通道中粘性激波的O（2）Hopf分岔
O(2) Hopf bifurcation of viscous shock waves in a channel
摘要：
推广Texier和Zumbrun的工作，在半线性非反射对称情形下，研究了一类含热粘弹性方程的拟线性双曲抛物方程组在通道中粘性激波的O（2）横向Hopf分支，或称“\cellular不稳定性”.主要困难是（i）通过适当的双曲抛物能量估计获得时间-T解算子的Fr'echet可微性，（ii）在没有任何中心流形约化的情况下处理O（2）对称性（由于缺乏频谱间隙）或（由于非标准的准线性双曲-抛物形式）在时间空间上通过空间动力学处理的必要框架-周期函数，这是解决这个问题的两种标准方法后一个问题通过时间-T映射的李雅普诺夫{施密特}约化来解决，得到一个具有O（2）加上近似S1对称性的四维问题，我们使用直接的隐函数定理参数来处理\by手。前者通过平衡拉格朗日坐标系中的信息与增广系统中的信息来处理。有趣的是，这一论点并不适用于气体动力学或磁流体力学（MHD），因为拉格朗日对称性的无限维族对应于任意体积保持的非线性同态下的不变性。
英文摘要：
Extending work of Texier and Zumbrun in the semilinear non-re ection symmetric case, we study O(2) transverse Hopf bifurcation, or \cellular instability," of viscous shock waves in a channel, for a class of quasilinear hyperbolic{parabolic systems including the equations of thermoviscoelasticity. The main difficulties are to (i) obtain Fr'echet differentiability of the time-T solution operator by appropriate hyperbolic{parabolic energy estimates, and (ii) handle O(2) symmetry in the absence of either center manifold reduction (due to lack of spectral gap) or (due to nonstandard quasilinear hyperbolic-parabolic form) the requisite framework for treatment by spatial dynamics on the space of time-periodic functions, the two standard treatments for this problem. The latter issue is resolved by Lyapunov{Schmidt reduction of the time-T map, yielding a four-dimensional problem with O(2) plus approximate S1 symmetry, which we treat \by hand" using direct Implicit Function Theorem arguments. The former is treated by balancing information obtained in Lagrangian coordinates with that from an augmented system. Interestingly, this argument does not apply to gas dynamics or magnetohydrodynamics (MHD), due to the infinite-dimensional family of Lagrangian symmetries corresponding to invariance under arbitrary volume-preserving diffeomorphisms.
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