Mathematical analysis of sideband instabilities with application to Rayleigh-Bénard convection
type: Item
instance of: scholarly article
title: Mathematical analysis of sideband instabilities with application to Rayleigh-Bénard convection
zbMATH Open document ID: 0871.76028
DOI: 10.1007/BF02679126
author: Alexander Mielke
published in: Journal of Nonlinear Science
publication date: 1997-10-08T00:00:00Z
review text: The author proposes a novel method for the analysis of sideband instabilities of convection rolls in the Rayleigh-Bénard problem with stress-free boundary conditions. In contrast to other methods, the present theory is based on the Lyapunov-Schmidt reduction (LSR) rather than on the spatial center manifold reduction. The key observation is that the spectral problem obtained from linearization at the given steady state can also be reduced by the LSR method to a finite dimensional eigenvalue problem which contains all the necessary information on possible unstable sidebands. The author comes to the following main conclusions: (i) the method provides a simple standard approach to the nonlinear bifurcation problem as well as to the sideband instability theory; (ii) the effects seen in the lower order terms really persist for the full problem; (iii) the method can be to set up a proper functional analytic framework.
Mathematics Subject Classification ID: 76E15
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 994023
zbMATH Keywords: stress-free boundary conditions
zbMATH Keywords: Lyapunov-Schmidt reduction
zbMATH Keywords: spectral problem
zbMATH Keywords: linearization
zbMATH Keywords: finite dimensional eigenvalue problem
Wikidata QID: Q59902099
reviewed by: Ioan Pop
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02679126
OpenAlex ID: W4243337117
description: scientific article; zbMATH DE number 994023