Existence and stability of curved multidimensional detonation fronts

The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck. 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 planar profile that grow exponentially with time. Later, he was able to prove that for large classes of steady ZND profiles, unstable zeros of $V$ always exist in the high-frequency regime, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense (later) defined by Majda; subsequent numerical work has shown that unstable zeros usually exist in the medium frequency regime as well. In this paper we begin a rigorous study of the implications for non-linear stability of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of $V$, one can prove the finite (but arbitrarily long) time existence of slightly curved, non-steady, multidimensional detonation fronts for ideal polytropic gases in both the ZND and Chapman-Jouguet models. In the ZND case we show that this nonlinear stability problem is actually governed by a different stability function, $\Delta_{\mathrm{ZND}}((\hat\lambdaλ, \hat\eta)$, which turns out to coincide with the high frequency limit of a scaled version of $V (\lambda, \eta)$ in $\Re\hat\lambda > 0$. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where $\Delta_{\mathrm{ZND}}(\hat\lambda,\hat\eta)$ is bounded away from zero in $\Re\hat\lambda > 0$. We also revisit Erpenbeck's arguments in order to simplify and complete some of the analysis in the proof of his main result.

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Recommended citation: Nicola Costanzino, H. Kristian Jenssen, Gregory Lyng, & Mark Williams, Existence and stability of curved multidimensional detonation fronts. Indiana University Mathematics Journal, 56 (2007): 1405–1462.