Existence of bifurcating quasipatterns in steady Bénard-Rayleigh convection
type: Item
instance of: scholarly article
title: Existence of bifurcating quasipatterns in steady Bénard-Rayleigh convection
zbMATH Open document ID: 1428.35032
author: Gérard Iooss
author: Boele Braaksma
published in: Archive for Rational Mechanics and Analysis
publication date: 2019-01-24T00:00:00Z
review text: The authors prove the existence of bifurcating quasipatterns in the steady Bénard-Rayleigh convection problem. The proof adapts a similar method, which was used to establish quasipatterns for the Swift-Hohenberg equation. The solution is approximated by a truncated power series expansion, which was formally obtained in a previous paper, but which diverges in general. After short sections introducing the physical problem, the quasilattices and the corresponding function spaces and operators, the properties of the convection problem for different boundary conditions are worked out in detail. Next the critical parameter value of the system and the corresponding wave vector are obtained and the initial series expansion for the solution is carried out. At fourth order of the series expansion the small divisor problem occurs and the authors have to use sophisticated methods to obtain valid solutions. They have to overcome the loss of self-adjointness of the linearization and to obtain a set of ``good parameter'' values, for which the ``range equation'' is solvable. Finally they obtain the solution by solving the bifurcation equation for ``good parameter'' values.
reviewed by: Alois Steindl
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76E06
Mathematics Subject Classification ID: 76E15
Mathematics Subject Classification ID: 76E30
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 7006095
zbMATH Keywords: wave vector
zbMATH Keywords: rotational symmetry
zbMATH Keywords: Nash-Moser process
zbMATH Keywords: small divisor problem
zbMATH Keywords: truncated power series expansion
MaRDI profile type: Publication
full work available at URL: s00205-018-1313-6
OpenAlex ID: W2893469872
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DOI: 10.1007/S00205-018-1313-6
description: scientific article; zbMATH DE number 7006095