Turning points at infinity and stability of detonations
Williams, Mark1
1 University of North Carolina Mathematics Department CB 3250, Phillips Hall Chapel Hill NC 27599, USA
Journées équations aux dérivées partielles (2015), article no. 12, 8 p.

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.

DOI : 10.5802/jedp.641
Williams, Mark 1

1 University of North Carolina Mathematics Department CB 3250, Phillips Hall Chapel Hill NC 27599, USA 
@incollection{JEDP_2015____A12_0,
     author = {Williams, Mark},
     title = {Turning points at infinity and stability of detonations},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {12},
     pages = {1--8},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2015},
     doi = {10.5802/jedp.641},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.641/}
}
TY  - JOUR
AU  - Williams, Mark
TI  - Turning points at infinity and stability of detonations
JO  - Journées équations aux dérivées partielles
PY  - 2015
SP  - 1
EP  - 8
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.641/
DO  - 10.5802/jedp.641
LA  - en
ID  - JEDP_2015____A12_0
ER  - 
%0 Journal Article
%A Williams, Mark
%T Turning points at infinity and stability of detonations
%J Journées équations aux dérivées partielles
%D 2015
%P 1-8
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.641/
%R 10.5802/jedp.641
%G en
%F JEDP_2015____A12_0
Williams, Mark. Turning points at infinity and stability of detonations. Journées équations aux dérivées partielles (2015), article  no. 12, 8 p. doi : 10.5802/jedp.641. http://www.numdam.org/articles/10.5802/jedp.641/

[1] Erpenbeck, J. J. Stability of steady-state equilibrium detonations, Phys. Fluids, Volume 5 (1962), pp. 604-614

[2] Erpenbeck, J. J. Stability of step shocks, Phys. Fluids, Volume 5 (1962) no. 10, pp. 1181-1187 | MR | Zbl

[3] Erpenbeck, J. J. Stability of idealized one-reaction detonations, Phys. Fluids, Volume 7 (1964), pp. 684-695 | Zbl

[4] Erpenbeck, J. J. Stability of detonations for disturbances of small transverse wavelength (1965) (Los Alamos Preprint, LA-3306)

[5] Erpenbeck, J. J. Detonation stability for disturbances of small transverse wave length, Phys. Fluids, Volume 9 (1966), pp. 1293-1306 | Zbl

[6] Fickett, W.; Davis, W. Detonation: Theory and Experiment, Univ. California Press, Berkeley, 1979

[7] Lafitte, O.; Williams, M.; Zumbrun, K. The Erpenbeck high frequency instability theorem for ZND detonations, Archive for Rational Mechanics and Analysis, Volume 204 (2012), pp. 141-187 | MR

[8] Lafitte, O.; Williams, M.; Zumbrun, K. Block-diagonalization of ODEs in the semiclassical limit and C ω vs. C ∞ stationary phase (2015) (submitted, http://arxiv.org/abs/1507.03116) | MR

[9] Lafitte, O.; Williams, M.; Zumbrun, K. High-frequency stability of detonations and turning points at infinity, SIAM J. Math. Analysis, Volume 47-3 (2015), pp. 1800-1878 (http://arxiv.org/abs/1312.6906) | MR

[10] Olver, F. W. J. Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter, Philos. Trans. Roy. Soc. London. Ser. A, Volume 250 (1958), pp. 479-517 | MR | Zbl

[11] Olver, F. W. J. Asymptotics and special functions, Academic Press, New York-London, 1974, pp. xvi+572 (Computer Science and Applied Mathematics) | MR | Zbl

[12] Short, M. Theory and modeling of detonation wave stability: A brief look at the past and toward the future (Proceedings, ICDERS 2005)

[13] Sibuya, Y. Uniform simplification in a full neighborhood of a transition point, American Mathematical Society, Providence, R. I., 1974, pp. vi+106 (Memoirs of the American Mathematical Society, No. 149) | MR | Zbl

[14] Wasow, W. Linear turning point theory, Applied Mathematical Sciences, 54, Springer-Verlag, New York, 1985, pp. ix+246 | DOI | MR | Zbl

Cité par Sources :