A periodic parabolic Cauchy problem: Homogenization with corrector

Author:
E. S. Vasilevskaya

Translated by:
the author

Original publication:
Algebra i Analiz, tom **21** (2009), nomer 1.

Journal:
St. Petersburg Math. J. **21** (2010), 1-41

MSC (2000):
Primary 35B27, 35K30

Published electronically:
November 4, 2009

MathSciNet review:
2553050

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Abstract | References | Similar Articles | Additional Information

Abstract: A wide class of matrix elliptic second-order differential operators with periodic coefficients, acting in , is studied. The operator is assumed to admit a factorization of the form , where is a homogeneous first-order differential operator. An approximation for the operator exponential as in the -operator norm is obtained, with error estimate of the order of . In the approximation, a corrector is taken into account. The result is applied to the study of homogenization for solutions of the Cauchy problem , where . An approximation with corrector for in the -norm is obtained for fixed , with error estimate of the order of .

**1.**N. S. Bakhvalov and G. P. Panasenko,*Osrednenie protsessov v periodicheskikh sredakh*, “Nauka”, Moscow, 1984 (Russian). Matematicheskie zadachi mekhaniki kompozitsionnykh materialov. [Mathematical problems of the mechanics of composite materials]. MR**797571**

N. Bakhvalov and G. Panasenko,*Homogenisation: averaging processes in periodic media*, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials; Translated from the Russian by D. Leĭtes. MR**1112788****2.**Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou,*Asymptotic analysis for periodic structures*, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR**503330****3.**Michael Birman and Tatyana Suslina,*Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics*, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71–107. MR**1882692****4.**M. Sh. Birman and T. A. Suslina,*Periodic second-order differential operators. Threshold properties and averaging*, Algebra i Analiz**15**(2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**15**(2004), no. 5, 639–714. MR**2068790**, 10.1090/S1061-0022-04-00827-1**5.**M. Sh. Birman and T. A. Suslina,*Threshold approximations for the resolvent of a factorized selfadjoint family taking a corrector into account*, Algebra i Analiz**17**(2005), no. 5, 69–90 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**17**(2006), no. 5, 745–762. MR**2241423**, 10.1090/S1061-0022-06-00927-7**6.**M. Sh. Birman and T. A. Suslina,*Averaging of periodic elliptic differential operators taking a corrector into account*, Algebra i Analiz**17**(2005), no. 6, 1–104 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**17**(2006), no. 6, 897–973. MR**2202045**, 10.1090/S1061-0022-06-00935-6**7.**M. Sh. Birman and T. A. Suslina,*Averaging of periodic differential operators taking a corrector into account. Approximation of solutions in the Sobolev class 𝐻²(ℝ^{𝕕})*, Algebra i Analiz**18**(2006), no. 6, 1–130 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**18**(2007), no. 6, 857–955. MR**2307356**, 10.1090/S1061-0022-07-00977-6**8.**V. V. Zhikov,*Spectral approach to asymptotic diffusion problems*, Differentsial′nye Uravneniya**25**(1989), no. 1, 44–50, 180 (Russian); English transl., Differential Equations**25**(1989), no. 1, 33–39. MR**986395****9.**V. V. Zhikov,*On some estimates from homogenization theory*, Dokl. Akad. Nauk**406**(2006), no. 5, 597–601 (Russian). MR**2347318****10.**V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik,*Usrednenie differentsialnykh operatorov*, “Nauka”, Moscow, 1993 (Russian, with English and Russian summaries). MR**1318242**

V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik,*Homogenization of differential operators and integral functionals*, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR**1329546****11.**V. V. Zhikov and S. E. Pastukhova,*On operator estimates for some problems in homogenization theory*, Russ. J. Math. Phys.**12**(2005), no. 4, 515–524. MR**2201316****12.**V. V. Zhikov and S. E. Pastukhova,*Estimates of homogenization for a parabolic equation with periodic coefficients*, Russ. J. Math. Phys.**13**(2006), no. 2, 224–237. MR**2262826**, 10.1134/S1061920806020087**13.**O. A. Ladyženskaja and N. N. Ural′ceva,*Lineinye i kvazilineinye uravneniya ellipticheskogo tipa*, Izdat. “Nauka”, Moscow, 1964 (Russian). MR**0211073**

Olga A. Ladyzhenskaya and Nina N. Ural′tseva,*Linear and quasilinear elliptic equations*, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR**0244627****14.**T. A. Suslina,*On the averaging of periodic parabolic systems*, Funktsional. Anal. i Prilozhen.**38**(2004), no. 4, 86–90 (Russian); English transl., Funct. Anal. Appl.**38**(2004), no. 4, 309–312. MR**2117512**, 10.1007/s10688-005-0010-z**15.**T. A. Suslina,*Homogenization of a periodic parabolic Cauchy problem*, Nonlinear equations and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 201–233. MR**2343612**

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Additional Information

**E. S. Vasilevskaya**

Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Petrodvorets, 198504 St. Petersburg, Russia

Email:
vasilevskaya-e@yandex.ru

DOI:
https://doi.org/10.1090/S1061-0022-09-01083-8

Keywords:
Parabolic Cauchy problem,
homogenization,
effective operator,
corrector

Received by editor(s):
September 1, 2009

Published electronically:
November 4, 2009

Additional Notes:
Supported by RFBR (grant no. 08-01-00209-a) and by a “Scientific schools” grant (no. 816.2008.1)

Article copyright:
© Copyright 2009
American Mathematical Society