Abstract
The spatially periodic, steady-state solutions to systems of partial differential equations (PDE) are calledplanforms. There already exists a partial classification of the planforms for Euclidean equivariant systems of PDE inR 2 (see [6, 7]), In this article we attempt to give such a classification for Euclidean equivariant systems of PDE inR 3. Based on the symmetry and spatial periodicity of each planform, 59 different planforms are found.
We attempt to find the planforms on all lattices inR 3 that are forced to exist near a steady-state bifurcation from a trivial solution. The proof of our classification uses Liapunov-Schmidt reduction with symmetry (which can be used if we assume spatial periodicity of the solutions) and the Equivariant Branching Lemma. The analytical problem of finding planforms for systems of PDE is reduced to the algebraic problem of computing isotropy subgroups with one dimensional fixed point subspaces.
The Navier-Stokes equations and reaction-diffusion equations (with constant diffusion coefficients) are examples of systems of PDE that satisfy the conditions of our classifications. In this article, we show that our classification applies to the Kuramoto-Sivashinsky equation.
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References
F. H. Busse,Das Stabilitätsverhalten der Zellularkonvektion bei endlicher Amplitude Thesis, University of Munich, 1962 (Engl. Transl, by S. H. Davis, Rand Rep. LT-69-19, Rand Corporation, Santa Monica, Calif.).
F. H. Busse,Patterns of convection in spherical shells. J. Fluid Mech.72, 65–85 (1975).
E. Buzano and M. Golubitsky,Bifurcation on the hexagonal lattice and the planar Bénard problem. Phil. Trans. R Soc. Lond.A308, 617–687 (1983).
P. Chossat,Solutions avec symétrie diédrale dans les problèmes de bifurcation invariants par symétrie sphériques. C.R. Acad. Sci. Paris300 Serie I, No. 8, 639–642 (1983).
P. Chossat, R. Lauterbach and I. Melbourne,Steady-state bifurcation with 0(3)-symmetry. Arch. Rat. Mech. Anal.113, 313–376 (1990).
B. Dionne,Spatially Periodic Patterns in Two and Three Dimensions. Thesis, University of Houston, August 1990.
B. Dionne and M. Golubitsky,Planforms in two and three dimensions. ZAMP43, 36–62 (1992).
M. Golubitsky, I. N. Stewart and D. G. Schaeffer,Singularities and Groups in Bifurcation Theory: Vol. II, Appl. Math. Sci. Ser. 69, Springer-Verlag, New York 1988.
K. Kirchgässner,Exotische Lösungen des Bénardschen Problems. Math. Meth. Appl. Sci.1, 453–467 (1979).
W. Miller Jr.,Symmetry Groups and their Applications. Academic Press, New York 1972.
A. Schlüter, D. Lortz and F. Busse,On the stability of steady finite amplitude convection. J. Fluid Mech.23, 129–144 (1965).
A. Vanderbauwhede,Local Bifrucation and Symmetry. Habilitation Thesis, Rijsuniversiteit Gent 1980.
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Dionne, B. Planforms in three dimensions. Z. angew. Math. Phys. 44, 673–694 (1993). https://doi.org/10.1007/BF00948482
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DOI: https://doi.org/10.1007/BF00948482