Bifurcating fronts for the Taylor-Couette problem in infinite cylinders
Bifurcating fronts for the Taylor-Couette problem in infinite cylinders
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无限圆柱体中 Taylor-Couette 问题的分叉前沿
DOI：
10.1007/s000330050142
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
G. Schneider
中科院分区：
文献类型：
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作者：
M. Hǎrǎguş;G. Schneider
我们证明了无限圆柱上弱不稳定Taylor-Couette问题分岔锋的存在性。这些锋面连接了一个固定的分岔模式，这里是泰勒涡，这里是平凡的基态，这里是库埃特流。为了证明存在性结果，我们改进了Collet和Eckmann（1986）和Eckmann和Wayne（1991）在建立Swift-Hohenberg方程分岔锋存在性时已经使用的方法。存在性证明基于空间动力学和中心流形理论。中心流形理论应用的难点之一是在分岔参数消失的虚轴上存在无穷多个特征值。但是，如果分岔参数增加，那么有限维的减少是可能的，因为特征值以不同的速度离开虚轴。与以前的工作相反，我们必须使用规范化方法和非标准截止函数来获得足够大的中心流形，以包含分岔锋。
We show the existence of bifurcating fronts for the weakly unstable Taylor—Couette problem in an infinite cylinder. These fronts connect a stationary bifurcating pattern, here the Taylor vortices, with the trivial ground state, here the Couette flow. In order to show the existence result we improve a method which was already used in establishing the existence of bifurcating fronts for the Swift—Hohenberg equation by Collet and Eckmann, 1986, and by Eckmann and Wayne, 1991. The existence proof is based on spatial dynamics and center manifold theory. One of the difficulties in applying center manifold theory comes from an infinite number of eigenvalues on the imaginary axis for vanishing bifurcation parameter. But nevertheless, a finite dimensional reduction is possible, since the eigenvalues leave the imaginary axis with different velocities, if the bifurcation parameter is increased. In contrast to previous work we have to use normalform methods and a non–standard cut–off function to obtain a center manifold which is large enough to contain the bifurcating fronts.