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Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.ORCID iD: 0000-0003-2318-1716
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.ORCID iD: 0000-0003-2942-6841
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems. At the end of the paper an example is given to concretize the use of the main result.

National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-30685OAI: oai:DiVA.org:miun-30685DiVA, id: diva2:1092384
Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2020-02-17Bibliographically approved
In thesis
1. Homogenization Results for Parabolic and Hyperbolic-Parabolic Problems and Further Results on Homogenization in Perforated Domains
Open this publication in new window or tab >>Homogenization Results for Parabolic and Hyperbolic-Parabolic Problems and Further Results on Homogenization in Perforated Domains
2017 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is based on four papers. The main focus is on homogenization of selected parabolic problems with time oscillations, and hyperbolic-parabolic problems without time oscillations. The approaches are prepared by means of certain methods, such as two-scale convergence, multiscale convergence and evolution multiscale convergence. We also discuss further results on homogenization of evolution problems in perforated domains.

Place, publisher, year, edition, pages
Sundsvall: Mid Sweden University, 2017. p. 36
Series
Mid Sweden University licentiate thesis, ISSN 1652-8948 ; 131
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-30683 (URN)978-91-88527-15-8 (ISBN)
Presentation
2017-05-30, Q221, Akademigatan 1, Östersund, 10:00 (English)
Opponent
Supervisors
Note

Vid tidpunkten för försvar av avhandlingen var följande delarbeten opublicerade: delarbete 1 inskickat, delarbete 2 accepterat, delarbete 4 inskickat.

At the time of the defence the following papers were unpublished: paper 1 submitted, paper 2 accepted, paper 4 submitted.

Available from: 2017-05-03 Created: 2017-05-02 Last updated: 2017-05-03Bibliographically approved
2. Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales
Open this publication in new window or tab >>Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is based on six papers. We study the homogenization of selected parabolic problems with one or more microscopic scales in space and time, respectively. The approaches are prepared by means of certain methods, like two-scale convergence, multiscale convergence and also the evolution setting of multiscale convergence and very weak multiscale convergence. Paper I treats a linear parabolic homogenization problem with rapid spatial and temporal oscillations in perforated domains. Suitable results of two-scale convergence type are established. Paper II deals with further development of compactness results which can be used in the homogenization procedure engaging a certain limit condition. The homogenization procedure deals with a parabolic problem with a certain matching between a fast spatial and a fast temporal scale and a coefficient passing to zero that the time derivative is multiplied with. Papers III and IV are further generalizations of Paper II and investigate homogenization problems with different types of matching between the microscopic scales. Papers III and IV deal with one and two rapid scales in both space and time respectively. Paper V treats the nonlinearity of monotone parabolic problems with an arbitrary number of spatial and temporal scales by applying the perturbed test functions method together with multiscale convergence and very weak multiscale convergence adapted to the evolution setting. In Paper VI we discuss the relation between two-scale convergence and the unfolding method and potential extensions of existing results. The papers above are summarized in Chapter 4. Chapter 1 gives a brief introduction to the topic and Chapters 2 and 3 are surveys over some important previous results.

Place, publisher, year, edition, pages
Sundsvall: Mid Sweden University, 2020. p. 54
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 314
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:miun:diva-38423 (URN)978-91-88947-36-9 (ISBN)
Public defence
2020-03-16, Q 221, Akademigatan 1, Östersund, 13:00 (English)
Opponent
Supervisors
Note

Vid tidpunkten för disputationen var följande delarbeten opublicerade: delarbete 4 (inskickat), delarbete 5 (manuskript), delarbete 6 (manuskript).

At the time of the doctoral defence the following papers were unpublished: paper 4 (submitted), paper 5 (manuscript), paper 6 (manuscript). 

Available from: 2020-02-17 Created: 2020-02-17 Last updated: 2020-02-17Bibliographically approved

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Other links

https://arxiv.org/abs/1704.01375

Authority records BETA

Flodén, LiselottJonasson, PernillaOlsson Lindberg, MarianneLobkova, Tatiana

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Citation style
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