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 Izv. RAN. Ser. Mat., 2006, Volume 70, Issue 2, Pages 25–56 (Mi izv555)

This article is cited in 11 scientific papers (total in 11 papers)

Justification of the averaging method for parabolic equations containing rapidly oscillating terms with large amplitudes

V. B. Levenshtam

Rostov State University

Abstract: We justify the averaging method for abstract parabolic equations with stationary principal part that contain non-linearities (subordinate to the principal part) some of whose terms are rapidly oscillating in time with zero mean and are proportional to the square root of the frequency of oscillation. Our interest in the exponent 1/2 is motivated by the fact that terms proportional to lower powers of the frequency have no influence on the average. For linear equations of the same type, we justify an algorithm for the study of the stability of solutions in the case when the stationary averaged problem has eigenvalues on the imaginary axis (the critical case).

DOI: https://doi.org/10.4213/im555

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English version:
Izvestiya: Mathematics, 2006, 70:2, 233–263

Bibliographic databases:

UDC: 517.43, 517.948, 513.881
MSC: 35R20, 35G05, 35K10
Received: 11.09.2003
Revised: 22.09.2004

Citation: V. B. Levenshtam, “Justification of the averaging method for parabolic equations containing rapidly oscillating terms with large amplitudes”, Izv. RAN. Ser. Mat., 70:2 (2006), 25–56; Izv. Math., 70:2 (2006), 233–263

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Levenshtam V.B., “Some questions of averaging theory for parabolic equations with large high-frequency summands”, Doklady Mathematics, 74:3 (2006), 827–830
2. A. K. Kapikyan, V. B. Levenshtam, “First-order partial differential equations with large high-frequency terms”, Comput. Math. Math. Phys., 48:11 (2008), 2059–2076
3. V. B. Levenshtam, “O vzaimosvyazi dvukh klassov reshenii uravnenii Nave–Stoksa. II”, Vladikavk. matem. zhurn., 14:4 (2012), 52–62
4. Ivleva N.S., Levenshtam V.B., “Asimptoticheskoe integrirovanie parabolicheskikh sistem s bolshim parametrom”, Izvestiya vysshikh uchebnykh zavedenii. severo-kavkazskii region. seriya: estestvennye nauki, 2012, no. 6, 26–31
5. Khatlamadzhiyan G.L., “Asymptotic Integration of Some Evolution Problems with Large High-Frequency Terms”, Differ. Equ., 49:12 (2013), 1596–1608
6. V. B. Levenshtam, “Justification of the method of averaging for the system of equations with the Navier–Stokes operator in the principal part”, St. Petersburg Math. J., 26:1 (2015), 69–90
7. V. V. Gusachenko, V. B. Levenshtam, “Asymptotic analysis of linear parabolic problems with singularities”, Comput. Math. Math. Phys., 55:1 (2015), 71–84
8. Levenshtam V.B., Ishmeev M.R., “Asymptotic Integration of Linear System With High-Frequency Coefficients and Stokes Operator in the Main Part”, Asymptotic Anal., 92:3-4 (2015), 363–376
9. V. L. Khatskevich, “On the Homogenization Principle in a Time-Periodic Problem for the Navier–Stokes Equations with Rapidly Oscillating Mass Force”, Math. Notes, 99:5 (2016), 757–768
10. L. I. Sazonov, “High-frequency asymptotics of solutions of ODE in a Banach space”, Izv. Math., 81:6 (2017), 1234–1252
11. Ishmeev M.R., Levenshtam V.B., “A System of Partial Differential Equations With High-Frequency Coefficients and Stokes Operator in the Main Part. Asymptotic Integration in the Case of Multiple Degeneration”, Russ. J. Math. Phys., 25:3 (2018), 284–299
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