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Johnsen, P. & Lobkova, T. (2018). Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales. Applications of Mathematics, 63(5), 503-521
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: 2019-03-25Bibliographically approved
Flodén, L., Holmbom, A., Jonasson, P., Lobkova, T., Olsson Lindberg, M. & Zhang, Y. (2017). A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem. In: Sivasundaram, S (Ed.), AIP Conference Proceedings: . Paper presented at 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016, 4 July 2016 through 8 July 2016, University of La Rochelle La Rochelle; France. American Institute of Physics (AIP), 1798, Article ID UNSP 020177.
Open this publication in new window or tab >>A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem
2017 (English)In: AIP Conference Proceedings / [ed] Sivasundaram, S, American Institute of Physics (AIP), 2017, Vol. 1798, article id UNSP 020177Conference paper, Published paper (Refereed)
##### Abstract [en]

We study the homogenization of a hyperbolic-parabolic PDE with oscillations in one fast spatial scale. Moreover, the first order time derivative has a degenerate coefficient passing to infinity when ϵ→0. We obtain a local problem which is of elliptic type, while the homogenized problem is also in some sense an elliptic problem but with the limit for ϵ-1∂tuϵ as an undetermined extra source term in the right-hand side. The results are somewhat surprising and work remains to obtain a fully rigorous treatment. Hence the last section is devoted to a discussion of the reasonability of our conjecture including numerical experiments.

##### Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2017
##### Series
AIP Conference Proceedings, ISSN 0094-243X
Mathematics
##### Identifiers
urn:nbn:se:miun:diva-30455 (URN)10.1063/1.4972769 (DOI)000399203000176 ()2-s2.0-85013657168 (Scopus ID)9780735414648 (ISBN)
##### Conference
11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016, 4 July 2016 through 8 July 2016, University of La Rochelle La Rochelle; France
Available from: 2017-03-13 Created: 2017-03-13 Last updated: 2018-12-05Bibliographically approved
Flodén, L., Holmbom, A., Jonasson, P., Olsson Lindberg, M., Lobkova, T. & Persson, J. (2017). Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales. In: Quintela, P., Barral, P., Gómez, D., Pena, F.J., Rodríguez, J., Salgado, P., Vázquez-Mendéz, M.E. (Ed.), Progress in Industrial Mathematics at ECMI 2016: . Paper presented at ECMI-16, the 19th European Conference on Mathematics for Industry at Santiago de Compostela, 13-17th June 2016. (pp. 617-623). Springer
Open this publication in new window or tab >>Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales
2017 (English)In: Progress in Industrial Mathematics at ECMI 2016 / [ed] Quintela, P., Barral, P., Gómez, D., Pena, F.J., Rodríguez, J., Salgado, P., Vázquez-Mendéz, M.E., Springer, 2017, p. 617-623Conference paper, Published paper (Refereed)
##### Abstract [en]

We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {ε; ε2}. The homogenization is performed by means of evolution multiscale convergence, a generalization of the concept of two-scale convergence to include any number of scales in both space and time. In particular we apply a compactness result for gradients. The outcome of the homogenization procedure is that we obtain a homogenized problem of hyperbolic-parabolic type together with two elliptic local problems, one for each rapid scale, for the correctors.

Springer, 2017
##### Series
Mathematics in Industry, ISSN 1612-3956 ; 26
Mathematics
##### Identifiers
urn:nbn:se:miun:diva-30694 (URN)10.1007/978-3-319-63082-3_94 (DOI)978-3-319-63081-6 (ISBN)
##### Conference
ECMI-16, the 19th European Conference on Mathematics for Industry at Santiago de Compostela, 13-17th June 2016.
Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2019-06-20Bibliographically approved
Flodén, L., Jonasson, P., Olsson Lindberg, M. & Lobkova, T. 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
(English)In: Article in journal (Other academic) Submitted
##### 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: 2018-12-05Bibliographically approved
##### Identifiers
ORCID iD: orcid.org/0000-0003-2318-1716

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