Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales
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_tu_{\varepsil...
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Institute of Mathematics of the Czech Academy of Science
2021
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oai:doaj.org-article:901588ce3f1340a2a6a03ac519dd46bb2021-11-08T09:59:13ZHomogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales0862-79592464-713610.21136/MB.2021.0087-19https://doaj.org/article/901588ce3f1340a2a6a03ac519dd46bb2021-12-01T00:00:00Zhttp://mb.math.cas.cz/full/146/4/mb146_4_8.pdfhttps://doaj.org/toc/0862-7959https://doaj.org/toc/2464-7136In 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_tu_{\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.Tatiana DanielssonPernilla JohnsenInstitute of Mathematics of the Czech Academy of Sciencearticle homogenization parabolic problem multiscale convergence very weak multiscale convergence two-scale convergenceMathematicsQA1-939ENMathematica Bohemica, Vol 146, Iss 4, Pp 483-511 (2021) |
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homogenization parabolic problem multiscale convergence very weak multiscale convergence two-scale convergence Mathematics QA1-939 |
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homogenization parabolic problem multiscale convergence very weak multiscale convergence two-scale convergence Mathematics QA1-939 Tatiana Danielsson Pernilla Johnsen Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales |
| description |
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_tu_{\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. |
| format |
article |
| author |
Tatiana Danielsson Pernilla Johnsen |
| author_facet |
Tatiana Danielsson Pernilla Johnsen |
| author_sort |
Tatiana Danielsson |
| title |
Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales |
| title_short |
Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales |
| title_full |
Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales |
| title_fullStr |
Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales |
| title_full_unstemmed |
Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales |
| title_sort |
homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales |
| publisher |
Institute of Mathematics of the Czech Academy of Science |
| publishDate |
2021 |
| url |
https://doaj.org/article/901588ce3f1340a2a6a03ac519dd46bb |
| work_keys_str_mv |
AT tatianadanielsson homogenizationoflinearparabolicequationswiththreespatialandthreetemporalscalesforcertainmatchingsbetweenthemicroscopicscales AT pernillajohnsen homogenizationoflinearparabolicequationswiththreespatialandthreetemporalscalesforcertainmatchingsbetweenthemicroscopicscales |
| _version_ |
1718442732072992768 |