manifolds $S_\varepsilon$ that converge to a critical manifold $S_0$ under a spectral-gap framework and small cross-diffusion, with strong convergence in $L^\infty(0,T;H^2)$ shown in 1D. The Lyapunov-Perron approach yields explicit slow-manifold graphs, differentiability, and exponential attraction of trajectories, and the slow flow is shown to approximate the reduced slow dynamics. In the linear case, explicit expressions and Galerkin-based approximations for slow manifolds are derived via Fourier modes, illustrating the geometry and enabling practical approximations. Together, these results provide a rigorous reduction framework for fast-reaction PDEs with cross diffusion and clarify the roles of initial layers and cross-diffusion in the slow dynamics.

Abstract

Multiple time scales problems are investigated by combining geometrical and analytical approaches. More precisely, for fast-slow reaction-diffusion systems, we first prove the existence of slow manifolds for the abstract problem under the assumption that cross diffusion is small. This is done by extending the Fenichel theory to an infinite dimensional setting, where a main idea is to introduce a suitable space splitting corresponding to a small parameter, which controls additional fast contributions of the slow variable. These results require a strong convergence in $L^\infty(0,T;H^2(Ω))$, which is the subtle analytical issue in fast reaction problems in comparison to previous works. By considering a nonlinear fast reversible reaction in one dimension, we successfully prove this convergence and therefore obtain the slow manifold for a PDE with fast reactions. Moreover, the obtained convergence also shows the influence of the cross diffusion term and illuminates the role of the initial layer. Explicit approximations of the slow manifold are also carried out in the case of linear systems.

Slow Manifolds for PDE with Fast Reactions and Small Cross Diffusion

TL;DR

The paper develops a rigorous slow-manifold theory for fast-slow PDEs with small cross diffusion by extending Fenichel-type ideas to infinite dimensions. It combines geometric and analytic methods, proving the existence of attracting slow manifolds that converge to a critical manifold under a spectral-gap framework and small cross-diffusion, with strong convergence in shown in 1D. The Lyapunov-Perron approach yields explicit slow-manifold graphs, differentiability, and exponential attraction of trajectories, and the slow flow is shown to approximate the reduced slow dynamics. In the linear case, explicit expressions and Galerkin-based approximations for slow manifolds are derived via Fourier modes, illustrating the geometry and enabling practical approximations. Together, these results provide a rigorous reduction framework for fast-reaction PDEs with cross diffusion and clarify the roles of initial layers and cross-diffusion in the slow dynamics.

Abstract

Multiple time scales problems are investigated by combining geometrical and analytical approaches. More precisely, for fast-slow reaction-diffusion systems, we first prove the existence of slow manifolds for the abstract problem under the assumption that cross diffusion is small. This is done by extending the Fenichel theory to an infinite dimensional setting, where a main idea is to introduce a suitable space splitting corresponding to a small parameter, which controls additional fast contributions of the slow variable. These results require a strong convergence in , which is the subtle analytical issue in fast reaction problems in comparison to previous works. By considering a nonlinear fast reversible reaction in one dimension, we successfully prove this convergence and therefore obtain the slow manifold for a PDE with fast reactions. Moreover, the obtained convergence also shows the influence of the cross diffusion term and illuminates the role of the initial layer. Explicit approximations of the slow manifold are also carried out in the case of linear systems.

Paper Structure

This paper contains 22 sections, 24 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Problem Setting and Main results
  3. Abstract Existence Theory
  4. Slow Manifolds for Fast Reversible Reactions
  5. Existence of Slow Manifolds
  6. Lyapunov-Perron Operator
  7. Distance to the Critical Manifold
  8. Differentiability of the Slow Manifold
  9. Attraction of Trajectories
  10. Approximation of the slow flow
  11. Slow Manifolds for Fast Chemical Reaction systems
  12. Sketch of the Proof
  13. Uniform-in- Bounds and Fast Reaction Limit
  14. Initial Layer
  15. Convergence in $L^{\infty}(0,T;L^2(I))$
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let the assumptions of S1--S3, A1--A2, N1--N4, X1--X3, G and C hold. Then, there exists a family of slow manifolds $\{S_\varepsilon\}_{\varepsilon\in (0,\varepsilon_0]} \subset X_1\times X_1$ of system general system in the sense of Definition Def.slow.manifold.

Theorems & Definitions (69)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.5
  • Remark 3.1
  • Remark 3.2
  • ...and 59 more