Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1415 - 1446, Volume 27, Issue 4, August 2010      doi:10.3934/dcds.2010.27.1415

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Claude-Michel Brauner - Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence Cedex, France (email)
Josephus Hulshof - Faculty of Sciences – Mathematics and Computer Science division, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081HV Amsterdam, Netherlands (email)
Luca Lorenzi - Dipartimento di Matematica, Università degli Studi di Parma, Viale G. P. Usberti 53/A, 43124 Parma, Italy (email)
Gregory I. Sivashinsky - School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email)

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$.

Keywords:  Front dynamics, stability, Kuramoto-Sivashinsky equation, fully nonlinear equations, pseudo-differential operators.
Mathematics Subject Classification:  Primary: 35K55; Secondary: 35B25, 35B35, 80A25.

Received: October 2009;      Revised: February 2010;      Available Online: March 2010.