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

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November 2010, 27(4): 1415-1446. doi: 10.3934/dcds.2010.27.1415

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

Claude-Michel Brauner ^1, , Josephus Hulshof ^2, , Luca Lorenzi ^3, and Gregory I. Sivashinsky ^4,

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, Italy
4.	School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received October 2009 Revised February 2010 Published March 2010

Abstract
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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$.

Keywords: Front dynamics, stability, Kuramoto-Sivashinsky equation, pseudo-differential operators., fully nonlinear equations.

Mathematics Subject Classification: Primary: 35K55; Secondary: 35B25, 35B35, 80A2.

Citation: Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1415-1446. doi: 10.3934/dcds.2010.27.1415

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