# Article

Full entry | PDF   (0.3 MB)
Keywords:
reiterated homogenization; multiscale convergence; parabolic equation
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 $\partial u/\partial t -\div (b(x,t)\nabla u)=f$.
References:
[1] G.  Allaire, M.  Briane: Multiscale convergence and reiterated homogenisation. Proc. R. Soc. Edinb. 126 (1996), 297–342. DOI 10.1017/S0308210500022757 | MR 1386865
[2] M  Avellaneda: Iterated homogenization, differential effective medium theory and applications. Commun. Pure Appl. Math. 40 (1987), 527–554. DOI 10.1002/cpa.3160400502 | MR 0896766 | Zbl 0629.73010
[3] A.  Bensoussan, J.-L.  Lions, and G.  Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam-New York-Oxford, 1978. MR 0503330
[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
[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
[6] A  Holmbom: Homogenization of parabolic equations—an alternative approach and some corrector-type results. Appl. Math. 42 (1997), 321–343. DOI 10.1023/A:1023049608047 | MR 1467553 | Zbl 0898.35008
[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. DOI 10.1142/S0252959901000024 | MR 1823125
[8] N.  Svanstedt, N.  Wellander: A note on two-scale convergence of differential operators. Submitted.
[9] R.  Temam: Navier-Stokes equations. Theory and Numerical Analysis. North-Holland, Amsterdam-New York-Oxford, 1977. MR 0609732 | Zbl 0383.35057

Partner of