A stability index for detonation waves in Majda’s model for reacting flow

Abstract

Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [SIAM J. Math. Anal. 32 (2001) 929; Commun. Pure Appl. Math. 51 (7) (1998) 797], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier–Stokes equations of reacting flow in [G. Lyng, One dimensional stability of detonation waves, Doctoral Thesis, Indiana University, 2002; G. Lyng, K. Zumbrun, Stability of detonation waves, Preprint, 2003]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [SIAM J. Sci. Statist. Comput. 7u (1986) 1059] and analytical results of [Commun. Math. Phys. 204 (3) (1999) 551; Commun. Math. Phys. 202 (3) (1999) 547] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, “bump-type” ignition function) deflagration profiles, as discussed in [SIAM J. Math. Anal. 24 (1993) 968; SIAM J. Appl. Math. 55 (1995) 175] for the full equations.