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Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
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
ISBN: 978-91-88947-36-9 (print)OAI: oai:DiVA.org:miun-38423DiVA, id: diva2:1393527
##### Public defence
2020-03-16, Q 221, Akademigatan 1, Östersund, 13:00 (English)
##### 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
##### List of papers
1. Homogenization of linear parabolic equations with a certain resonant matching between rapid spatial and temporal oscillations in periodically perforated domains
Open this publication in new window or tab >>Homogenization of linear parabolic equations with a certain resonant matching between rapid spatial and temporal oscillations in periodically perforated domains
2019 (English)In: Acta Mathematicae Applicatae Sinica (English Series), ISSN 0168-9673, E-ISSN 1618-3932, Vol. 35, no 2, p. 340-358Article in journal (Refereed) Published
##### Abstract [en]

In this article, we study homogenization of a parabolic linear problem governed by a coefficient matrix with rapid spatial and temporal oscillations in periodically perforated domains with homogeneous Neumann data on the boundary of the holes. We prove results adapted to the problem for a characterization of multiscale limits for gradients and very weak multiscale convergence.

##### Keywords
Homogenization, two-scale convergence, multiscale convergence, periodically perforated domains ack, norm approximation
Mathematics
##### Identifiers
urn:nbn:se:miun:diva-30695 (URN)10.1007/s10255-019-0810-1 (DOI)000467899800008 ()2-s2.0-85065704100 (Scopus ID)
Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2020-02-17Bibliographically approved
2. Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
Open this publication in new window or tab >>Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
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 $\varepsilon\partial_tu_{\varepsilon}\left(x,t\right)-\nabla\cdot\left(a\left(x/\varepsilon,t/\varepsilon^3\right)\nabla u_{\varepsilon}\left(x,t\right)\right)=f\left(x,t\right)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\varepsilon$ 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 $\left{u_{\varepsilon}\right}$ different from the standard setting are used, which means that these results are also of independent interest.

##### Keywords
homogenization, parabolic problem, multiscale convergence, very weak multiscale convergence, two-scale convergence
Mathematics
##### Identifiers
urn:nbn:se:miun:diva-35061 (URN)10.21136/AM.2018.0350-17 (DOI)000448719900002 ()2-s2.0-85055682025 (Scopus ID)
Available from: 2018-12-05 Created: 2018-12-05 Last updated: 2020-02-17Bibliographically approved
3. Homogenization of the heat equation with a vanishing volumetric heat capacity
Open this publication in new window or tab >>Homogenization of the heat equation with a vanishing volumetric heat capacity
2019 (English)In: Progress in Industrial Mathematics at ECMI 2018 / [ed] Faragó, István, Izsák, Ferenc, Simon, Péter L. (Eds.), 2019Conference paper, Published paper (Refereed)
##### Series
Mathematics in Industry
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:miun:diva-38424 (URN)10.1007/978-3-030-27550-1_43 (DOI)978-3-030-27550-1 (ISBN)
##### Conference
20th European Conference on Mathematics for Industry, ECMI 2018, Budapest, Hungary, June 2018
Available from: 2020-02-17 Created: 2020-02-17 Last updated: 2020-02-17Bibliographically approved
4. Homogenization of linear parabolic equations with three spatial and three temporal microscopic scales for certain matching between the microscopic scales
Open this publication in new window or tab >>Homogenization of linear parabolic equations with three spatial and three temporal microscopic scales for certain matching between the microscopic scales
(English)In: Article in journal (Refereed) Submitted
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:miun:diva-38425 (URN)
Available from: 2020-02-17 Created: 2020-02-17 Last updated: 2020-02-17Bibliographically approved
5. Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales
Open this publication in new window or tab >>Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales
##### 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.

Mathematics
##### Identifiers
urn:nbn:se:miun:diva-30685 (URN)
Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2020-02-17Bibliographically approved
6. On some concepts of convergence and their connections
Open this publication in new window or tab >>On some concepts of convergence and their connections
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:miun:diva-38426 (URN)
Available from: 2020-02-17 Created: 2020-02-17 Last updated: 2020-02-17Bibliographically approved

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Danielsson, Tatiana

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Cite
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