| Title:
|
Multiscale convergence and reiterated homogenization of parabolic problems (English) |
| Author:
|
Holmbom, Anders |
| Author:
|
Svanstedt, Nils |
| Author:
|
Wellander, Niklas |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
50 |
| Issue:
|
2 |
| Year:
|
2005 |
| Pages:
|
131-151 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Reiterated homogenization is studied for divergence structure parabolic problems of the form $\partial u_{\varepsilon }/\partial t - \div \bigl (a\bigl (x,x/\varepsilon ,x/\varepsilon ^2, t,t/\varepsilon ^{k}\bigr )\nabla u_{\varepsilon }\bigr )=f$. It is shown that under standard assumptions on the function $a(x,y_1,y_2,t,\tau )$ the sequence $\lbrace u_\epsilon \rbrace $ of solutions converges weakly in $L^2(0,T;H^1_0(\Omega ))$ to the solution $u$ of the homogenized problem $\partial u/\partial t -\div (b(x,t)\nabla u)=f$. (English) |
| Keyword:
|
reiterated homogenization |
| Keyword:
|
multiscale convergence |
| Keyword:
|
parabolic equation |
| MSC:
|
35B27 |
| MSC:
|
35K20 |
| idZBL:
|
Zbl 1099.35011 |
| idMR:
|
MR2125155 |
| DOI:
|
10.1007/s10492-005-0009-z |
| . |
| Date available:
|
2009-09-22T18:21:20Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134597 |
| . |
| Reference:
|
[1] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenisation.Proc. R. Soc. Edinb. 126 (1996), 297–342. MR 1386865, 10.1017/S0308210500022757 |
| Reference:
|
[2] M Avellaneda: Iterated homogenization, differential effective medium theory and applications.Commun. Pure Appl. Math. 40 (1987), 527–554. Zbl 0629.73010, MR 0896766, 10.1002/cpa.3160400502 |
| Reference:
|
[3] A. Bensoussan, J.-L. Lions, and G. Papanicolaou: Asymptotic Analysis for Periodic Structures.North-Holland, Amsterdam-New York-Oxford, 1978. MR 0503330 |
| Reference:
|
[4] D Cioranescu, P. Donato: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications.Oxford Univ. Press, New York, 1999. MR 1765047 |
| Reference:
|
[5] A. Dall’Aglio, F. Murat: A corrector result for $H$-converging parabolic problems with time-dependent coefficients. Dedicated to Ennio De Giorgi.Ann. Sc. Norm. Super. Pisa Cl. Sci. IV 25 (1997), 329–373. MR 1655521 |
| Reference:
|
[6] A Holmbom: Homogenization of parabolic equations—an alternative approach and some corrector-type results.Appl. Math. 42 (1997), 321–343. Zbl 0898.35008, MR 1467553, 10.1023/A:1023049608047 |
| Reference:
|
[7] J.-L. Lions, D. Lukkassen, L. E. Persson, and P. Wall: Reiterated homogenization of nonlinear monotone operators.Chin. Ann. Math. Ser. B 22 (2001), 1–12. MR 1823125, 10.1142/S0252959901000024 |
| Reference:
|
[8] N. Svanstedt, N. Wellander: A note on two-scale convergence of differential operators.Submitted. |
| Reference:
|
[9] R. Temam: Navier-Stokes equations. Theory and Numerical Analysis.North-Holland, Amsterdam-New York-Oxford, 1977. Zbl 0383.35057, MR 0609732 |
| . |
