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Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för matematik och ämnesdidaktik.ORCID-id: 0000-0003-2318-1716
Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för matematik och ämnesdidaktik.
2018 (engelsk)Inngår i: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 63, nr 5, s. 503-521Artikkel i tidsskrift (Fagfellevurdert) 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.

sted, utgiver, år, opplag, sider
2018. Vol. 63, nr 5, s. 503-521
Emneord [en]
homogenization, parabolic problem, multiscale convergence, very weak multiscale convergence, two-scale convergence
HSV kategori
Identifikatorer
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
Tilgjengelig fra: 2018-12-05 Laget: 2018-12-05 Sist oppdatert: 2020-02-17bibliografisk kontrollert
Inngår i avhandling
1. Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales
Åpne denne publikasjonen i ny fane eller vindu >>Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales
2020 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Sundsvall: Mid Sweden University, 2020. s. 54
Serie
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 314
HSV kategori
Identifikatorer
urn:nbn:se:miun:diva-38423 (URN)978-91-88947-36-9 (ISBN)
Disputas
2020-03-16, Q 221, Akademigatan 1, Östersund, 13:00 (engelsk)
Opponent
Veileder
Merknad

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). 

Tilgjengelig fra: 2020-02-17 Laget: 2020-02-17 Sist oppdatert: 2020-02-17bibliografisk kontrollert

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