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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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: A wide class of matrix elliptic second-order differential operators $ \mathcal{A}=\mathcal{A}(\mathbf{x},\mathbf{D})$ with periodic coefficients, acting in $ L_2(\mathbb{R}^d;\mathbb{C}^n)$, is studied. The operator $ \mathcal{A}$ is assumed to admit a factorization of the form $ \mathcal{A}=\mathcal{X}^*\mathcal{X}$, where $ \mathcal{X}$ is a homogeneous first-order differential operator. An approximation for the operator exponential $ e^{-\mathcal{A}\tau}$ as $ \tau\rightarrow\infty$ in the $ (L_2(\mathbb{R}^d;\mathbb{C}^n))$-operator norm is obtained, with error estimate of the order of $ \tau^{-1}$. In the approximation, a corrector is taken into account. The result is applied to the study of homogenization for solutions of the Cauchy problem $ \partial_\tau\mathbf{u}_\varepsilon= -\mathcal{A}_\varepsilon\mathbf{u}_\varepsilon$, where $ \mathcal{A}_\varepsilon=\mathcal{A}(\mathbf{x}/\varepsilon,\mathbf{D})$. An approximation with corrector for $ \mathbf{u}_\varepsilon$ in the $ (L_2(\mathbb{R}^d;\mathbb{C}^n))$-norm is obtained for fixed $ \tau>0$, with error estimate of the order of $ \varepsilon^2$.


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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: http://dx.doi.org/10.1090/S1061-0022-09-01083-8
PII: S 1061-0022(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