Stability and attractors for the quasi-steady equation of cellular flames
type: Item
instance of: scholarly article
title: Stability and attractors for the quasi-steady equation of cellular flames
zbMATH Open document ID: 1109.35097
DOI: 10.4171/IFB/145
published in: Interfaces and Free Boundaries
publication date: 2006-11-21T00:00:00Z
review text: Summary: We continue to study a simple integro-differential equation: the quasi-steady equation \[ \Phi_t+ \tfrac12 \Phi_+x^2= (I-\alpha A)\Phi_{xx}, \qquad A:= (I-\partial_x^2)^{-1}, \] of flame front dynamics. This equation is dynamically similar to the Kuramoto-Sivashinsky (KS) equation. In [\textit{M. L. Frankel, P. V. Gordon} and \textit{G. I. Sivashinsky} [On disintegration of near-limit cellular flames, Phys. Lett. A 310, 389--392 (2003)], where it was introduced, its well-posedness and proximity for finite time intervals to the KS equation in Sobolev spaces of periodic functions were established. Here we demonstrate that QS possesses a universal absorbing set, and a compact attractor. Furthermore we show that the attractor is of a finite Hausdorff dimension, and give an estimate on it. We discuss relationship with the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 80A25
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L30
zbMATH DE Number: 5074423
zbMATH Keywords: Kuramoto-Sivashinsky equation
zbMATH Keywords: absorbing set
zbMATH Keywords: attractor
zbMATH Keywords: Hausdorff dimension
author: C.-M. Brauner
author: M. Frankel
author: Josephus Hulshof
author: Victor Roytburd
MaRDI profile type: Publication
cites work: Metastability in a flame front evolution equation
cites work: Weakly nonlinear asymptotics of the \(k-\theta\) model of cellular flames: the Q-S equation
cites work: Instabilities in a two-dimensional combustion model with free boundary
cites work: A global attracting set for the Kuramoto-Sivashinsky equation
cites work: New bounds for the Kuramoto‐Sivashinsky equation
full work available at URL: https://doi.org/10.4171/ifb/145
OpenAlex ID: W1990502362
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description: scientific article; zbMATH DE number 5074423