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The averaging method in the convection problem with high-frequency oblique vibrations. (English. Russian original) Zbl 0874.35094

Sib. Math. J. 37, No. 5, 970-982 (1996); translation from Sib. Mat. Zh. 37, No. 5, 1103-1116 (1996).
In former papers the averaging method was used for studying the problem of the onset of a convection flow in a fluid subject to high-frequency vertical vibrations, oblique vibrations, or vibrations in weightlessness. Mathematical justification of applicability of the method to such problems was carried out for vertical oscillations of a particular form and later for a broad class of vertical oscillations and oscillations in weightlessness.
In the present article, we extend the justification to a broad class of vibrations acting in arbitrary directions. The results are exposed for some system of evolution differential equations which involves the above-indicated convection problems (as particular cases). We especially point out that here we essentially improve the earlier results in the framework of a more general problem. For instance, we considerably widen the scale that characterizes smoothness of the initial data of the problem, find out conditions under which (weak) solutions are classical, and study instability of solutions in more detail.

MSC:

35Q35 PDEs in connection with fluid mechanics
76E15 Absolute and convective instability and stability in hydrodynamic stability
76M25 Other numerical methods (fluid mechanics) (MSC2010)
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[1] S. M. Zen’kovskaya and I. B. Simonenko, ”On the influence of high frequency vibrations on the onset of convection,” Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza, No. 5, 51–55 (1966).
[2] S. M. Zen’kovskaya, ”Study of convection in a fluid layer under oblique vibrations,” submitted to VINITI on 1978, No. 2437-78.
[3] G. Z. Gershuni and E. M. Zhukhovitskii, ”On free heat convection in a vibration field in weightlessness,” Dokl. Akad. Nauk SSSR,249, No. 3, 580–584 (1979).
[4] G. Z. Gershuni and E. M. Zhukhovitskii, ”On convective instability of a fluid in a vibration field,” Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza, No. 4, 12–19 (1981).
[5] N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory on Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1955).
[6] I. B. Simonenko, ”Justification of the averaging method for the convection problem in a field of rapidly oscillating forces and for other parabolic equations,” Mat. Sb.,87, No. 2, 236–253 (1972). · Zbl 0253.35049
[7] V. B. Levenshtam, ”Justification of the averaging method for the convection problem with highfrequency vibrations,” Sibirsk. Mat. Zh.,34, No. 2, 92–109 (1993) · Zbl 0834.35012
[8] L. D. Landau and E. M. Lifshits, Hydrodynamics [in Russian], Nauka, Moscow (1986). · Zbl 0664.76001
[9] V. I. Yudovich, The Linearization Method in Hydrodynamic Stability Theory [in Russian], Rostov. Univ., Rostov-on-Don (1984). · Zbl 0553.76038
[10] M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).
[11] V. B. Levenshtam, ”On the range of the restriction of the projection П of hydrodynamics to a Hölder space,” in: Abstracts: Proceedings of the VII School-Seminar ”Nonlinear Problems of Stability Theory” [in Russian], Moscow Univ., Moscow, 1992, p. 35.
[12] I. B. Simonenko, The Averaging Method in the Theory of Nonlinear Equations of Parabolic Type with Applications to the Problems of Hydrodynamic Stability [in Russian], Rostov. Univ., Rostov-on-Don (1989). · Zbl 0707.35072
[13] Yu. L. Daletskii and M. G. Krein, Stability of Solutions to Differential Equations in Banach Space [in Russian], Nauka, Moscow (1970).
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