Boundary conditions as symmetry constraints. (English) Zbl 0735.35007
Singularity theory and its applications. Pt. II: Singularities, bifurcations and dynamics, Proc. Symp., Warwick/UK 1988-89, Lect. Notes Math. 1463, 63-79 (1991).
Summary: [For the entire collection see Zbl 0723.00029.]
H. Fujii, M. Mimura and Y. Nishiura [Physica D 5, 1-42 (1982)] and D. Armbruster and G. Dangelmayr [Contemp. Math. 56, 53-68 (1986; Zbl 0602.58039) and Math. Proc. Camb. Philos. Soc. 101, 167-192 (1987; Zbl 0633.58011)] have observed that reaction-diffusion equations on the interval with Neumann boundary conditions can be viewed as restrictions of similar problems with periodic boundary conditions; and that this extension reveals the presence of additional symmetry constraints which affect the generic bifurcation phenomena. We show that, more generally, similar observations hold for multi-dimensional rectangular domains with either Neumann or Dirichlet boundary conditions, and analyse the group-theoretic restrictions that this structure imposes upon bifurcations. We discuss a number of examples of these phenomena that arise in applications, including the Taylor-Couette experiment, Rayleigh-Bénard convection, and the Faraday experiment.
H. Fujii, M. Mimura and Y. Nishiura [Physica D 5, 1-42 (1982)] and D. Armbruster and G. Dangelmayr [Contemp. Math. 56, 53-68 (1986; Zbl 0602.58039) and Math. Proc. Camb. Philos. Soc. 101, 167-192 (1987; Zbl 0633.58011)] have observed that reaction-diffusion equations on the interval with Neumann boundary conditions can be viewed as restrictions of similar problems with periodic boundary conditions; and that this extension reveals the presence of additional symmetry constraints which affect the generic bifurcation phenomena. We show that, more generally, similar observations hold for multi-dimensional rectangular domains with either Neumann or Dirichlet boundary conditions, and analyse the group-theoretic restrictions that this structure imposes upon bifurcations. We discuss a number of examples of these phenomena that arise in applications, including the Taylor-Couette experiment, Rayleigh-Bénard convection, and the Faraday experiment.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 35Q35 | PDEs in connection with fluid mechanics |
