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Stability of Viscous Weak Detonation Waves for Majda’s Model

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Abstract

Continuing the program initiated by Humpherys et al. (Physica D 259:63–80, 2013) for strong detonation waves, we use a combination of analytical and numerical Evans-function techniques to analyze the spectral stability of weak detonation waves in a simplified model for gas-dynamical combustion. Combining these new spectral stability results with the pointwise Green function analysis of Lyng et al. (J Differ Equ 233(2):654–698, 2007), we conclude that these waves are nonlinearly stable. The principal novelty of this analysis is the treatment of weak detonation waves. In contrast to the case of strong detonation waves, weak detonation waves are under compressive and the stability of these waves is delicate and has not been treated by standard weighted-norm techniques. The present analysis thus provides a case study illustrating the flexibility and power of the Evans-function-based approach to stability. As in the case of strong detonations, we find that all tested waves are spectrally stable, hence nonlinearly stable.

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Notes

  1. The CJ detonation is sonic behind the front.

  2. In the introduction of [18] there is a careful description of a number of the many variations of the Majda model (scalar balance law coupled to reaction equation) that have appeared in the literature since Majda’s original paper. We call all of these models, “Majda models.”

  3. The precise form of \(L\) can be seen in (3.1).

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Acknowledgments

Humpherys was partially supported by NSF Grant DMS-0847074 (CAREER). Lyng was partially supported by NSF Grant DMS-0845127 (CAREER). Zumbrun was partially supported by NSF Grant DMS-0801745.

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Jeffrey Hendricks & Jeffrey Humpherys

  2. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA

    Gregory Lyng

  3. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Kevin Zumbrun

Authors
  1. Jeffrey Hendricks
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  2. Jeffrey Humpherys
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  3. Gregory Lyng
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  4. Kevin Zumbrun
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Correspondence to Jeffrey Humpherys.

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Hendricks, J., Humpherys, J., Lyng, G. et al. Stability of Viscous Weak Detonation Waves for Majda’s Model. J Dyn Diff Equat 27, 237–260 (2015). https://doi.org/10.1007/s10884-015-9440-3

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  • DOI: https://doi.org/10.1007/s10884-015-9440-3

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