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Slow-fast system in Rosales-Majda combustion model with fractional order kinetics

Claude-Michel Brauner, Jinlong Jing, Robert Roussarie

Abstract

We consider traveling wave solutions of a one-dimensional model for detonation waves derived by Rosales and Majda, when the reaction order $α$ belongs to $[0,1)$. The chemical kinetics is a simplified Arrhenius law or a Heaviside function. The model in the reaction zone is a slow-fast dynamical system for a vector representing temperature and mass fraction, which depends on the velocity $c$ and small viscosity $β$. Our goal in this paper is to study the bifurcation diagram in the $(β,c)$ parameter space and identify the nature of the trajectories corresponding to viscous shock waves. The demonstrations are based on a variety of techniques including the Poincaré-Bendixson theorem and the Fenichel theory. Theoretical results are confirmed by numerical computations.

Slow-fast system in Rosales-Majda combustion model with fractional order kinetics

Abstract

We consider traveling wave solutions of a one-dimensional model for detonation waves derived by Rosales and Majda, when the reaction order belongs to . The chemical kinetics is a simplified Arrhenius law or a Heaviside function. The model in the reaction zone is a slow-fast dynamical system for a vector representing temperature and mass fraction, which depends on the velocity and small viscosity . Our goal in this paper is to study the bifurcation diagram in the parameter space and identify the nature of the trajectories corresponding to viscous shock waves. The demonstrations are based on a variety of techniques including the Poincaré-Bendixson theorem and the Fenichel theory. Theoretical results are confirmed by numerical computations.
Paper Structure (26 sections, 21 theorems, 68 equations, 25 figures)

This paper contains 26 sections, 21 theorems, 68 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Background results
  4. Organization of the paper
  5. Phase portraits
  6. Phase portrait of $\tilde{X}^U$
  7. Phase portrait of $\tilde{X}$
  8. The Chapman-Jouguet case
  9. Existence of a solution in function of $(\beta,c)$
  10. Property for $c$ large
  11. Properties for $\beta$ near $0$ and $+\infty$.
  12. Slow analysis for $\beta\rightarrow 0_{+}$
  13. Limit for $\beta\rightarrow +\infty$
  14. The transition map $Z(T,\beta,c)$
  15. Rotating property
  16. ...and 11 more sections

Key Result

Theorem 1.2

(i) There exists a function $\beta_0(c) :[c^\mathit{cj},+\infty)\rightarrow {\mathbb{R}}^+,$ of class $[\frac{1}{1-\alpha}],$ such that there is a strong reactive solution of special type for $\beta=\beta_0(c)$; (ii) there exists a function $\beta_1(c) :[c^\mathit{cj},c_{\ast})\rightarrow {\mathbb{R (iv) for $c>c^\mathit{cj}, 0<\beta<\beta_0(c)$, there is a strong reactive solution with a bump; fo

Figures (25)

  • Figure 1: This figure provides a qualitative representation of the bifurcation diagram. Note that the only unproven point in this paper concerns the monotonicity of the branch of reactive waves of special type. Thus far, it has only been proven that this branch is locally the graph of $\beta$ as a function of $c$.
  • Figure 2: Phase portrait of $\tilde{X}^U$ for $c>c^\mathit{cj}$
  • Figure 3: Phase portrait of $\tilde{X}$ for $c>c^\mathit{cj}$
  • Figure 4: Phase portraits for $c=c^\mathit{cj}$
  • Figure 5: The limit vector field for $\beta=0$
  • ...and 20 more figures

Theorems & Definitions (39)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Claim 2.1
  • Claim 2.2
  • Claim 2.3
  • Claim 2.4
  • Claim 2.5
  • ...and 29 more