Geodesic
Asymptotic analysis of the generalized convection problem
Eurasian mathematical journal, Tome 6 (2015) no. 1, pp. 41-55 
N. Ivleva; V. Levenshtam. Asymptotic analysis of the generalized convection problem. Eurasian mathematical journal, Tome 6 (2015) no. 1, pp. 41-55. http://geodesic.mathdoc.fr/item/EMJ_2015_6_1_a2/
@article{EMJ_2015_6_1_a2,
     author = {N. Ivleva and V. Levenshtam},
     title = {Asymptotic analysis of the generalized convection problem},
     journal = {Eurasian mathematical journal},
     pages = {41--55},
     year = {2015},
     volume = {6},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2015_6_1_a2/}
}
TY  - JOUR
AU  - N. Ivleva
AU  - V. Levenshtam
TI  - Asymptotic analysis of the generalized convection problem
JO  - Eurasian mathematical journal
PY  - 2015
SP  - 41
EP  - 55
VL  - 6
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/EMJ_2015_6_1_a2/
LA  - en
ID  - EMJ_2015_6_1_a2
ER  - 
%0 Journal Article
%A N. Ivleva
%A V. Levenshtam
%T Asymptotic analysis of the generalized convection problem
%J Eurasian mathematical journal
%D 2015
%P 41-55
%V 6
%N 1
%U http://geodesic.mathdoc.fr/item/EMJ_2015_6_1_a2/
%G en
%F EMJ_2015_6_1_a2

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

The averaging method is justified and the complete asymptotics of a solution periodic in time is constructed and justified for an evolutional system of partial differential equations with quickly oscillating in time junior terms, some of which are proportional to the frequency of oscillations. The considered system generalizes the well-known thermal liquid convection problem (in Oberdeck–Boussinesc approach) when a vessel with a liquid vibrates with high frequency.

[1] N. N. Bogoliubov, Y. A. Mitropolski, Asymptotic methods in the theory of non-linear oscillations, Gordon and Breach, New York, 1961 | MR

[2] N. E. Kochin, I. A. Kibel', N. W. Rose, Theoretical hydromechanics, Interscience, London, 1964 | MR | Zbl

[3] O. A. Ladyzhenskaya, Mathematical problems in the dynamics of a viscous, incompressible fluids, Nauka, M., 1970 (in Russian)

[4] V. B. Levenshtam, “Asymptotic integration of a vibrational convection problem”, Differential Equations, 34:4 (1998), 522–531 | MR | Zbl

[5] V. B. Levenshtam, “Asymptotic expansion of the solution to the problem of vibrational convection”, Comput. Math. Math. Phys., 40:9 (2000), 1416–1424 | MR | Zbl

[6] I. B. Simonenko, “A justification of the averaging method for a problem of convection in a field of rapidly oscilating forces and for other parabolic equations”, Mathematics of the USSR-Sbornik, 87(129):2 (1972), 236–253 (in Russian) | MR | Zbl

[7] V. A. Solonnikov, “On the differential properties of the solution of the first boundary-value problem for a non-stationary system of Navier–Stokes equations”, Trudy Mat. Inst. Steklov, 73, 1964, 221–291 (in Russian) | MR | Zbl

[8] R. M. Temam, Navier–Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1977 | MR | Zbl

[9] M. I. Vishik, L. A. Lyusternik, “Regular degeneracy and the boundary layer for nonlinear differential equations with a small parameter”, Uspekhi Mat. Nauk, 15:4 (1960), 3–95 (in Russian) | MR

[10] V. I. Yudovich, The linearization method in hydrodynamical stability theory, Translation of Mathematical Monographs, 74, AMS Providence, 1989, 170 pp. | MR | Zbl

[11] S. M. Zenkovskaya, I. B. Simonenko, “On influence of high-frequency vibrations on the onset of convection”, Izvestiya AN USSR. Mech. of Fluid and Gas, 1966, no. 5, 51–55 (in Russian)