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

Claude-Michel Brauner (1) , Josephus Hulshof (2) , Luca Lorenzi (3) , G.I. Sivashinsky (4)

1 IMB - Institut de Mathématiques de Bordeaux
2 Department of Computer Science [Amsterdam]
3 Dipartimento di Matematica
4 School of Mathematical Sciences

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 $\varepsilon$, we rescale both the dependent and independent variables and prove rigourously the convergence to the solution of the Kuramoto-Sivashinsky equation as $\varepsilon \to 0$.

Mots clés: Front dynamics; stability; Kuramoto-Sivashinsky equation; fully nonlinear equations; pseudo-differential operators

Domaines: Equations aux dérivées partielles [math.AP]

Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, G.I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. 2009. ⟨hal-00426859⟩