Geodesic
Stability criterion for small perturbations for a quasi-gasdynamic system of equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 2, pp. 262-269 
A. A. Zlotnik; I. A. Zlotnik. Stability criterion for small perturbations for a quasi-gasdynamic system of equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 2, pp. 262-269. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_2_a6/
@article{ZVMMF_2006_46_2_a6,
     author = {A. A. Zlotnik and I. A. Zlotnik},
     title = {Stability criterion for small perturbations for a~quasi-gasdynamic system of equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {262--269},
     year = {2006},
     volume = {46},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_2_a6/}
}
TY  - JOUR
AU  - A. A. Zlotnik
AU  - I. A. Zlotnik
TI  - Stability criterion for small perturbations for a quasi-gasdynamic system of equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2006
SP  - 262
EP  - 269
VL  - 46
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_2_a6/
LA  - ru
ID  - ZVMMF_2006_46_2_a6
ER  - 
%0 Journal Article
%A A. A. Zlotnik
%A I. A. Zlotnik
%T Stability criterion for small perturbations for a quasi-gasdynamic system of equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2006
%P 262-269
%V 46
%N 2
%U http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_2_a6/
%G ru
%F ZVMMF_2006_46_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

The stability of small perturbations against a constant background is studied for a system of quasi-gasdynamic equations in an arbitrary number of space variables. It is established that, for a fixed adiabatic exponent $\gamma$, the stability is determined only by the background Mach number, and a necessary and sufficient condition for stability at any Mach number is $\gamma\le\bar\gamma$, where $\bar\gamma\approx6.2479$. The proof is based on a direct analysis of the corresponding complex characteristic numbers depending on several parameters. The multidimensional case is successfully reduced to the one-dimensional one. Then, the generalized Routh-Hurwitz criterion is applied in conjunction with analytical calculations based on Mathematica.

[1] Chetverushkin B. N., Kineticheskie skhemy i kvazigazodinamicheskaya sistema uravnenii, MAKS Press, M., 2004

[2] Zlotnik A. A., “Klassifikatsiya nekotorykh modifikatsii sistemy uravnenii Eilera”, Dokl. RAN, 407:6 (2006), 747–751 | MR

[3] Eidelman S. D., Parabolicheskie sistemy, Nauka, M., 1964 | MR

[4] Gantmakher F. R., Teoriya matrits, Nauka, M., 1988 | MR | Zbl

[5] Dorodnitsyn L. V., “Ob ustoichivosti malykh kolebanii v kvazigazodinamicheskoi sisteme uravnenii”, Zh. vychisl. matem. i matem. fiz., 44:7 (2004), 1299–1305 | MR | Zbl