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Homogenization of parabolic equations with an arbitrary number of scales in both space and time
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Tillämpad matematik)ORCID iD: 0000-0001-6742-5781
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Tillämpad matematik)ORCID iD: 0000-0001-6742-5781
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Tillämpad matematik)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. (Tillämpad matematik)ORCID iD: 0000-0001-9984-2424
2014 (English)In: Journal of Applied Mathematics, ISSN 1110-757X, E-ISSN 1687-0042, Art. no. 101685- p.Article in journal (Refereed) Published
Abstract [en]

The main contribution of this paper is the homogenization of the linearparabolic equationtu (x, t) − ·axq1, ...,xqn,tr1, ...,trmu (x, t)= f(x, t)exhibiting an arbitrary finite number of both spatial and temporal scales.We briefly recall some fundamentals of multiscale convergence and providea characterization of multiscale limits for gradients in an evolution settingadapted to a quite general class of well-separated scales, which we nameby jointly well-separated scales (see Appendix for the proof). We proceedwith a weaker version of this concept called very weak multiscale convergence.We prove a compactness result with respect to this latter typefor jointly well-separated scales. This is a key result for performing thehomogenization of parabolic problems combining rapid spatial and temporaloscillations such as the problem above. Applying this compactnessresult together with a characterization of multiscale limits of sequences ofgradients we carry out the homogenization procedure, where we togetherwith the homogenized problem obtain n local problems, i.e. one for eachspatial microscale. To illustrate the use of the obtained result we apply itto a case with three spatial and three temporal scales with q1 = 1, q2 = 2and 0 < r1 < r2.MSC: 35B27; 35K10

Place, publisher, year, edition, pages
Boston: Hindawi Publishing Corporation, 2014. Art. no. 101685- p.
Keyword [en]
Multiscale convergence, very weak multiascale convergence, homogenization theory, parabolic partial differential equations, evolution
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-20903DOI: 10.1155/2014/101685ISI: 000332561700001ScopusID: 2-s2.0-84896941846OAI: oai:DiVA.org:miun-20903DiVA: diva2:682633
Note

Publ online Dec 2013

Available from: 2013-12-28 Created: 2013-12-28 Last updated: 2016-10-20Bibliographically approved

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Flodén, LiselottHolmbom, AndersOlsson Lindberg, MariannePersson, Jens
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