## Journal of Applied Mathematics

### Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time

#### Abstract

The main contribution of this paper is the homogenization of the linear parabolic equation ${\partial }_{t}{u}^{\epsilon }(x,t)-\nabla ·(a(x/{\epsilon }^{{q}_{\mathrm{1}}},...,x/{\epsilon }^{{q}_{n}},t/{\epsilon }^{{r}_{\mathrm{1}}},...,t/{\epsilon }^{{r}_{m}})\nabla {u}^{\epsilon }(x,t))=f(x,t)$ 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 ${q}_{\mathrm{1}}=\mathrm{1}$, ${q}_{\mathrm{2}}=\mathrm{2}$, and $\mathrm{0}<{r}_{\mathrm{1}}<{r}_{\mathrm{2}}$.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 101685, 16 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305550

Digital Object Identifier
doi:10.1155/2014/101685

Mathematical Reviews number (MathSciNet)
MR3176810

Zentralblatt MATH identifier
07010542

#### Citation

Flodén, Liselott; Holmbom, Anders; Olsson Lindberg, Marianne; Persson, Jens. Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time. J. Appl. Math. 2014 (2014), Article ID 101685, 16 pages. doi:10.1155/2014/101685. https://projecteuclid.org/euclid.jam/1425305550

#### References

• A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1978.
• G. Nguetseng, “A general convergence result for a functional related to the theory of homogenization,” SIAM Journal on Mathematical Analysis, vol. 20, no. 3, pp. 608–623, 1989.
• D. Lukkassen, G. Nguetseng, and P. Wall, “Two-scale convergence,” International Journal of Pure and Applied Mathematics, vol. 2, no. 1, pp. 35–86, 2002.
• G. Allaire, “Homogenization and two-scale convergence,” SIAM Journal on Mathematical Analysis, vol. 23, no. 6, pp. 1482–1518, 1992.
• A. Holmbom, “Homogenization of parabolic equations: an alternative approach and some corrector-type results,” Applications of Mathematics, vol. 42, no. 5, pp. 321–343, 1997.
• L. Flodén, A. Holmbom, M. Olsson, and J. Persson, “Very weak multiscale convergence,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1170–1173, 2010.
• G. Nguetseng and J. L. Woukeng, “$\Sigma$-convergence of nonlinear parabolic operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 968–1004, 2007.
• L. Flodén and M. Olsson, “Homogenization of some parabolic operators with several time scales,” Applications of Mathematics, vol. 52, no. 5, pp. 431–446, 2007.
• J. L. Woukeng, “Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales,” Annali di Matematica Pura ed Applicata, vol. 189, no. 3, pp. 357–379, 2010.
• J. L. Woukeng, “$\Sigma$-convergence and reiterated homogenization of nonlinear parabolic operators,” Communications on Pure and Applied Analysis, vol. 9, no. 6, pp. 1753–1789, 2010.
• J. Persson, “Homogenization of monotone parabolic problems with several temporal scales,” Applications of Mathematics, vol. 57, no. 3, pp. 191–214, 2012.
• L. Flodén, A. Holmbom, M. Olsson Lindberg, and J. Persson, “Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence,” Annals of Functional Analysis, vol. 2, no. 1, pp. 84–99, 2011.
• G. Nguetseng, H. Nnang, and N. Svanstedt, “Deterministic homogenization of quasilinear damped hyperbolic equations,” Acta Mathematica Scientia B, vol. 31, no. 5, pp. 1823–1850, 2011.
• J. L. Woukeng and D. Dongo, “Multiscale homogenization of nonlinear hyperbolic equations with several time scales,” Acta Mathematica Scientia B, vol. 31, no. 3, pp. 843–856, 2011.
• D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, New York, NY, USA, 1999.
• G. Allaire and M. Briane, “Multiscale convergence and reiterated homogenisation,” Proceedings of the Royal Society of Edinburgh A, vol. 126, no. 2, pp. 297–342, 1996.
• J. Persson, Selected topics in homogenization [Ph.D. thesis], Mid Sweden University, 2012.
• E. Zeidler, Nonlinear Functional Analysis and Its Applications IIA, Springer, New York, NY, USA, 1990.
• L. Flodén, A. Holmbom, M. Olsson Lindberg, and J. Persson, “Homogenization of parabolic equations with multiple spatial and temporal scales,” Tech. Rep. 2011:6, Mid Sweden University, 2011.
• S. Spagnolo, “Convergence of parabolic equations,” Bollettino della Unione Matematica Italiana B, vol. 14, no. 2, pp. 547–568, 1977.
• P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, Ill, USA, 1988. \endinput