Affordable Access

Galloping instability of viscous shock waves

Authors
  • Texier, Benjamin
  • Zumbrun, Kevin
Type
Preprint
Publication Date
Feb 07, 2007
Submission Date
Sep 12, 2006
Identifiers
arXiv ID: math/0609331
Source
arXiv
License
Unknown
External links

Abstract

Motivated by physical and numerical observations of time oscillatory ``galloping'', ``spinning'', and ``cellular'' instabilities of detonation waves, we study Poincar\'e--Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional center manifold. We overcome this by direct Lyapunov--Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap. The key computation is a space-time stability estimate on the transverse linearized solution operator reminiscent of Duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves.

Report this publication

Statistics

Seen <100 times