The Erpenbeck high frequency instability theorem for ZND detonations

TL;DR

This paper rigorously analyzes the spectral stability of ZND detonations, confirming the existence of high-frequency instabilities predicted by Erpenbeck, and provides detailed asymptotic behavior of solutions to related ODE systems.

Contribution

It offers the first rigorous proof of the high-frequency instability phenomena in ZND detonations, extending and simplifying Erpenbeck's earlier theoretical insights.

Findings

Confirmed existence of unstable zeros of the stability function at high frequencies.

Provided detailed asymptotic analysis of solutions to the governing ODEs.

Extended the theoretical framework for spectral stability of detonations.

Abstract

The rigorous study of spectral stability for strong detonations was begun by J.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-D\"oring (ZND) model, which assumes a finite reaction rate but ignores effects like viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function $V(\tau,\eps)$ whose zeros in $\Re \tau>0$ correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper [Er3] he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of $V$ exist for perturbations of sufficiently large transverse wavenumber $\eps$, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly twenty years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of $V(\tau,\eps)$, the paper \cite{Er3} remains the only work we know of that presents a detailed and convincing theoretical argument for detecting them. The analysis in [Er3] points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in [Er3] and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as $\eps\to \infty$ of solutions to a linear system of ODEs in $x$, depending on $\eps$ and a complex frequency $\tau$ as parameters, with turning points $x_*$ on the half-line $[0,\infty)$.