Transition to Longitudinal Instability of Detonation Waves is Generically Associated with Hopf Bifurcation to Time-Periodic Galloping Solutions

Communications in Mathematical Physics

Volumn 302, Issue 1, 2011, Pages 1-51

  Texier, Benjamin ^a   Zumbrun, Kevin ^b  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b INDIANA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 78650928876     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-010-1175-8     Document Type: Article

References (70)

1. Abouseif G., Toong T. Y.: Theory of unstable one-dimensional detonations. Combust. Flame 45, 67-94 (1982).
2. Alexander J., Gardner R., Jones C. K. R. T.: A topological invariant arising in the analysis of traveling waves. J. Reine Angew. Math. 410, 167-212 (1990).
3. Alpert R. L., Toong T. Y.: Periodicity in exothermic hypersonic flows about blunt projectiles. Acta Astron. 17, 538-560 (1972).
4. Barker B., Humpherys J., Rudd K., Zumbrun K.: Stability of viscous shocks in isentropic gas dynamics. Commun. Math. Phys 281(1), 231-249 (2008).
5. Batchelor G. K.: An introduction to fluid dynamics. Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1999).
6. Beck M., Sandstede B., Zumbrun K.: Nonlinear stability of time-periodic shocks. Arc. Rat. Mech. Anal. 196, 1011-1076 (2010).
7. Bourlioux A., Majda A.: Theoretical and numerical structure of unstable detonations. Proc. R. Soc. Lond. A 350, 29-68 (1995).
8. Bourlioux A., Majda A., Roytburd V.: Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51, 303-343 (1991).
9. Bridges T. J., Derks G., Gottwald G.: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys. D 172(1-4), 190-216 (2002).
10. Brin, L.: Numerical testing of the stability of viscous shock waves. Doctoral thesis, Indiana University, 1998.
11. Brin L. Q.: Numerical testing of the stability of viscous shock waves. Math. Comp. 70(235), 1071-1088 (2001).
12. Brin, L., Zumbrun, K.: Analytically varying eigenvectors and the stability of viscous shock waves. In: Proc. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). Mat. Contemp. 22, 19-32, 2002.
13. Buckmaster, J. D.: An introduction to combustion theory. The mathematics of combustion, Frontiers in App. Math. Philadelphia, SIAM, 1985, pp. 3-46.
14. Buckmaster J., Neves J.: One-dimensional detonation stability: the spectrum for infinite activation energy. Phys. Fluids 31(12), 3572-3576 (1988).
15. Carr J.,: Applications of centre manifold theory. Applied Mathematical Sciences, 35. New York-Berlin: Springer-Verlag, 1981.
16. Chen G. Q.: Global solutions to the compressible Navier-Stokes equations for a reacting mixture. SIAM J. Math. Anal. 23(3), 609-634 (1992).
17. Courant R., Friedrichs K. O.: Supersonic flow and shock waves. Springer-Verlag, New York (1976).
18. Edmunds D. E., Evans W. D.: Spectral theory and differential operators. Oxford University Press, Oxford (1987).
19. Erpenbeck J. J.: Stability of steady-state equilibrium detonations. Phys. Fluids 5, 604-614 (1962).
20. Erpenbeck J. J.: Stability of idealized one-reaction detonations. Phys. Fluids 7, 684 (1964).
21. Erpenbeck J. J.: Detonation stability for disturbances of small transverse wave length. Phys. Fluids 9, 1293-1306 (1966).
22. Erpenbeck J. J.: Nonlinear theory of unstable one-dimensional detonations. Phys. Fluids 10(2), 274-289 (1967).
23. Fickett W.: Stability of the square wave detonation in a model system. Physica 16D, 358-370 (1985).
24. Fickett, W.: Detonation in miniature. In The mathematics of combustion, Frontiers in App. Math. Philadelphia: SIAM, 1985, pp. 133-182.
25. Fickett, W., Davis, W. C.: Detonation, Berkeley, CA: University of California Press, 1979, reissued as Detonation: Theory and experiment Mineola, New York: Dover Press, 2000.
26. Fickett W., Wood W. W.: Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9, 903-916 (1966).
27. Gardner R.: On the detonation of a combustible gas. Trans. Amer. Math. Soc. 277(2), 431-468 (1983).
28. Gohberg, I., Krein, M. G.: Introduction to the theory of linear nonselfadjoint operators. Translations of mathematical monographs, Volume 18, Providence, RI: Amer, Math. Soc., 1969.
29. Gardner R., Zumbrun K.: The Gap Lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797-855 (1998).
30. Gasser I., Szmolyan P.: A geometric singular perturbation analysis of detonation and deflagration waves. SIAM J. Math. Anal. 24, 968-986 (1993).
31. Gasser I., Szmolyan P.: Detonation and deflagration waves with multistep reaction schemes. SIAM J. Appl. Math. 55, 175-191 (1995).
32. Hale, J., Koçak, H.: Dynamics and bifurcations. Texts in Applied Mathematics, 3. New York: Springer-Verlag, 1991.
33. Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, Volume 840, Berlin: Springer-Verlag, 1981.
34. Howard P., Zumbrun K.: Stability of undercompressive viscous shock waves. J. Diff. Eq. 225(1), 308-360 (2006).
35. Humpherys J., Lafitte O., Zumbrun K.: Stability of viscous shock profiles in the high Mach number limit. Commun. Math. Phys. 293(1), 1-36 (2010).
36. Humpherys J., Lyng G., Zumbrun K.: Spectral stability of ideal-gas shock layers. Arch. Rat. Mech. Anal. 194(3), 1029-1079 (2009).
37. Humpherys J., Zumbrun K.: An efficient shooting algorithm for Evans function calculations in large systems. Phys. D 220(2), 116-126 (2006).
38. Humpherys J., Zumbrun K.: Spectral stability of small amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems. Z. Angew. Math. Phys. 53, 20-34 (2002).
39. Jenssen H. K., Lyng G., Williams M.: Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion. Indiana Univ. Math. J. 54(1), 1-64 (2005).
40. Kato T.: Perturbation theory for linear operators. Springer-Verlag, Berlin Heidelberg (1985).
41. Kasimov A. R., Stewart D. S.: Spinning instability of gaseous detonations. J. Fluid Mech. 466, 179-203 (2002).
42. Lee H. I., Stewart D. S.: Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech. 216, 102-132 (1990).
43. Lyng G., Zumbrun K.: A stability index for detonation waves in Majda's model for reacting flow. Phys. D 194(1-2), 1-29 (2004).
44. Lyng G., Zumbrun K.: One-dimensional stability of viscous strong detonation waves. Arch. Rat. Mech. Anal. 173(2), 213-277 (2004).
45. Lyng G., Raoofi M., Texier B., Zumbrun K.: Pointwise Green Function Bounds and stability of combustion waves. J. Diff. Eqs. 233(2), 654-698 (2007).
46. Marsden, J. E., McCracken, M.: The Hopf bifurcation and its applications. Applied Mathematical Sciences 19, Berlin-Heidelberg-New York: Springer, 1976.
47. Mascia C., Zumbrun K.: Pointwise Green's function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51(4), 773-904 (2002).
48. Mascia C., Zumbrun K.: Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Comm. Pure Appl. Math. 57(7), 841-876 (2004).
49. Mascia C., Zumbrun K.: Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Rat. Mech. Anal. 169(3), 177-263 (2003).
50. Mascia C., Zumbrun K.: Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Rat. Mech. Anal. 172(1), 93-131 (2004).
51. Mascia C., Zumbrun K.: Stability of large-amplitude shock profiles of general relaxation systems. SIAM J. Math. Anal. 37(3), 889-913 (2005).
52. Métivier, G., Zumbrun, K.: Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc. 175(826) (2005).
53. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. New York: Springer-Verlag, 1983.
54. McVey U. B., Toong T. Y.: Mechanism of instabilities in exothermic blunt-body flows. Combus. Sci. Tech. 3, 63-76 (1971).
55. Raoofi R., Zumbrun K.: Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems. J. Diff. Eqs. 246(4), 1539-1567 (2009).
56. Sandstede B., Scheel A.: Hopf bifurcation from viscous shock waves. SIAM J. Math. Anal. 39(6), 2033-2052 (2008).
57. Shizuta S., Kawashima Y.: On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. Tohoku Math. J. (2) 40(3), 449-464 (1988).
58. Short M.: 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).
59. Short M.: Multidimensional linear stability of a detonation wave at high activation energy. Siam J. Appl. Math. 57(2), 307-326 (1997).
60. Tan D., Tesei A.: Nonlinear stability of strong detonation waves in gas dynamical combustion. Nonlinearity 10, 355-376 (1997).
61. Texier B., Zumbrun K.: Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves. Methods Appl. Anal. 12(4), 349-380 (2005).
62. Texier B., Zumbrun K.: Galloping instability of viscous shock waves. Physica D 237(10-12), 1553-1601 (2008).
63. Texier B., Zumbrun K.: Hopf bifurcation of viscous shock waves in compressible gas-dynamics and MHD. Arch. Rat. Mech. Anal. 190(1), 107-140 (2008).
64. Vanderbauwhede, A., Iooss, G.: Center manifold theory in infinite dimensions. In: Dynamics reported: expositions in dynamical systems, Dynam. Report. Expositions Dynam. Systems (N. S.) 1, Berlin: Springer, 1992, pp. 125-163.
65. Zumbrun, K.: Multidimensional stability of planar viscous shock waves. In: Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., 47, Boston, MA: Birkhäuser Boston, 2001, pp. 307-516.
66. Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of mathematical fluid dynamics. Vol. III, Amsterdam: North-Holland, 2004, pp. 311-533.
67. Zumbrun, K.: Planar stability criteria for viscous shock waves of systems with real viscosity. In: Hyperbolic systems of balance laws, Lecture Notes in Math., 1911, Berlin: Springer, 2007, pp. 229-326.
68. Zumbrun K.,: Stability of viscous detonations in the ZND limit. To appear, Arch. Ration. Mech. Anal. doi: 10. 1007/s00205-101-03426, 2010.
69. Zumbrun, K., Howard, P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Mathematics Journal 47, 741-871 (1998); Errata, Indiana Univ. Math. J. 51(4), 1017-1021 (2002)
70. Zumbrun K., Serre D.: Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48, 937-992 (1999).