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

TL;DR

This paper derives a fully nonlinear, fourth-order pseudodifferential equation for flame front dynamics in a simplified combustion model and proves convergence to the Kuramoto-Sivashinsky equation as a parameter approaches zero.

Contribution

It introduces a simplified quasi-steady NEF model, explicitly derives the nonlinear front equation, and establishes convergence to the Kuramoto-Sivashinsky equation.

Findings

Derived a fully nonlinear pseudodifferential equation for the flame front.

Proved stability of the null solution.

Established convergence to the Kuramoto-Sivashinsky equation as a parameter tends to zero.

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