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Funktsional. Anal. i Prilozhen., 2015, Volume 49, Issue 1, Pages 88–93 (Mi faa3177)  

This article is cited in 7 scientific papers (total in 7 papers)

Brief communications

Homogenization of Solutions of Initial Boundary Value Problems for Parabolic Systems

Yu. M. Meshkovaa, T. A. Suslinab

a Chebyshev Laboratory, St. Petersburg State University, Department of Mathematics and Mechanics
b St. Petersburg State University, Faculty of Physics

Abstract: Let ${\mathcal O}\subset {\mathbb R}^d$ be a bounded $C^{1,1}$ domain. In $L_2({\mathcal O};{\mathbb C}^n)$ we consider strongly elliptic operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ given by the differential expression $b({\mathbf D})^*g({\mathbf x}/\varepsilon)b({\mathbf D})$, $\varepsilon>0$, with Dirichlet and Neumann boundary conditions, respectively. Here $g({\mathbf x})$ is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and $b({\mathbf D})$ is a first-order differential operator. We find approximations of the operators $\exp(-A_{D,\varepsilon} t)$ and $\exp(-A_{N,\varepsilon} t)$ for fixed $t>0$ and small $\varepsilon$ in the $L_2 \to L_2$ and $L_2 \to H^1$ operator norms with error estimates depending on $\varepsilon$ and $t$. The results are applied to homogenize the solutions of initial boundary value problems for parabolic systems.

Keywords: homogenization of periodic differential operators, parabolic systems, initial boundary value problems, effective operator, corrector, operator error estimates

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00760
Ministry of Education and Science of the Russian Federation 11.G34.31.0026
Gazprom Neft


DOI: https://doi.org/10.4213/faa3177

Full text: PDF file (171 kB)
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English version:
Functional Analysis and Its Applications, 2015, 49:1, 72–76

Bibliographic databases:

UDC: 517.956.4
Received: 07.02.2014

Citation: Yu. M. Meshkova, T. A. Suslina, “Homogenization of Solutions of Initial Boundary Value Problems for Parabolic Systems”, Funktsional. Anal. i Prilozhen., 49:1 (2015), 88–93; Funct. Anal. Appl., 49:1 (2015), 72–76

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. A. Suslina, “Homogenization of elliptic operators with periodic coefficients depending on the spectral parameter”, St. Petersburg Math. J., 27:4 (2016), 651–708  mathnet  crossref  mathscinet  isi  elib
    2. Yu. M. Meshkova, T. A. Suslina, “Two-parametric error estimates in homogenization of second-order elliptic systems in $\mathbb{R}^d$”, Appl. Anal., 95:7 (2016), 1413–1448  crossref  mathscinet  zmath  isi  elib  scopus
    3. Yu. M. Meshkova, T. A. Suslina, “Homogenization of initial boundary value problems for parabolic systems with periodic coefficients”, Appl. Anal., 95:8 (2016), 1736–1775  crossref  mathscinet  zmath  isi  elib  scopus
    4. J. Geng, Zh. Shen, “Convergence rates in parabolic homogenization with time-dependent periodic coefficients”, J. Funct. Anal., 272:5 (2017), 2092–2113  crossref  mathscinet  zmath  isi  scopus
    5. W. Niu, Ya. Xu, “Convergence rates in homogenization of higher-order parabolic systems”, Discret. Contin. Dyn. Syst., 38:8 (2018), 4203–4229  crossref  mathscinet  zmath  isi  scopus
    6. Niu W., Xu Ya., “A Refined Convergence Result in Homogenization of Second Order Parabolic Systems”, J. Differ. Equ., 266:12 (2019), 8294–8319  crossref  mathscinet  zmath  isi  scopus
    7. N. N. Senik, “On homogenization for locally periodic elliptic and parabolic operators”, Funct. Anal. Appl., 54:1 (2020), 68–72  mathnet  crossref  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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