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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

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