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Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.ORCID iD: 0000-0003-2318-1716
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
2018 (English)In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 63, no 5, p. 503-521Article in journal (Refereed) Published
Abstract [en]

This paper is devoted to the study of the linear parabolic problem  by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient  in front of the timederivative. First, we have an elliptic homogenized problem although the problem studiedis parabolic. Secondly, we get a parabolic local problem even though the problem has adifferent 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  different from the standard setting are used, which means that these results are also of independent interest.

Place, publisher, year, edition, pages
2018. Vol. 63, no 5, p. 503-521
Keywords [en]
homogenization, parabolic problem, multiscale convergence, very weak multiscale convergence, two-scale convergence
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-35061DOI: 10.21136/AM.2018.0350-17ISI: 000448719900002Scopus ID: 2-s2.0-85055682025OAI: oai:DiVA.org:miun-35061DiVA, id: diva2:1268383
Available from: 2018-12-05 Created: 2018-12-05 Last updated: 2020-02-17Bibliographically approved
In thesis
1. 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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Publisher's full textScopushttps://articles.math.cas.cz/?type=A&v=63&n=5

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Johnsen, PernillaLobkova, Tatiana

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