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Recent Results on Stability of Planar Detonations

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Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics

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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.

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Notes

  1. 1.

    Condition [44, (5.13)] comparing relative sizes of oscillatory modes in the first-order expansion of decaying solution \(\tilde{Z}\), depending on the geometry of background profile \(\bar{W}\); see [44, Prop. 5.1] and discussion just below.

  2. 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

  1. 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.

    Google Scholar 

  2. 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)

    MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MATH  Google Scholar 

  4. B. Barker, J. Humpherys, K. Zumbrun, STABLAB: a MATLAB-based numerical library for Evans function computation (2015). Available at: http://impact.byu.edu/stablab/

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Barker, K. Zumbrun, A numerical investigation of stability of ZND detonations for Majda’s model, preprint. arxiv:1011.1561

    Google Scholar 

  7. B. Barker K. Zumbrun, Numerical stability of ZND detonations, in preparation

    Google Scholar 

  8. G. K. Batchelor. An Introduction to Fluid Dynamics, paperback edn. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1999)

    MATH  Google Scholar 

  9. M. Beck, B. Sandstede, K. Zumbrun, Nonlinear stability of time-periodic viscous shocks. Arch. Ration. Mech. Anal. 196 (3), 1011–1076 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Bourlioux, A. Majda, V. Roytburd, Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51, 303–343 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. L.Q. Brin, Numerical testing of the stability of viscous shock waves. Math. Comput. 70 (235), 1071–1088 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Buckmaster, The contribution of asymptotics to combustion. Phys. D 20 (1), 91–108 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Buckmaster, J. Neves, One-dimensional detonation stability: the spectrum for infinite activation energy. Phys. Fluids 31 (12), 3572–3576 (1988)

    Article  MATH  Google Scholar 

  14. D.L. Chapman, VI. On the rate of explosion in gases. Philos. Mag. Ser. 5 47 (284), 90–104 (1899). doi: 10.1080/14786449908621243

  15. P. Clavin, L. He, Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases. J. Fluid Mech. 306, 306–353 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. E.A. Coddington, M. Levinson, Theory of Ordinary Differential Equations (McGraw–Hill Book Company, Inc., New York, 1955)

    MATH  Google Scholar 

  17. R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Springer–Verlag, New York, 1976), xvi+464pp.

    Google Scholar 

  18. W. Döring, Über Detonationsvorgang in Gasen [On the detonation process in gases]. Ann. Phys. 43, 421–436 (1943). doi: 10.1002/andp.19434350605

    Article  Google Scholar 

  19. J.J. Erpenbeck, Stability of steady-state equilibrium detonations. Phys. Fluids 5, 604–614 (1962)

    Article  MathSciNet  Google Scholar 

  20. J.J. Erpenbeck, Stability of step shocks. Phys. Fluids 5 (10), 1181–1187 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  21. J.J. Erpenbeck, Stability of idealized one-reaction detonations. Phys. Fluids 7, 684 (1964)

    Article  MATH  Google Scholar 

  22. 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

    Google Scholar 

  23. J.J. Erpenbeck, Detonation stability for disturbances of small transverse wave length. Phys. Fluids 9, 1293–1306 (1966)

    Article  MATH  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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

    Google Scholar 

  26. W. Fickett, W. Wood, Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9, 903–916 (1966)

    Article  Google Scholar 

  27. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. I. Gasser, P. Szmolyan, A geometric singular perturbation analysis of detonation and deflagration waves. SIAM J. Math. Anal. 24, 968–986 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342 (1), 177–243 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Humpherys, G. Lyng, K. Zumbrun, Multidimensional spectral stability of large-amplitude Navier-Stokes shocks, preprint. arxiv:1603.03955

    Google Scholar 

  34. J. Humpherys, K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems. Physica D 220 (2), 116–126 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Humpherys, G. Lyng, K. Zumbrun, Spectral stability of ideal gas shock layers. Arch. Ration. Mech. Anal. 194 (3), 1029–1079 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Humpherys, K. Zumbrun, Efficient numerical stability analysis of detonation waves in ZND. Q. Appl. Math. 70 (4), 685–703 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    MATH  Google Scholar 

  39. 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)

    MATH  Google Scholar 

  40. 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)

    Article  MathSciNet  MATH  Google Scholar 

  41. A.R. Kasimov, D.S. Stewart, Spinning instability of gaseous detonations. J. Fluid Mech. 466, 179–203 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1985)

    Google Scholar 

  43. H.O. Kreiss, Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–298 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  44. 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)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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)

    Article  MathSciNet  MATH  Google Scholar 

  46. 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)

    Google Scholar 

  47. P.D. Lax, Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math. 10, 537–566 (1957)

    Article  MATH  Google Scholar 

  48. 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.

    Google Scholar 

  49. H.I. Lee, D.S. Stewart, Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech. 216, 103–132 (1990)

    Article  MATH  Google Scholar 

  50. J.H.S. Lee, The Detonation Phenomenon (Cambridge University Press, Cambridge/New York, 2008). ISBN-13:978-0521897235

    Google Scholar 

  51. N. Levinson, The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  52. T.-P. Liu, S.-H. Yu, Nonlinear stability of weak detonation waves for a combustion model. Commun. Math. Phys. 204 (3), 551–586 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  53. G. Lyng, K. Zumbrun, One-dimensional stability of viscous strong detonation waves. Arch. Ration. Mech. Anal. 173 (2), 213–277 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  54. G. Lyng, K. Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow. Physica D 194, 1–29 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  55. G. Lyng, M. Raoofi, B. Texier, K. Zumbrun, Pointwise Green function bounds and stability of combustion waves. J. Differ. Equ. 233, 654–698 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. A. Martinez, An Introduction to Semiclassical and Microlocal Analysis (Springer, New York, 2002)

    Book  MATH  Google Scholar 

  57. G. Métivier, K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Am. Math. Soc. 175 (826), vi+107pp (2005)

    Google Scholar 

  58. A. Majda, A qualitative model for dynamic combustion. SIAM J. Appl. Math. 41, 70–91 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  59. A. Majda, The stability of multi-dimensional shock fronts – a new problem for linear hyperbolic equations. Mem. Am. Math. Soc. 275, 1–95 (1983)

    Google Scholar 

  60. 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)

    MATH  Google Scholar 

  61. C. Mascia, K. Zumbrun, Pointwise Green’s function bounds for shock profiles with degenerate viscosity. Arch. Ration. Mech. Anal. 169, 177–263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  62. G. Métivier, The block structure condition for symmetric hyperbolic problems. Bull. Lond. Math. Soc. 32, 689–702 (2000)

    Article  MATH  Google Scholar 

  63. 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

    Google Scholar 

  64. T. Nguyen, Stability of multi-dimensional viscous shocks for symmetric systems with variable multiplicities. Duke Math. J. 150 (3), 577–614 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  65. F.W.U. Olver, Asymptotics and Special Functions (Academic Press, New York, 1974)

    MATH  Google Scholar 

  66. R. L. Pego, M. I. Weinstein, Eigenvalues, and instabilities of solitary waves. Philos. Trans. R. Soc. Lond. Ser. A 340 (1656), 47–94 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  67. 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

    Google Scholar 

  68. 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)

    Article  MathSciNet  MATH  Google Scholar 

  69. J.M. Powers, S. Paolucci, Accurate spatial resolution estimates for reactive supersonic flow with detailed chemistry. AIAA J. 43 (5), 1088–1099 (2005)

    Article  Google Scholar 

  70. 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

  71. 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)

    Google Scholar 

  72. J.-M. Roquejoffre, J.-P. Vila, Stability of ZND detonation waves in the Majda combustion model. Asymptot. Anal. 18 (3–4), 329–348 (1998)

    MathSciNet  MATH  Google Scholar 

  73. B. Sandstede, A. Scheel, Hopf bifurcation from viscous shock waves. SIAM J. Math. Anal. 39, 2033–2052 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  74. 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

    Google Scholar 

  75. 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)

    Article  MathSciNet  MATH  Google Scholar 

  76. M. Short, Multidimensional linear stability of a detonation wave at high activation energy. SIAM J. Appl. Math. 57 (2), 307–326 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  77. 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

    Google Scholar 

  78. D.S. Stewart, A.R. Kasimov, State of detonation stability theory and its application to propulsion. J. Propuls. Power 22 (6), 1230–1244 (2006)

    Article  Google Scholar 

  79. A. Szepessy, Dynamics and stability of a weak detonation wave. Commun. Math. Phys. 202 (3), 547–569 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  80. B. Texier, K. Zumbrun, Galloping instability of viscous shock waves. Physica D. 237, 1553–1601 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  81. B. Texier, K. Zumbrun, Hopf bifurcation of viscous shock waves in gas dynamics and MHD. Arch. Ration. Mech. Anal. 190, 107–140 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  82. 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)

    Article  MathSciNet  MATH  Google Scholar 

  83. 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

    Google Scholar 

  84. J. von Neumann, in John von Neumann, Collected Works, vol. 6. ed. by A.J. Taub (Permagon Press, Elmsford, 1963) [1942], pp. 178–218

    Google Scholar 

  85. W. Wasow, Linear Turning Point Theory. Applied Mathematical Sciences, vol. 54 (Springer-Verlag, New York, 1985), ix+246pp.

    Google Scholar 

  86. M. Williams, Heteroclinic orbits with fast transitions: a new construction of detonation profiles. Indiana Univ. Math. J. 59 (3), 1145–1209 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  87. 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)

    Google Scholar 

  88. 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

    Google Scholar 

  89. 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

    Google Scholar 

  90. K. Zumbrun, Stability of detonation waves in the ZND limit. Arch. Ration. Mech. Anal. 200 (1), 141–182 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  91. 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)

    Article  MathSciNet  MATH  Google Scholar 

  92. 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)

    Google Scholar 

  93. K. Zumbrun, D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48, 937–992 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Special thanks to the anonymous and extraordinarily attentive referee, whose many thoughtful suggestions and comments greatly improved the exposition.

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

    Kevin Zumbrun

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  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Ferruccio Colombini

  2. Dipartimento di Matematica e Geoscienze, Università di Trieste, Trieste, Italy

    Daniele Del Santo

  3. Institut de Mathématiques de Bordeaux, Université de Bordeaux - UMR5251, Talence, France

    David Lannes

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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

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