Annali di Matematica Pura ed Applicata

, Volume 183, Issue 2, pp 173–239

# Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

• Claudia Lederman
• Jean-Michel Roquejoffre
• Noemi Wolanski
Article

## Abstract

This paper is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [17]. The paper [17] derives, by means of a three-scale matched asymptotics, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis Number – i.e. the ratio between thermal and molecular diffusion – to be strictly less than unity. If ε is the inverse of the – reduced – activation energy, the idea underlying the construction of [17] is that (i) the time scale of the radius motion is ε-2, and that (ii) at each time step, the solution is ε-close to a steady solution.

In this paper, we give a rigorous proof of the validity of this model under the restriction that the Lewis number is close to 1 – independently of the order of magnitude of the activation energy. The method used comprises three steps: (i) a linear stability analysis near a steady – or quasi-steady – solution, which justifies the fact that the relevant time scale is ε-2; (ii) the rigorous construction of an approximate solution; (iii) a nonlinear stability argument.

## Keywords

half derivatives high activation energies linear and nonlinear stability combustion

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## Authors and Affiliations

• Claudia Lederman
• 1
Email author
• Jean-Michel Roquejoffre
• 2
• Noemi Wolanski
• 1
1. 1.Departamento de Matemática, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
2. 2.Université Paul SabatierToulouseFrance