Existence and Stability of Curved Multidimensional Detonation Fronts

• Published in: Indiana University mathematics journal 2007-01, Vol.56 (3), p.1405-1461
• Main Authors: Costanzino, N., Jenssen, H.K., Lyng, G., Williams, Mark
• Format: Article
• Language: English
• Subjects: Boundary conditions; Coefficients; Determinants; Detonation; Eigenvalues; Eigenvectors; Exact sciences and technology; Gases; General mathematics; General, history and biography; High frequencies; Limit theorems; Mathematical analysis; Mathematics; Partial differential equations; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Shock waves
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.2007.56.2972

The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck in [9]. He used a normal mode analysis to define a stability function V(λ, η), whose zeros in ℜλ > 0 correspond to multidimensional perturbations of a steady planar profile that grow exponentially with time. In [11] 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 nonlinear stability of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of V(λ, η), 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, ΔZND(λ̂, η̂), which turns out to coincide with the high frequency limit of V(λ, η)/|λ, η| in ℜλ̂ > 0. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where ΔZND(λ̂, η̂) is bounded away from zero in ℜλ̂ > 0. We also revisit the argument of [11] in order to simplify and complete some of the analysis in the proof of the main result there.