Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space H1 (ℝd)
Mathematical Modelling of Natural Phenomena (2010)
- Volume: 5, Issue: 4, page 390-447
- ISSN: 0973-5348
Access Full Article
top
Abstract
topHow to cite
topSuslina, T.. "Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space H1 (ℝd)." Mathematical Modelling of Natural Phenomena 5.4 (2010): 390-447. <http://eudml.org/doc/197654>.
@article{Suslina2010,
abstract = {In L2(ℝd;
ℂn), we consider a wide class of matrix elliptic second
order differential operators $\mathcal\{A\}$ε
with rapidly oscillating coefficients (depending on x/ε).
For a fixed τ > 0 and small ε > 0, we find
approximation of the operator exponential exp(− $\mathcal\{A\}$ετ) in the
(L2(ℝd;
ℂn) →
H1(ℝd;
ℂn))-operator norm with an error term of order
ε. In this approximation, the corrector is taken into account. The
results are applied to homogenization of a periodic parabolic Cauchy problem. },
author = {Suslina, T.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {periodic differential operators; parabolic Cauchy problem; homogenization; effective operator; corrector; exponential operator},
language = {eng},
month = {5},
number = {4},
pages = {390-447},
publisher = {EDP Sciences},
title = {Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space H1 (ℝd)},
url = {http://eudml.org/doc/197654},
volume = {5},
year = {2010},
}
TY - JOUR
AU - Suslina, T.
TI - Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space H1 (ℝd)
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 390
EP - 447
AB - In L2(ℝd;
ℂn), we consider a wide class of matrix elliptic second
order differential operators $\mathcal{A}$ε
with rapidly oscillating coefficients (depending on x/ε).
For a fixed τ > 0 and small ε > 0, we find
approximation of the operator exponential exp(− $\mathcal{A}$ετ) in the
(L2(ℝd;
ℂn) →
H1(ℝd;
ℂn))-operator norm with an error term of order
ε. In this approximation, the corrector is taken into account. The
results are applied to homogenization of a periodic parabolic Cauchy problem.
LA - eng
KW - periodic differential operators; parabolic Cauchy problem; homogenization; effective operator; corrector; exponential operator
UR - http://eudml.org/doc/197654
ER -
References
top- N. S. Bakhvalov, G. P. Panasenko. Homogenization: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials. "Nauka", Moscow, 1984; English transl., Math. Appl. (Soviet Ser.), vol. 36, Kluwer Acad. Publishers Group, Dordrecht, 1989.
- A. Bensoussan, J. L. Lions, G. Papanicolaou. Asymptotic analysis for periodic structures. Stud. Math. Appl., vol. 5, North-Holland Publishing Company, Amsterdam–New York, 1978, 700 pp.
- M. Birman, T. 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.
- M. Sh. Birman, T. A. Suslina. Second order periodic differential operators. Threshold properties and homogenization. Algebra i Analiz, 15 (2003), no. 5, 1–108; English transl., St. Petersburg Math. J.,15 (2004), no. 5, 639–714.
- M. Sh. Birman, T. A. Suslina. Threshold approximations with corrector term for the resolvent of a factorized selfadjoint operator family. Algebra i Analiz, 17 (2005), no. 5, 69–90; English transl., St. Petersburg Math. J.,17 (2006), no. 5, 745–762.
- M. Sh. Birman, T. A. Suslina.Homogenization with corrector term for periodic elliptic differential operators, Algebra i Analiz, 17 (2005), no. 6, 1–104.
- M. Sh. Birman, T. A. Suslina. Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev classH1(d), Algebra i Analiz, 18 (2006), no. 6, 1–130; English transl., St. Petersburg Math. J., 18 (2007), no. 6, 857–955.
- T. Kato. Perturbation theory for linear operators, 2nd ed. Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin-New York, 1976.
- E. Sanchez-Palencia. Nonhomogeneous media and vibration theory. Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin–New York, 1980.
- T. A. Suslina. Homogenization of periodic parabolic systems. Funktsional. Anal. i Prilozhen., 38 (2004), no. 4, 86-90; English transl., Funct. Anal. Appl.,38 (2004), no. 4, 309–312.
- T. A. Suslina. Homogenization of periodic parabolic Cauchy problem. Amer. Math. Soc. Transl., ser.2, 220 (2007), 201–233.
- E. S. Vasilevskaya. A periodic parabolic Cauchy problem: homogenization with corrector. Algebra i Analiz, 20 (2009), no. 1, 3–60; English transl., St. Petersburg Math. J., 20 (2010), no. 1.
- V. V. Zhikov, On some estimates of homogenization theory. Dokl. Ros. Akad. Nauk, 406 (2006), no. 5, 597–601; English transl., Dokl. Math., 73 (2006), 96–99.
- V. V. Zhikov, S. M. Kozlov, O. A. Oleinik. Homogenization of differential operators. "Nauka", Moscow, 1993; English transl., Springer-Verlag, Berlin, 1994.
- V. V. Zhikov, S. E. Pastukhova. On operator estimates for some problems in homogenization theory. Russ. J. Math. Phys., 12 (2005), no. 4, 515–524.
- V. V. Zhikov, S. E. Pastukhova. Estimates of homogenization for a parabolic equation with periodic coefficients. Russ. J. Math. Phys., 13 (2006), no. 2, 251–265.
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.