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Reduced dynamics for models of pattern formation

Erika Hausenblas, Tsiry Avisoa Randrianasolo

Abstract

The goal of this work is to analyze the long-term behavior of reaction-diffusion systems arising in two-species chemical models and to identify the minimal set of modes that determine their dynamics. The models considered include, as particular cases, the Brusselator, the Gray--Scott, and the Glycolysis models. These systems are described by coupled reaction-diffusion equations and admit a finite-dimensional representation based on a limited number of spatial Fourier modes that capture their essential reduced dynamics. The concept of determining modes, introduced in this context, is closely related to other approaches that seek finite-dimensional representations of infinite-dimensional dynamics, such as the Proper Orthogonal Decomposition and the construction of Approximate Inertial Manifolds. We prove that the dynamics of the system can be completely characterized by a finite number of low modes, since all higher modes are asymptotically determined by them, thus providing an analytical foundation for reduced dynamics in models of pattern formation.

Reduced dynamics for models of pattern formation

Abstract

The goal of this work is to analyze the long-term behavior of reaction-diffusion systems arising in two-species chemical models and to identify the minimal set of modes that determine their dynamics. The models considered include, as particular cases, the Brusselator, the Gray--Scott, and the Glycolysis models. These systems are described by coupled reaction-diffusion equations and admit a finite-dimensional representation based on a limited number of spatial Fourier modes that capture their essential reduced dynamics. The concept of determining modes, introduced in this context, is closely related to other approaches that seek finite-dimensional representations of infinite-dimensional dynamics, such as the Proper Orthogonal Decomposition and the construction of Approximate Inertial Manifolds. We prove that the dynamics of the system can be completely characterized by a finite number of low modes, since all higher modes are asymptotically determined by them, thus providing an analytical foundation for reduced dynamics in models of pattern formation.
Paper Structure (7 sections, 8 theorems, 69 equations)

This paper contains 7 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Background materials and the main statement
  3. Determining modes
  4. The main statement
  5. Preliminaries
  6. Proof of the main result
  7. Discussion

Key Result

Lemma 1

We assume that the properties P1-P6 hold for the system eq:gs. If $u_0,v_0\in L^2(D)$ then for the solution to the system eq:gs it holds that $u(t),v(t)\ge 0$ and $u(t),v(t)\in C^2(D)$ for all $t\ge 0$.

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Definition 2.1
  • Theorem 2.1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 5 more