Majda and ZND models for detonation: nonlinear stability vs. formation of singularities

TL;DR

This paper compares the nonlinear stability and singularity formation in Majda and ZND models for detonation, showing singularities in ZND and stability in Majda models under certain conditions.

Contribution

It provides new insights into the stability and singularity formation in Majda and ZND detonation models using energy estimates and weighted norms.

Findings

ZND model exhibits singularity formation downstream.

Majda model remains stable and free of singularities near small-amplitude waves.

Energy estimates imply asymptotic orbital stability for small-amplitude waves in Majda model.

Abstract

For the ZND model, we show also singularity formation on the downstream side for arbitrary exponentially-growing weighted norms. For the Majda model on the other hand, we establish for appropriate such weighted norms a set of energy estimates implying not only non-formation of singularities near waves of arbitrary amplitude but also full asymptotic orbital stability for small-amplitude ones.