Stability of strong detonation waves for Majda’s model with general ignition functions
Authors:
Soyeun Jung, Zhao Yang and Kevin Zumbrun
Journal:
Quart. Appl. Math. 79 (2021), 357-365
MSC (2020):
Primary 76L05
DOI:
https://doi.org/10.1090/qam/1582
Published electronically:
October 6, 2020
MathSciNet review:
4246496
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Abstract: For strong detonation waves of the inviscid Majda model, spectral stability was established by Jung and Yao for waves with step-type ignition functions, by a proof based largely on explicit knowledge of wave profiles. In the present work, we extend their stability results to strong detonation waves with more general ignition functions where explicit profiles are unknown. Our proof is based on reduction to a generalized Sturm-Liouville problem, similar to that used by Sukhtayev, Yang, and Zumbrun to study spectral stability of hydraulic shock profiles of the Saint-Venant equations.
References
- B. Barker and K. Zumbrun, A numerical stability investigation of strong ZND detonations for Majda’s model, preprint; arXiv:1011.1561.
- J. J. Erpenbeck, Stability of steady-state equilibrium detonations, Phys. Fluids 5 (1962), 604–614.
- Jerome J. Erpenbeck, Stability of step shocks, Phys. Fluids 5 (1962), 1181–1187. MR 155515, DOI https://doi.org/10.1063/1.1706503
- J. J. Erpenbeck, Stability of idealized one-reaction detonations, Phys. Fluids 7 (1964).
- Helge Kristian Jenssen, Gregory Lyng, and Mark Williams, Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion, Indiana Univ. Math. J. 54 (2005), no. 1, 1–64. MR 2126075, DOI https://doi.org/10.1512/iumj.2005.54.2685
- Soyeun Jung and Jinghua Yao, Stability of ZND detonations for Majda’s model, Quart. Appl. Math. 70 (2012), no. 1, 69–76. MR 2920616, DOI https://doi.org/10.1090/S0033-569X-2011-01232-3
- Gregory Lyng and Kevin Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow, Phys. D 194 (2004), no. 1-2, 1–29. MR 2075662, DOI https://doi.org/10.1016/j.physd.2004.01.036
- Andrew Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math. 41 (1981), no. 1, 70–93. MR 622874, DOI https://doi.org/10.1137/0141006
- Alim Sukhtayev, Zhao Yang, and Kevin Zumbrun, Spectral stability of hydraulic shock profiles, Phys. D 405 (2020), 132360, 9. MR 4068612, DOI https://doi.org/10.1016/j.physd.2020.132360
- Alim Sukhtayev and Kevin Zumbrun, A Sturm-Liouville theorem for quadratic operator pencils, J. Differential Equations 268 (2020), no. 7, 3848–3879. MR 4053607, DOI https://doi.org/10.1016/j.jde.2019.10.010
- Zhao Yang and Kevin Zumbrun, Stability of hydraulic shock profiles, Arch. Ration. Mech. Anal. 235 (2020), no. 1, 195–285. MR 4062477, DOI https://doi.org/10.1007/s00205-019-01422-4
- Kevin Zumbrun, High-frequency asymptotics and one-dimensional stability of Zel’dovich–von Neumann–Döring detonations in the small-heat release and high-overdrive limits, Arch. Ration. Mech. Anal. 203 (2012), no. 3, 701–717. MR 2928130, DOI https://doi.org/10.1007/s00205-011-0457-4
- Kevin Zumbrun, Stability of detonation profiles in the ZND limit, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 141–182. MR 2781588, DOI https://doi.org/10.1007/s00205-010-0342-6
References
- B. Barker and K. Zumbrun, A numerical stability investigation of strong ZND detonations for Majda’s model, preprint; arXiv:1011.1561.
- J. J. Erpenbeck, Stability of steady-state equilibrium detonations, Phys. Fluids 5 (1962), 604–614.
- Jerome J. Erpenbeck, Stability of step shocks, Phys. Fluids 5 (1962), 1181–1187. MR 155515, DOI https://doi.org/10.1063/1.1706503
- J. J. Erpenbeck, Stability of idealized one-reaction detonations, Phys. Fluids 7 (1964).
- Helge Kristian Jenssen, Gregory Lyng, and Mark Williams, Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion, Indiana Univ. Math. J. 54 (2005), no. 1, 1–64. MR 2126075, DOI https://doi.org/10.1512/iumj.2005.54.2685
- Soyeun Jung and Jinghua Yao, Stability of ZND detonations for Majda’s model, Quart. Appl. Math. 70 (2012), no. 1, 69–76. MR 2920616, DOI https://doi.org/10.1090/S0033-569X-2011-01232-3
- Gregory Lyng and Kevin Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow, Phys. D 194 (2004), no. 1-2, 1–29. MR 2075662, DOI https://doi.org/10.1016/j.physd.2004.01.036
- Andrew Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math. 41 (1981), no. 1, 70–93. MR 622874, DOI https://doi.org/10.1137/0141006
- Alim Sukhtayev, Zhao Yang, and Kevin Zumbrun, Spectral stability of hydraulic shock profiles, Phys. D 405 (2020), 132360, 9. MR 4068612, DOI https://doi.org/10.1016/j.physd.2020.132360
- Alim Sukhtayev and Kevin Zumbrun, A Sturm-Liouville theorem for quadratic operator pencils, J. Differential Equations 268 (2020), no. 7, 3848–3879. MR 4053607, DOI https://doi.org/10.1016/j.jde.2019.10.010
- Zhao Yang and Kevin Zumbrun, Stability of hydraulic shock profiles, Arch. Ration. Mech. Anal. 235 (2020), no. 1, 195–285. MR 4062477, DOI https://doi.org/10.1007/s00205-019-01422-4
- Kevin Zumbrun, High-frequency asymptotics and one-dimensional stability of Zel’dovich–von Neumann–Döring detonations in the small-heat release and high-overdrive limits, Arch. Ration. Mech. Anal. 203 (2012), no. 3, 701–717. MR 2928130, DOI https://doi.org/10.1007/s00205-011-0457-4
- Kevin Zumbrun, Stability of detonation profiles in the ZND limit, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 141–182. MR 2781588, DOI https://doi.org/10.1007/s00205-010-0342-6
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Additional Information
Soyeun Jung
Affiliation:
Division of International Studies, Kongju National University, Chungcheongnam-Do 32588, Republic of Korea
MR Author ID:
970238
Email:
soyjung@kongju.ac.kr
Zhao Yang
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405; and Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois 61801
MR Author ID:
1315108
ORCID:
0000-0003-3878-945X
Email:
yangzha@indiana.edu, zhaouiuc@illinois.edu
Kevin Zumbrun
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
MR Author ID:
330192
Email:
kzumbrun@indiana.edu
Received by editor(s):
March 10, 2020
Received by editor(s) in revised form:
July 25, 2020
Published electronically:
October 6, 2020
Additional Notes:
The research of the first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2019R1F1A1063018).
The research of the second author was supported by the College of Arts and Sciences Dissertation Year Fellowship of Indiana University, Bloomington.
The research of the third author was partially supported under NSF grant no. DMS-1400555.
Article copyright:
© Copyright 2020
Brown University
