# Preprint 2006-009

# Existence and Stability of Curved Multidimensional Detonation Fronts

## N. Costanzino, H. K. Jenssen, G. Lyng, and M. Williams

**Abstract:**
In this paper we study the spectral and nonlinear stability of strong
detonations in the two most commonly studied inviscid models of combustion,
the ZND (finite reaction rate) and Chapman-Jouguet (instantaneous reaction)
models. The rigorous study of spectral stability for ZND detonations was begun
by J.J. Erpenbeck in \cite{E1}. He used a normal mode analysis to define a
stability function $V(\lambda,\eta)$, whose zeros in $\Re\lambda>0$ correspond
to multidimensional perturbations of a steady profile that grow exponentially
with time. The profile in the reaction zone, $x\leq 0$, is given by a
nonconstant function, say $\tw(x)$, of distance from the front, so it is
impossible to give an exact, explicit formula for $V(\lambda,\eta)$ from which
the unstable zeros, if they exist, can be determined easily. Numerical
computations (e.g., \cite{LS,Sh,SS}) have shown that unstable zeros usually do
exist for perturbations in the medium frequency range ($\rho_0\leq
|\lambda,\eta|\leq R$). In a remarkable paper Erpenbeck \cite{E5} was able to
show that for large classes of steady ZND profiles, unstable zeros always
exist in the high frequency regime, that is, for $|\lambda,\eta|\geq R$ for
$R$ arbitrarily large. We shall refer to this result as Erpenbeck's
Instability Theorem. An easily computable (by hand) necessary and sufficient
condition for the existence of zeros in the low frequency range was given in
\cite{JLW}; this criterion implies, for example, that for ideal polytropic
gases there are no unstable zeros in the range $0\leq |\lambda,\eta|\leq
\rho_0$ for $\rho_0>0$ sufficiently small.