A fully nonlinear equation for the flame front in a quasi-steadycombustion model

Claude-Michel Brauner; Josephus Hulshof; Luca Lorenzi; Gregory I. Sivashinsky

doi:10.3934/dcds.2010.27.1415

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2010, Volume 27, Issue 4: 1415-1446. Doi: 10.3934/dcds.2010.27.1415

This issue Previous Article Singular perturbation systems with stochastic forcing and the renormalization group method Next Article On the initial-value problem to the Degasperis-Procesi equation with linear dispersion

A fully nonlinear equation for the flame front in a quasi-steady combustion model

1.
Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence Cedex
2.
Faculty of Sciences – Mathematics and Computer Science division, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081HV Amsterdam
3.
Dipartimento di Matematica, Università degli Studi di Parma, Viale G. P. Usberti 53/A, 43124 Parma
4.
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978

Received: September 30, 2009

Revised: January 31, 2010

Published: November 2010

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Abstract

We revisit the Near Equidiffusional Flames (NEF) model introduced by Matkowsky and Sivashinsky in 1979 and consider a simplified, quasi-steady version of it. This simplification allows, near the planar front, an explicit derivation of the front equation. The latter is a pseudodifferential fully nonlinear parabolic equation of the fourth-order. First, we study the (orbital) stability of the null solution. Second, introducing a parameter ε, we rescale both the dependent and independent variables and prove rigourously the convergence to the solution of the Kuramoto-Sivashinsky equation as ε $ \to 0$.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35B25, 35B35, 80A25.

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