# Interfaces and Free Boundaries

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**Volume 7, Issue 2, 2005, pp. 131–146**

**DOI: 10.4171/IFB/117**

Published online: 2005-06-30

Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the Q-S equation

Claude-Michel Brauner^{[1]}, Michael L. Frankel

^{[2]}, Josephus Hulshof

^{[3]}and Gregory I. Sivashinsky

^{[4]}(1) Université de Bordeaux I, Talence, France

(2) Indiana University Purdue University Indianapolis, USA

(3) Vrije Universiteit, Amsterdam, Netherlands

(4) Tel-Aviv University, Israel

We consider a quasi-steady version of the $\kappa-\theta$ model of flame front dynamics introduced in \cite{FGS03} . In this case the mathematical model reduces to a single integro-differential equation. We show that a periodic problem for the latter equation is globally well-posed in Sobolev spaces of periodic functions. We prove that near the instability threshold the solutions of the equation are arbitrarily close to these of the Kuramoto-Sivashinsky equation on a fixed time interval if the evolution starts from close configurations. We present numerical simulations that illustrate the theoretical results, and also demonstrate the ability of the quasi-steady equation to generate chaotic cellular dynamics.

*Keywords: *Combustion, premixed flame, Kuramoto-Sivashinsky equation, front dynamics, Sobolev spaces of periodic functions, stretched coordinates

Brauner Claude-Michel, Frankel Michael, Hulshof Josephus, Sivashinsky Gregory: Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the Q-S equation. *Interfaces Free Bound.* 7 (2005), 131-146. doi: 10.4171/IFB/117