Abstract
We describe recent analytical and numerical results on stability and behavior of viscous and inviscid detonation waves obtained by dynamical systems/Evans function techniques like those used to study shock and reaction diffusion waves. In the first part, we give a broad description of viscous and inviscid results for 1D perturbations; in the second, we focus on inviscid high-frequency stability in multi-D and associated questions in turning point theory/WKB expansion.
Dedicated to Guy Métivier on the occasion of his 65th birthday.
Research of K.Z. was partially supported under NSF grants no. DMS-0300487 and DMS-0801745.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
As discussed in [45], Erpenbeck treated turning points/glancing modes at points x ∗ bounded away from 0 and ∞; however, these cases necessarily occur at certain boundary frequencies, so must be considered in a complete stability analysis, as must be issues not treated in [22] of uniformity for frequencies near but not at a glancing point.
References
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55 For sale by the Superintendent of Documents (U.S. Government Printing Office, Washington DC, 1964), xiv+1046pp.
J. Alexander, R. Gardner, C. Jones, A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990)
S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels (French) [Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems]. Commun. Partial Differ. Equ. 14 (2), 173–230 (1989)
B. Barker, J. Humpherys, K. Zumbrun, STABLAB: a MATLAB-based numerical library for Evans function computation (2015). Available at: http://impact.byu.edu/stablab/
B. Barker, J. Humpherys, G. Lyng, K. Zumbrun, Viscous hyperstabilization of detonation waves in one space dimension. SIAM J. Appl. Math. 75 (3), 885–906 (2015)
B. Barker, K. Zumbrun, A numerical investigation of stability of ZND detonations for Majda’s model, preprint. arxiv:1011.1561
B. Barker K. Zumbrun, Numerical stability of ZND detonations, in preparation
G. K. Batchelor. An Introduction to Fluid Dynamics, paperback edn. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1999)
M. Beck, B. Sandstede, K. Zumbrun, Nonlinear stability of time-periodic viscous shocks. Arch. Ration. Mech. Anal. 196 (3), 1011–1076 (2010)
A. Bourlioux, A. Majda, V. Roytburd, Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51, 303–343 (1991)
L.Q. Brin, Numerical testing of the stability of viscous shock waves. Math. Comput. 70 (235), 1071–1088 (2001)
J. Buckmaster, The contribution of asymptotics to combustion. Phys. D 20 (1), 91–108 (1986)
J. Buckmaster, J. Neves, One-dimensional detonation stability: the spectrum for infinite activation energy. Phys. Fluids 31 (12), 3572–3576 (1988)
D.L. Chapman, VI. On the rate of explosion in gases. Philos. Mag. Ser. 5 47 (284), 90–104 (1899). doi: 10.1080/14786449908621243
P. Clavin, L. He, Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases. J. Fluid Mech. 306, 306–353 (1996)
E.A. Coddington, M. Levinson, Theory of Ordinary Differential Equations (McGraw–Hill Book Company, Inc., New York, 1955)
R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Springer–Verlag, New York, 1976), xvi+464pp.
W. Döring, Über Detonationsvorgang in Gasen [On the detonation process in gases]. Ann. Phys. 43, 421–436 (1943). doi: 10.1002/andp.19434350605
J.J. Erpenbeck, Stability of steady-state equilibrium detonations. Phys. Fluids 5, 604–614 (1962)
J.J. Erpenbeck, Stability of step shocks. Phys. Fluids 5 (10), 1181–1187 (1962)
J.J. Erpenbeck, Stability of idealized one-reaction detonations. Phys. Fluids 7, 684 (1964)
J.J. Erpenbeck, Stability of Detonations for Disturbances of Small Transverse Wave-Length Los Alamos, LA-3306 (1965), 136pp. https://www.google.com/search?tbm=bks&hl=en&q=Stability+of+detonation+for+disturbances+of+small+transverse+wavelength+Report+of+los+Alamos+LA3306+1965
J.J. Erpenbeck, Detonation stability for disturbances of small transverse wave length. Phys. Fluids 9, 1293–1306 (1966)
L.M. Faria, A.R. Kasimov, R.R. Rosales, Study of a model equation in detonation theory. SIAM J. Appl. Math. 74 (2), 547–570 (2014)
W. Fickett, W.C. Davis, Detonation (University of California Press, Berkeley, 1979); reissued as Detonation: Theory and Experiment (Dover Press, Mineola, New York 2000). ISBN:0-486-41456-6
W. Fickett, W. Wood, Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9, 903–916 (1966)
R.A. Gardner, K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles. Commun. Pure Appl. Math. 51 (7), 797–855 (1998)
I. Gasser, P. Szmolyan, A geometric singular perturbation analysis of detonation and deflagration waves. SIAM J. Math. Anal. 24, 968–986 (1993)
O. Gues, G. Métivier, M. Williams, K. Zumbrun, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit. Arch. Ration. Mech. Anal. 175 (2), 151–244 (2005)
O. Gues, G. Métivier, M. Williams, K. Zumbrun, Existence and stability of noncharacteristic boundary layers for the compressible Navier-Stokes and MHD equations. Arch. Ration. Mech. Anal. 197 (1), 1–87 (2010)
M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342 (1), 177–243 (2008)
J. Hendricks, J. Humpherys, G. Lyng, K. Zumbrun, Stability of viscous weak detonation waves for Majda’s model. J. Dyn. Differ. Equ. 27 (2), 237–260 (2015)
J. Humpherys, G. Lyng, K. Zumbrun, Multidimensional spectral stability of large-amplitude Navier-Stokes shocks, preprint. arxiv:1603.03955
J. Humpherys, K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems. Physica D 220 (2), 116–126 (2006)
J. Humpherys, G. Lyng, K. Zumbrun, Spectral stability of ideal gas shock layers. Arch. Ration. Mech. Anal. 194 (3), 1029–1079 (2009)
J. Humpherys, O. Lafitte, K. Zumbrun, Stability of isentropic Navier-Stokes shocks in the high-Mach number limit. Commun. Math. Phys. 293 (1), 1–36 (2010)
J. Humpherys, K. Zumbrun, Efficient numerical stability analysis of detonation waves in ZND. Q. Appl. Math. 70 (4), 685–703 (2012)
E. Jouguet, Sur la propagation des réactions chimiques dans les gaz [On the propagation of chemical reactions in gases]. J. Math. Pures Appl. 6 (1), 347–425 (1905)
E. Jouguet, Sur la propagation des réactions chimiques dans les gaz [On the propagation of chemical reactions in gases]. J. Math. Pures Appl. 6 (2), 5–85 (1906)
H.K. Jenssen, G. Lyng, M. Williams, Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion. Indiana Univ. Math. J. 54, 1–64 (2005)
A.R. Kasimov, D.S. Stewart, Spinning instability of gaseous detonations. J. Fluid Mech. 466, 179–203 (2002)
T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1985)
H.O. Kreiss, Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–298 (1970)
O. Lafitte, M. Williams, K. Zumbrun, The Erpenbeck high frequency instability theorem for Zeldovitch-von Neumann-Döring detonations. Arch. Ration. Mech. Anal. 204 (1), 141–187 (2012)
O. Lafitte, M. Williams, K. Zumbrun, High-frequency stability of detonations and turning points at infinity. SIAM J. Math. Anal. 47 (3), 1800–1878 (2015)
O. Lafitte, M. Williams, K. Zumbrun, Block-diagonalization of ODEs in the semiclassical limit and C ω vs. C ∞ stationary phase. SIAM J. Math. Anal. 48 (3), 1773–1797 (2016)
P.D. Lax, Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math. 10, 537–566 (1957)
P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, vol. 11 (Society for Industrial and Applied Mathematics, Philadelphia, 1973), v+48pp.
H.I. Lee, D.S. Stewart, Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech. 216, 103–132 (1990)
J.H.S. Lee, The Detonation Phenomenon (Cambridge University Press, Cambridge/New York, 2008). ISBN-13:978-0521897235
N. Levinson, The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)
T.-P. Liu, S.-H. Yu, Nonlinear stability of weak detonation waves for a combustion model. Commun. Math. Phys. 204 (3), 551–586 (1999)
G. Lyng, K. Zumbrun, One-dimensional stability of viscous strong detonation waves. Arch. Ration. Mech. Anal. 173 (2), 213–277 (2004)
G. Lyng, K. Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow. Physica D 194, 1–29 (2004)
G. Lyng, M. Raoofi, B. Texier, K. Zumbrun, Pointwise Green function bounds and stability of combustion waves. J. Differ. Equ. 233, 654–698 (2007)
A. Martinez, An Introduction to Semiclassical and Microlocal Analysis (Springer, New York, 2002)
G. Métivier, K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Am. Math. Soc. 175 (826), vi+107pp (2005)
A. Majda, A qualitative model for dynamic combustion. SIAM J. Appl. Math. 41, 70–91 (1981)
A. Majda, The stability of multi-dimensional shock fronts – a new problem for linear hyperbolic equations. Mem. Am. Math. Soc. 275, 1–95 (1983)
A. Majda, R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts. I. The basic perturbation analysis. SIAM J. Appl. Math. 43, 1310–1334 (1983)
C. Mascia, K. Zumbrun, Pointwise Green’s function bounds for shock profiles with degenerate viscosity. Arch. Ration. Mech. Anal. 169, 177–263 (2003)
G. Métivier, The block structure condition for symmetric hyperbolic problems. Bull. Lond. Math. Soc. 32, 689–702 (2000)
G. Métivier, Stability of multidimensional shocks, in Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Applications, vol. 47 (Birkhäuser Boston, Boston, 2001), pp. 25–103
T. Nguyen, Stability of multi-dimensional viscous shocks for symmetric systems with variable multiplicities. Duke Math. J. 150 (3), 577–614 (2009)
F.W.U. Olver, Asymptotics and Special Functions (Academic Press, New York, 1974)
R. L. Pego, M. I. Weinstein, Eigenvalues, and instabilities of solitary waves. Philos. Trans. R. Soc. Lond. Ser. A 340 (1656), 47–94 (1992)
R. Pemantle, M.C. Wilson, Asymptotic expansions of oscillatory integrals with complex phase, in Algorithmic Probability and Combinatorics. Contemporary Mathematics, vol. 520 (American Mathematical Society, Providence, 2010), pp. 221–240
R. Plaza, K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles. J. Discret. Cont. Dyn. Sys. 10, 885–924 (2004)
J.M. Powers, S. Paolucci, Accurate spatial resolution estimates for reactive supersonic flow with detailed chemistry. AIAA J. 43 (5), 1088–1099 (2005)
C.M. Romick, T.D. Aslam, J.D. Powers, The Dynamics of Unsteady Detonation with Diffusion. 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando (2011), http://dx.doi.org/10.2514/6.2011-799
C.M. Romick, T.D. Aslam, J.D. Powers, The effect of diffusion on the dynamics of unsteady detonations. J. Fluid Mech. 699, 453–464 (2012)
J.-M. Roquejoffre, J.-P. Vila, Stability of ZND detonation waves in the Majda combustion model. Asymptot. Anal. 18 (3–4), 329–348 (1998)
B. Sandstede, A. Scheel, Hopf bifurcation from viscous shock waves. SIAM J. Math. Anal. 39, 2033–2052 (2008)
D. Serre, Systèmes de lois de conservation I–II (Fondations, Diderot Editeur, Paris, 1996), iv+308pp. ISBN:2-84134-072-4 and xii+300pp. ISBN:2-84134-068-6
M. Short, An Asymptotic Derivation of the Linear Stability of the Square-Wave Detonation using the Newtonian limit. Proc. R. Soc. Lond. A 452, 2203–2224 (1996)
M. Short, Multidimensional linear stability of a detonation wave at high activation energy. SIAM J. Appl. Math. 57 (2), 307–326 (1997)
J. Smoller, Shock Waves and Reaction–Diffusion Equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 258 (Springer, New York, 1994), xxiv+632pp. ISBN:0-387-94259-9
D.S. Stewart, A.R. Kasimov, State of detonation stability theory and its application to propulsion. J. Propuls. Power 22 (6), 1230–1244 (2006)
A. Szepessy, Dynamics and stability of a weak detonation wave. Commun. Math. Phys. 202 (3), 547–569 (1999)
B. Texier, K. Zumbrun, Galloping instability of viscous shock waves. Physica D. 237, 1553–1601 (2008)
B. Texier, K. Zumbrun, Hopf bifurcation of viscous shock waves in gas dynamics and MHD. Arch. Ration. Mech. Anal. 190, 107–140 (2008)
B. Texier, K. Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions. Commun. Math. Phys. 302 (1), 1–51 (2011)
J. von Neumann, Theory of Detonation Waves, Aberdeen Proving Ground, Maryland: Office of Scientific Research and Development, Report No. 549, Ballistic Research Laboratory File No. X-122 Progress Report to the National Defense Research Committee, Division B, OSRD-549 (April 1, 1942. PB 31090). (4 May 1942), 34 pages
J. von Neumann, in John von Neumann, Collected Works, vol. 6. ed. by A.J. Taub (Permagon Press, Elmsford, 1963) [1942], pp. 178–218
W. Wasow, Linear Turning Point Theory. Applied Mathematical Sciences, vol. 54 (Springer-Verlag, New York, 1985), ix+246pp.
M. Williams, Heteroclinic orbits with fast transitions: a new construction of detonation profiles. Indiana Univ. Math. J. 59 (3), 1145–1209 (2010)
Y.B. Zel’dovich, [On the theory of the propagation of detonation in gaseous systems]. J. Exp. Theor. Phys. 10, 542–568 (1940). Translated into English in: National Advisory Committee for Aeronautics Technical Memorandum No. 1261 (1950)
K. Zumbrun, Multidimensional stability of planar viscous shock waves, in Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations and Applications, vol. 47 (Birkhäuser Boston, Boston, 2001), pp. 307–516
K. Zumbrun, Stability of large-amplitude shock waves of compressible navier–stokes equations, with an appendix by Helge Kristian Jenssen and Gregory Lyng, in Handbook of Mathematical Fluid Dynamics, vol. III (North-Holland, Amsterdam, 2004), pp. 311–533
K. Zumbrun, Stability of detonation waves in the ZND limit. Arch. Ration. Mech. Anal. 200 (1), 141–182 (2011)
K. 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 (3), 701–717 (2012)
K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves. Indiana Math. J. 47, 741–871 (1998); Errata, Indiana Univ. Math. J. 51 (4), 1017–1021 (2002)
K. Zumbrun, D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48, 937–992 (1999)
Acknowledgements
Special thanks to the anonymous and extraordinarily attentive referee, whose many thoughtful suggestions and comments greatly improved the exposition.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Zumbrun, K. (2017). Recent Results on Stability of Planar Detonations. In: Colombini, F., Del Santo, D., Lannes, D. (eds) Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics. Springer INdAM Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-52042-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-52042-1_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-52041-4
Online ISBN: 978-3-319-52042-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)