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Existence and uniqueness of traveling fronts for a free interface model of autoignition in reactive jets

Mingxin Ma, Peter V. Gordon, Robert Roussarie, Peipei Shang, Claude-Michel Brauner

Abstract

In this paper we consider a one-dimensional reaction-diffusion model with piecewise continuous reaction term that describes propagation of autoignition fronts in reactive co-flow jets in a certain parametric regime. The model is reduced to a free boundary problem with two interfaces. It is shown that this problem admits permanent traveling front solution which is unique up to translations. The result is obtained using dynamical system approach employing Stable Manifold Theorem and the Melnikov integral as the main tools.

Existence and uniqueness of traveling fronts for a free interface model of autoignition in reactive jets

Abstract

In this paper we consider a one-dimensional reaction-diffusion model with piecewise continuous reaction term that describes propagation of autoignition fronts in reactive co-flow jets in a certain parametric regime. The model is reduced to a free boundary problem with two interfaces. It is shown that this problem admits permanent traveling front solution which is unique up to translations. The result is obtained using dynamical system approach employing Stable Manifold Theorem and the Melnikov integral as the main tools.

Paper Structure

This paper contains 10 sections, 11 theorems, 101 equations, 6 figures.

Table of Contents

  1. introduction
  2. Main results and organization of the paper
  3. Proof of Theorem \ref{['theorem2']}
  4. Preliminaries
  5. Application of the Implicit Function Theorem
  6. Dynamical system and proof of Theorem \ref{['theorem3']}
  7. Existence of the function $v_c(u)$
  8. Computation of $\partial_cv(u,c)$
  9. The function $\psi(R)$
  10. Proof of Theorem \ref{['t:1']}

Key Result

Theorem 1

There exists a triplet $(R^{\ast}, c^{\ast}, \theta^{\ast}) \in ({\mathbb{R}}^{+})^2 \times C^1({\mathbb{R}})$ that verifies system eq:p1-eq:p6. Moreover, $(R^{\ast}, c^{\ast})$ is unique, $\theta^*$ is increasing on $\mathbb{R}$ and is unique up to translations.

Figures (6)

  • Figure 1: Profile of the function $F(\theta)$ with $\theta_{ig}=0.1,~\theta_{hl}=0.2,~q=1, ~h=0.3.$
  • Figure 2: A sketch of a traveling front solution for problem \ref{['eq:i6']}-\ref{['eq:i8']}.
  • Figure 3: Profile of the function $\varphi(R)$. Here, $q = 1, \theta_{ig} = 0.2, \theta_{hl} = 0.25.$
  • Figure 4: Phase portraits of $X_{0}$ and $X_{c}, c>0$.
  • Figure 5: The vector field $X_c$ in $\mathcal{B}$.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 11 more