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Journées équations aux dérivées partielles

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Mark Williams
Turning points at infinity and stability of detonations
Journées équations aux dérivées partielles (2015), Exp. No. 12, 8 p., doi: 10.5802/jedp.641
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Résumé - Abstract

We begin by looking at a few simple examples of turning points in systems of ODEs depending on parameters, and then focus on the difficult case where the turning point occurs at infinity. We explain how turning points at infinity arise in a problem of detonation stability that was studied by J. J. Erpenbeck in the 1960s. In this problem the relevant system of ODEs describes the evolution of high frequency perturbations of a detonation profile, and the parameters on which the system depends are the perturbation frequencies. The resolution of the problem requires an analysis of the turning point at infinity that is uniform with respect to the parameters. This is joint work with Olivier Lafitte and Kevin Zumbrun.


[1] J. J. Erpenbeck, “Stability of steady-state equilibrium detonations”, Phys. Fluids 5 (1962), p. 604-614
[2] J. J. Erpenbeck, “Stability of step shocks”, Phys. Fluids 5 (1962) no. 10, p. 1181-1187  MR 155515 |  Zbl 0111.38403
[3] J. J. Erpenbeck, “Stability of idealized one-reaction detonations”, Phys. Fluids 7 (1964), p. 684-695  Zbl 0123.42901
[4] J. J. Erpenbeck, “Stability of detonations for disturbances of small transverse wavelength”, Los Alamos Preprint, LA-3306, 1965
[5] J. J. Erpenbeck, “Detonation stability for disturbances of small transverse wave length”, Phys. Fluids 9 (1966), p. 1293-1306  Zbl 0144.47204
[6] W. Fickett & W. Davis, Detonation: Theory and Experiment, Univ. California Press, Berkeley, 1979
[7] O. Lafitte, M. Williams & K. Zumbrun, “The Erpenbeck high frequency instability theorem for ZND detonations”, Archive for Rational Mechanics and Analysis 204 (2012), p. 141-187  MR 2898738
[8] O. Lafitte, M. Williams & K. Zumbrun, “Block-diagonalization of ODEs in the semiclassical limit and $C^\omega $ vs. $C^\infty $ stationary phase”, submitted, http://arxiv.org/abs/1507.03116, 2015  MR 3348117
[9] O. Lafitte, M. Williams & K. Zumbrun, “High-frequency stability of detonations and turning points at infinity”, SIAM J. Math. Analysis 47-3 (2015), p. 1800-1878, http://arxiv.org/abs/1312.6906  MR 3348117
[10] F. W. J. Olver, “Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter”, Philos. Trans. Roy. Soc. London. Ser. A 250 (1958), p. 479-517  MR 94496 |  Zbl 0083.05701
[11] F. W. J. Olver, Asymptotics and special functions, Academic Press, New York-London, 1974, Computer Science and Applied Mathematics  MR 435697 |  Zbl 0303.41035
[12] M. Short, “Theory and modeling of detonation wave stability: A brief look at the past and toward the future”, Proceedings, ICDERS 2005
[13] Y. Sibuya, Uniform simplification in a full neighborhood of a transition point, American Mathematical Society, Providence, R. I., 1974, Memoirs of the American Mathematical Society, No. 149  MR 440140 |  Zbl 0297.34051
[14] W. Wasow, Linear turning point theory, Applied Mathematical Sciences 54, Springer-Verlag, New York, 1985 Article |  MR 771669 |  Zbl 0558.34049
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