O(2) Hopf bifurcation of viscous shock waves in a channel

TL;DR

This paper investigates the O(2) transverse Hopf bifurcation of viscous shock waves in channels within quasilinear hyperbolic-parabolic systems, addressing complex symmetry and differentiability challenges with novel analytical techniques.

Contribution

It extends previous work by analyzing the Hopf bifurcation in a broader class of systems, introducing new methods to handle symmetry and differentiability without standard reductions.

Findings

Established Fréchet differentiability of the solution operator.

Developed a Lyapunov–Schmidt reduction approach for O(2) symmetry.

Identified limitations of the method for gas dynamics and MHD.

Abstract

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.