Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

Pernilla Johnsen; Tatiana Lobkova

Applications of Mathematics (2018)

  • Volume: 63, Issue: 5, page 503-521
  • ISSN: 0862-7940

Abstract

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This paper is devoted to the study of the linear parabolic problem ε ∂ t u ε ( x , t ) - ∇ · ( a ( x / ε , t / ε 3 ) ∇ u ε ( x , t ) ) = f ( x , t ) by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient ε in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence { u ε } different from the standard setting are used, which means that these results are also of independent interest.

How to cite

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Johnsen, Pernilla, and Lobkova, Tatiana. "Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales." Applications of Mathematics 63.5 (2018): 503-521. <http://eudml.org/doc/294528>.

@article{Johnsen2018,
abstract = {This paper is devoted to the study of the linear parabolic problem $\varepsilon \partial _\{t\}u_\{\varepsilon \}( x,t) -\nabla \cdot ( a( \{x\}/\{\varepsilon \},\{t\}/\{\varepsilon ^\{3\}\}) \nabla u_\{\varepsilon \}( x,t)) =f( x,t) $ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\varepsilon $ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence $\lbrace u_\{\varepsilon \}\rbrace $ different from the standard setting are used, which means that these results are also of independent interest.},
author = {Johnsen, Pernilla, Lobkova, Tatiana},
journal = {Applications of Mathematics},
keywords = {homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence},
language = {eng},
number = {5},
pages = {503-521},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales},
url = {http://eudml.org/doc/294528},
volume = {63},
year = {2018},
}

TY - JOUR
AU - Johnsen, Pernilla
AU - Lobkova, Tatiana
TI - Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 5
SP - 503
EP - 521
AB - This paper is devoted to the study of the linear parabolic problem $\varepsilon \partial _{t}u_{\varepsilon }( x,t) -\nabla \cdot ( a( {x}/{\varepsilon },{t}/{\varepsilon ^{3}}) \nabla u_{\varepsilon }( x,t)) =f( x,t) $ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\varepsilon $ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence $\lbrace u_{\varepsilon }\rbrace $ different from the standard setting are used, which means that these results are also of independent interest.
LA - eng
KW - homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence
UR - http://eudml.org/doc/294528
ER -

References

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