Galloping instability of viscous shock waves

TL;DR

This paper investigates the complex bifurcation behavior of viscous shock waves, particularly focusing on oscillatory instabilities like galloping, using advanced mathematical techniques to handle spectral challenges.

Contribution

It introduces a novel approach to analyze Poincaré--Hopf bifurcation in viscous conservation laws without relying on a spectral gap, employing direct Lyapunov--Schmidt reduction and detailed stability estimates.

Findings

Identification of conditions leading to galloping instability.

Development of a new analytical framework for bifurcation analysis without spectral gap.

Establishment of space-time stability estimates for linearized operators.

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.