Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Er] J. J. Erpenbeck, Stability of idealized one-reaction detonations, Phys. Fluids 7 (1964).
  • [GZ] 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)
  • [JLW] 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)
  • [KS] A.R. Kasimov and D.S. Stewart, Spinning instability of gaseous detonations. J. Fluid Mech. 466 (2002), 179-203. MR 1925152 (2003g:76093)
  • [LyZ1] 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)
  • [LyZ2] 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)
  • [LRTZ] 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)
  • [M] A. Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math., 41 (1981), 70-91. MR 622874 (82j:35096)
  • [RV] J.-M. Roquejoffre and J.-P. Vila, Stability of ZND detonation waves in the Majda combustion model, Asymptot. Anal. 18 (1998), no. 3-4, 329-348. MR 1668958 (99m:80016)
  • [Sa] D. Sattinger, On the stability of waves of nonlinear parabolic systems. Adv. Math. 22 (1976), 312-355. MR 0435602 (55:8561)
  • [TZ] 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).
  • [Z1] 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)
  • [Z2] K. Zumbrun, Stability of viscous detonation waves in the ZND limit, preprint (2009).

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 76L05, 76E99, 76N99, 80A32

Retrieve articles in all journals with MSC (2000): 76L05, 76E99, 76N99, 80A32


Additional Information

Soyeun Jung
Affiliation: Indiana University, Bloomington, Indiana 47405
Email: soyjung@indiana.edu

Jinghua Yao
Affiliation: Indiana University, Bloomington, Indiana 47405
Email: yaoj@indiana.edu

DOI: https://doi.org/10.1090/S0033-569X-2011-01232-3
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.

American Mathematical Society