Stability of viscous weak detonation waves for Majda's model

TL;DR

This paper investigates the spectral and nonlinear stability of weak detonation waves in a simplified combustion model using Evans-function techniques, highlighting the method's effectiveness for delicate undercompressive waves.

Contribution

It extends Evans-function stability analysis to weak detonation waves, a case previously unaddressed due to their delicate stability properties.

Findings

All tested weak detonation waves are spectrally stable.

Weak detonation waves are nonlinearly stable.

The Evans-function approach is effective for undercompressive waves.

Abstract

Continuing the program initiated by Humpherys, Lyng, & Zumbrun [17] for strong detonation waves, we use a combination of analytical and numerical Evans-function techniques to analyze the spectral stability of weak detonation waves in a simplified model for gas-dynamical combustion. Combining these new spectral stability results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [22], we conclude that these waves are nonlinearly stable. The principal novelty of this analysis is the treatment of weak detonation waves. In contrast to the case of strong detonation waves, weak detonation waves are undercompressive and the stability of these waves is delicate and has not been treated by standard weighted-norm techniques. The present analysis thus provides a case study illustrating the flexibility and power of the Evans-function- based approach to stability. As in the case of strong detonations, we find that all tested waves are spectrally stable, hence nonlinearly stable.