×

Homogenization of linear parabolic equations with a certain resonant matching between rapid spatial and temporal oscillations in periodically perforated domains. (English) Zbl 1416.35032

Summary: 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 characterization of multiscale limits for gradients and very weak multiscale convergence.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acerbi, E., Chiado Piat, V., Dal Maso, G., Percivale, D. An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal., 18(5): 481-496 (1992) · Zbl 0779.35011
[2] Allaire, G. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23(6): 1482-1518 (1992) · Zbl 0770.35005
[3] Allaire, G., Briane, M. Multi-scale convergence and reiterated homogenization. Proc. Roy. Soc. Edinburgh Sect. A, 126(2): 297-342 (1996) · Zbl 0866.35017
[4] Allaire, G., Murat, F. Homogenization of the Neumann problem with nonisolated holes. Asymptotic Anal., 7(2): 81-95 (1993) · Zbl 0823.35014
[5] Arbogast, T., Douglas, J., Hornung, U. Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal., 21(4): 823-836 (1990) · Zbl 0698.76106
[6] Bensoussan, A., Lions, J.L., Papanicolaou, G. Asymptotic analysis for periodic structures, studies in mathematics and its applications. North-Holland, Amsterdam, 1978 · Zbl 0404.35001
[7] Casado-Díaz, J., Gayte, I. A general compactness result and its application to the two-scale convergence of almost periodic functions. C.R. Acad.Sci. Paris Sér. I Math., 323(4): 329-334 (1996) · Zbl 0865.46003
[8] Cioranescu, D., Saint Jean Paulin, J. Homogenization in open sets with holes. J. Math. Anal. Appl., 71(2): 590-607 (1979) · Zbl 0427.35073
[9] Cioranescu, D., Damlamian, A., Donato, G., Griso, G., Zaki, R. The periodic unfolding method in domains with holes. SIAM J. Math. Anal., 44(2): 718-760 (2012) · Zbl 1250.49017
[10] Colombini, F., Spagnolo, S. Sur la convergence de solutions d equations paraboliques [On the convergence of solutions of parabolic equations]. J. Math. Pures Appl., (9) 56(3): 263-305 (1977) · Zbl 0354.35009
[11] Conca, C. On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl., (9) 64(2): 31-75 (1985) · Zbl 0566.35080
[12] Conca, C., Donato, P. Nonhomogeneous Neumann problems in domains with small holes. RAIRO Modél. Math. Anal. Numér., 22(4): 561-607 (1988) · Zbl 0669.35028
[13] Donato, P., Nabil, A. Homogenization and correctors for the heat equation in perforated domains. Ricerche di Matematica., 2(1): 115-144 (2001) · Zbl 1102.35305
[14] Donato, P., Nabil, A. Homogenization of semilinear parabolic equations in perforated domains. Chinese Ann. Math. Ser B., 25(2): 143-156 (2004) · Zbl 1085.35022
[15] Donato, P., Saint Jean Paulin, J. Homogenization of the Poisson equation in a porous medium with double periodicity. Japan J. Appl. Math., 10(2): 333-349 (1993) · Zbl 0811.35019
[16] Douanla, H., Tetsadjio, E. Reiterated homogenization of hyperbolic-parabolic equations in domains with tiny holes. Electron. J Differential Equations, (2017)(59): 1-22 (2017) · Zbl 1370.35038
[17] Douanla, H., Woukeng, J. L. Homogenization of reaction-diffusion equations in fractured porous media. Electron J Differential Equations., 2015(253): 1-23 (2015) · Zbl 1336.35046
[18] Flodén, L., Holmbom, A., Olsson, M., Persson, J. Very weak multiscale convergence. Appl. Math. Lett., 23(10): 1170-1173 (2010) · Zbl 1198.35023
[19] Flodén, L., Holmbom, A., Olsson Lindberg, M., Persson, J. Homogenization of parabolic equations with an arbitrary number of scales in both space and time. J. Appl. Math., Vol. 2014 (2014) p.??? · Zbl 1406.35140
[20] Flodén, L., Olsson, M. Homogenization of some parabolic operators with several time scales. Appl. Math., 52(5): 431-446 (2007) · Zbl 1164.35315
[21] Griffel, D.H. Applied Functional Analysis. Ellis Horwood Limited, Publishers-Chichester, John Wiley and Sons. New York, Toronto, 1981 · Zbl 0461.46001
[22] Holmbom, A. Homogenization of parabolic equations: an alternative approach and some corrector-type results. Appl. Math., 42(5): 321-343 (1997) · Zbl 0898.35008
[23] Holmbom, A., Silfver, J., Svanstedt, N., Wellander, N. On two-scale convergence and related sequential compactness topics. Appl. Math., 51(3): 247-262 (2006) · Zbl 1164.40304
[24] Lukassen, D., Nguetseng, G., Wall, P. Two-scale convergence. Int. J. Pure Appl. Math., 2(1): 33-81 (2002)
[25] Mascarenhas, M.L., Toader, A-M. Scale convergence in homogenization. Numer. Funct. Anal. Optim., 22(1-2): 127-158 (2001) · Zbl 0995.49013
[26] Nandakumaran, AK, Two-scale Convergence and Homogenization (2010)
[27] Nechvatal, L. Alternative approaches to the two-scale convergence. Appl. Math., 49(2): 97-110 (2004) · Zbl 1099.35012
[28] Nguetseng, G. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal., 20(3): 608-623 (1989) · Zbl 0688.35007
[29] Nguetseng, G., Woukeng, J.L. Σ-convergence of nonlinear parabolic operators. Nonlinear Anal., 66(4): 968-1004 (2007) · Zbl 1116.35011
[30] Nguetseng, G., Svanstedt, N. Σ-convergence. Banach J. Math. Anal., 5(1): 101-135 (2011) · Zbl 1229.46035
[31] Pak, H.C. Geometric two-scale convergence on forms and its applications to maxwell’s equations. Proc. Roy. Soc. Edinburgh Sect. A, 135(1): 133-147 (2005) · Zbl 1064.35020
[32] Oleinik, O.A., Shaposhnikova, T.A. On the homogenization of the Poisson equation in partially perforated domain with the arbitrary density of cavities and mixed conditions on their boundary. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 7(3): 129-146 (1996) · Zbl 0878.35011
[33] Persson, J. Homogenization of some selected elliptic and parabolic problems employing suitable generalized modes of two-scale convergence. Mid Sweden University Licentiate thesis 45, Mid Sweden University, 2010
[34] Persson, J. Selected Topics in Homogenization. Mid Sweden University Doctoral Thesis 127, Östersund, 2012
[35] Persson, J. Homogenization of monotone parabolic problems with several temporal scales. Appl. Math., 57(3): 191-214 (2012) · Zbl 1265.35018
[36] Piatnitski, A., Rybalko, V. Homogenization of boundary value problems for monotone operators in perforated domains with rapidly oscillating boundary conditions of fourier type. J. Math. Scien., 177(1): 109-140 (2011) · Zbl 1290.35019
[37] Silfver, J. G-convergence and homogenization involving operators compatible with two-scale convergence. PhD thesis, Mid Sweden University Doctoral Thesis 23, 2007 · Zbl 1164.35318
[38] Temam, R. Navier-Stokes Equation. North-Holland, Amsterdam, 1984 · Zbl 0568.35002
[39] Woukeng, J.L. Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales. Ann. Mat. Pura Appl., (4) 189(3): 357-379 (2010) · Zbl 1213.35067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.