On a strongly damped wave equation for the flame front
type: Item
instance of: scholarly article
title: On a strongly damped wave equation for the flame front
zbMATH Open document ID: 1221.35252
author: C.-M. Brauner
author: Luca Lorenzi
author: G. Sivashinsky
author: Chuanju Xu
published in: Chinese Annals of Mathematics. Series B
publication date: 2011-01-25T00:00:00Z
review text: This article is concerned with a nonlinear one-dimensional wave equation (second order in time, fourth order in space) including a damping operator. The equation studied is shown to arise heuristically from the two-dimensional model of near equidiffusive flames and describes the evolution of the perturbation of the planar front of a moving flame. Solutions are sought for periodic boundary conditions. The authors investigate the stability of the null solution with respect to the relevant model parameter \(\alpha\) related to the Lewis number. This analysis is based on the analyticity of the underlying linear semigroup of the linearized equation. Next it is shown that for \(\alpha\) close to \(1\), solutions of the nonlinear wave equation are close to solutions of the Kuramoto-Sivashinsky equation for the same initial data up to renormalization. This result holds true over a fixed time scale. Finally, numerical experiments illustrate and confirm the behavior of solutions in the asymptotic limit. An additional appendix gives a systematic derivation of the linear part of the governing equation.
Mathematics Subject Classification ID: 35L76
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 5841064
zbMATH Keywords: front dynamics
zbMATH Keywords: Kuramoto-Sivashinsky equation
zbMATH Keywords: analytic semigroups
zbMATH Keywords: spectral method
zbMATH Keywords: periodic boundary conditions
MaRDI profile type: Publication
full work available at URL: s11401-010-0616-1
OpenAlex ID: W2040266540
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DOI: 10.1007/S11401-010-0616-1
description: scientific article; zbMATH DE number 5841064