M. Golubitsky
The Benard problem, symmetry and the lattice of isotropy subgroups
In: Bifurcation Theory, Mechanics and Physics. (C.P. Bruter et al, eds.) D. Reidel Publishing Co., 1983, 225-256.
In this lecture I would like to describe some of the effects that are forced on steady state bifurcation problems by the existence of a group of symmetries. I shall discuss this relationship between symmetry and bifurcation by describing several mathematical problems which are motivated by the Benard problem.
The Benard problem in its simplest form is the study of the transition from pure conduction to convective motion in a contained fluid heated on (part of) its boundary. The model equations which lie behind the analysis are the Navier-Stokes equations in the Boussinesq approximation. The purpose of this exposition is to indicate the type of information that can be obtained through the use of singularity theory and group theory. For this reason the exact form of the Boussinesq equations is not needed and they will not be presented. The interested reader can consult the paper by Fauvre and Libchaber [1983] in this volume or the extensive and very interesting work of Busse [1962, 1975, 1978].
The specific results outlined below rely for their proofs on the machinery of group theory and singularity theory plus extensive calculations. What is remarkable is that after this effort has been expended the final answer has a delightful and compelling organization based on the lattice of isotropy subgroups of the given group representation. It is our intention to emphasize this relationship throughout.
The paper is divided into four sections. The first three sections concern specific realizations of the Benard problem while the last section presents certain general results concerning bifurcation with symmetry.