Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales

Tatiana Danielsson; Pernilla Johnsen

Mathematica Bohemica (2021)

  • Volume: 146, Issue: 4, page 483-511
  • ISSN: 0862-7959

Abstract

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In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in L 2 ( 0 , T ; H 0 1 ( Ω ) ) , fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation ε p ∂ t u ε ( x , t ) - ∇ · ( a ( x ε - 1 , x ε - 2 , t ε - q , t ε - r ) ∇ u ε ( x , t ) ) = f ( x , t ) , where 0 < p < q < r . The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by p , compared to the standard matching that gives rise to local parabolic problems.

How to cite

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Danielsson, Tatiana, and Johnsen, Pernilla. "Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales." Mathematica Bohemica 146.4 (2021): 483-511. <http://eudml.org/doc/297605>.

@article{Danielsson2021,
abstract = {In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^\{2\}(0,T;H_\{0\}^\{1\}(\Omega ))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $\varepsilon ^\{p\}\partial _\{t\}u_\{\varepsilon \}(x,t) -\nabla \cdot ( a( x\varepsilon ^\{-1\} ,x\varepsilon ^\{-2\},t\varepsilon ^\{-q\},t\varepsilon ^\{-r\}) \nabla u_\{\varepsilon \}(x,t) ) = f(x,t) $, where $0<p<q<r$. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by $p$, compared to the standard matching that gives rise to local parabolic problems.},
author = {Danielsson, Tatiana, Johnsen, Pernilla},
journal = {Mathematica Bohemica},
keywords = {homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence},
language = {eng},
number = {4},
pages = {483-511},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales},
url = {http://eudml.org/doc/297605},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Danielsson, Tatiana
AU - Johnsen, Pernilla
TI - Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 4
SP - 483
EP - 511
AB - In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2}(0,T;H_{0}^{1}(\Omega ))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $\varepsilon ^{p}\partial _{t}u_{\varepsilon }(x,t) -\nabla \cdot ( a( x\varepsilon ^{-1} ,x\varepsilon ^{-2},t\varepsilon ^{-q},t\varepsilon ^{-r}) \nabla u_{\varepsilon }(x,t) ) = f(x,t) $, where $0<p<q<r$. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by $p$, compared to the standard matching that gives rise to local parabolic problems.
LA - eng
KW - homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence
UR - http://eudml.org/doc/297605
ER -

References

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