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2014 Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time
Liselott Flodén, Anders Holmbom, Marianne Olsson Lindberg, Jens Persson
J. Appl. Math. 2014: 1-16 (2014). DOI: 10.1155/2014/101685


The main contribution of this paper is the homogenization of the linear parabolic equation tuε(x,t)-·(a(x/εq1,...,x/εqn,t/εr1,...,t/εrm)uε(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 provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtain n local problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales with q1=1, q2=2, and 0<r1<r2.


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Liselott Flodén. Anders Holmbom. Marianne Olsson Lindberg. Jens Persson. "Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time." J. Appl. Math. 2014 1 - 16, 2014.


Published: 2014
First available in Project Euclid: 2 March 2015

zbMATH: 07010542
MathSciNet: MR3176810
Digital Object Identifier: 10.1155/2014/101685

Rights: Copyright © 2014 Hindawi


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