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Lobkova, Tatiana
##### Publications (7 of 7) Show all publications
Lobkova, T. (2019). Homogenization of linear parabolic equations with a certain resonant matching between rapid spatial and temporal oscillations in periodically perforated domains. Acta Mathematicae Applicatae Sinica (English Series), 35(2), 340-358
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 (Other academic) 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)
##### Note

Finns även tillgänglig via https://arxiv.org/abs/1704.01483

Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2019-07-09Bibliographically approved
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
Lobkova, T. (2017). Homogenization Results for Parabolic and Hyperbolic-Parabolic Problems and Further Results on Homogenization in Perforated Domains. (Licentiate dissertation). Sundsvall: Mid Sweden University
Open this publication in new window or tab >>Homogenization Results for Parabolic and Hyperbolic-Parabolic Problems and Further Results on Homogenization in Perforated Domains
2017 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis is based on four papers. The main focus is on homogenization of selected parabolic problems with time oscillations, and hyperbolic-parabolic problems without time oscillations. The approaches are prepared by means of certain methods, such as two-scale convergence, multiscale convergence and evolution multiscale convergence. We also discuss further results on homogenization of evolution problems in perforated domains.

##### Place, publisher, year, edition, pages
Sundsvall: Mid Sweden University, 2017. p. 36
##### Series
Mid Sweden University licentiate thesis, ISSN 1652-8948 ; 131
Mathematics
##### Identifiers
urn:nbn:se:miun:diva-30683 (URN)978-91-88527-15-8 (ISBN)
##### Presentation
2017-05-30, Q221, Akademigatan 1, Östersund, 10:00 (English)
##### Note

Vid tidpunkten för försvar av avhandlingen var följande delarbeten opublicerade: delarbete 1 inskickat, delarbete 2 accepterat, delarbete 4 inskickat.

At the time of the defence the following papers were unpublished: paper 1 submitted, paper 2 accepted, paper 4 submitted.

Available from: 2017-05-03 Created: 2017-05-02 Last updated: 2017-05-03Bibliographically approved
Holmbom, A. & Lobkova, T. (2016). Homogenization of a hyperbolic-parabolic problem in a perforated domain. In: : . Paper presented at 19th European Conference on Mathematics for Industry, Santiago de Compostela, Spain, June 13-17, 2006..
Open this publication in new window or tab >>Homogenization of a hyperbolic-parabolic problem in a perforated domain
2016 (English)Conference paper, Oral presentation with published abstract (Other academic)
##### Keywords
Homogenization, two-scale convergence, multiscale convergence, perforated domains
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:miun:diva-28656 (URN)
##### Conference
19th European Conference on Mathematics for Industry, Santiago de Compostela, Spain, June 13-17, 2006.
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2016-12-29Bibliographically 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

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