Stability of ZND detonations for Majda’s model
Authors:
Soyeun Jung and Jinghua Yao
Journal:
Quart. Appl. Math. 70 (2012), 69-76
MSC (2000):
Primary 76L05; Secondary 76E99, 76N99, 80A32
DOI:
https://doi.org/10.1090/S0033-569X-2011-01232-3
Published electronically:
August 26, 2011
MathSciNet review:
2920616
Full-text PDF Free Access
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Abstract: We evaluate by direct calculation the Lopatinski determinant for ZND detonations in Majda’s model for reacting flow and show that on the nonstable (nonnegative real part) complex half-plane it has a single zero at the origin of multiplicity one, implying stability. Together with results of Zumbrun on the inviscid limit, this recovers the result of Roquejoffre–Vila that viscous detonations of Majda’s model also are stable for sufficiently small viscosity, for any fixed detonation strength, heat release, and rate of reaction.
References
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References
- J. J. Erpenbeck, Stability of idealized one-reaction detonations, Phys. Fluids 7 (1964).
- R. Gardner and K. Zumbrun, The Gap Lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51 (1998), no. 7, 797–855. MR 1617251 (99c:35152)
- H.K. Jenssen, G. Lyng, and M. Williams. Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion, Indiana Univ. Math. J. 54 (2005), 1–64. MR 2126075 (2006a:35249)
- A.R. Kasimov and D.S. Stewart, Spinning instability of gaseous detonations. J. Fluid Mech. 466 (2002), 179–203. MR 1925152 (2003g:76093)
- G. Lyng and K. Zumbrun, One-dimensional stability of viscous strong detonation waves, Arch. Ration. Mech. Anal. 173 (2004), no. 2, 213–277. MR 2081031 (2005f:76061)
- G. Lyng and K. Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow, Physica D, 194 (2004), 1–29. MR 2075662 (2005d:35134)
- G. Lyng, M. Raoofi, B. Texier, and K. Zumbrun, Pointwise Green Function Bounds and stability of combustion waves, J. Differential Equations 233 (2007), 654–698. MR 2292522 (2007m:35147)
- A. Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math., 41 (1981), 70–91. MR 622874 (82j:35096)
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- D. Sattinger, On the stability of waves of nonlinear parabolic systems. Adv. Math. 22 (1976), 312–355. MR 0435602 (55:8561)
- B. Texier and K. Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions, preprint (2008).
- K. Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves, 307–516, Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser Boston, Boston, MA, 2001. MR 1842778 (2002k:35200)
- K. Zumbrun, Stability of viscous detonation waves in the ZND limit, preprint (2009).
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Additional Information
Soyeun Jung
Affiliation:
Indiana University, Bloomington, Indiana 47405
MR Author ID:
970238
Email:
soyjung@indiana.edu
Jinghua Yao
Affiliation:
Indiana University, Bloomington, Indiana 47405
Email:
yaoj@indiana.edu
Received by editor(s):
May 21, 2010
Published electronically:
August 26, 2011
Additional Notes:
The research of S.J. and J.Y. was partially supported under NSF grants number DMS-0070765 and DMS-0300487. Thanks to Kevin Zumbrun for suggesting the problem and for helpful discussions.
Article copyright:
© Copyright 2011
Brown University
The copyright for this article reverts to public domain 28 years after publication.
