Spectral Stability of Weak Detonations in the Majda Model

• Main Author: Hendricks, Jeffrey James
• Format: Others
• Published: BYU ScholarsArchive 2013
• Subjects: Majda; Evans Function; combustion; shockwave; differential equations; Mathematics
• Online Access: https://scholarsarchive.byu.edu/etd/3626; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=4625&context=etd

Using analytical and numerical Evans-function techniques, we examine the spectral stability of weak-detonation-wave solutions of Majda's scalar model for a reacting gas mixture. We provide a proof of monotonicity of solutions. Using monotonicity we obtain a bound on possible unstable eigenvalues for weak-detonation-wave solutions that improves on the more general bound given by Humpherys, Lyng, and Zumbrun. We use a numerical approximation of the Evans function to search for possible unstable eigenvalues in the bounded region obtained by the energy estimate. For the parameter values tested, our results combined with the result of Lyng, Raoofi, Texier, and Zumbrun demonstrate that these waves are nonlinearly phase-asymptotically orbitally stable throughout the parameter space for which solutions were obtainable.