PLANFORMS IN 3 DIMENSIONS

被引:12
作者
DIONNE, B
机构
[1] The Fields Institute for Research in Mathematical, Sciences, Waterloo, N2L 5Z5, Ontario
来源
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK | 1993年 / 44卷 / 04期
关键词
D O I
10.1007/BF00948482
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The spatially periodic, steady-state solutions to systems of partial differential equations (PDE) are called planforms. There already exists a partial classification of the planforms for Euclidean equivariant systems of PDE in R2 (see [6, 7]). In this article we attempt to give such a classification for Euclidean equivariant systems of PDE in R3. Based on the symmetry and spatial periodicity of each planform, 59 different planforms are found. We attempt to find the planforms on all lattices in R3 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.
引用
收藏
页码:673 / 694
页数:22
相关论文
共 12 条
[1]  
BUSSE FH, 1975, J FLUID MECH, V72, P65
[2]  
BUSSE FH, LT6919 RAND CORP REP
[3]   BIFURCATION ON THE HEXAGONAL LATTICE AND THE PLANAR BENARD-PROBLEM [J].
BUZANO, E ;
GOLUBITSKY, M .
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 1983, 308 (1505) :617-667
[4]  
Chossat P., 1990, ARCH RATIONAL MECH A, V113, P313
[5]  
CHOSSAT P, 1983, CR ACAD SCI I-MATH, V300, P639
[6]   PLANFORMS IN 2 AND 3 DIMENSIONS [J].
DIONNE, B ;
GOLUBITSKY, M .
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, 1992, 43 (01) :36-62
[7]  
DIONNE B, 1990, THESIS U HOUSTON
[8]  
GOLUBITSKY M, 1988, APPL MATH SCI SER, V69
[9]  
Kirchgassner K., 1979, MATH METHOD APPL SCI, V1, P453
[10]  
Miller W., 1972, SYMMETRY GROUPS THEI