Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
homogenization; parabolic; monotone; two-scale convergence; multiscale convergence; very weak multiscale convergence
homogenization; parabolic; monotone; two-scale convergence; multiscale convergence; very weak multiscale convergence
Summary:
We prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of the scale parameter $\varepsilon $. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems.
References:
[1] Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482-1518. DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[2] Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenisation. Proc. R. Soc. Edinb., Sect. A 126 (1996), 297-342. DOI 10.1017/S0308210500022757 | MR 1386865 | Zbl 0866.35017
[3] Amar, M., Andreucci, D., Bellaveglia, D.: The time-periodic unfolding operator and applications to parabolic homogenization. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 28 (2017), 663-700. DOI 10.4171/RLM/781 | MR 3729583 | Zbl 1383.35015
[4] Amar, M., Andreucci, D., Gianni, R., Timofte, C.: Homogenization results for a class of parabolic equations with a non-local interface condition via time-periodic unfolding. NoDEA, Nonlinear Differ. Equ. Appl. 26 (2019), Article ID 52, 28 pages. DOI 10.1007/s00030-019-0592-4 | MR 4029530 | Zbl 1435.35035
[5] Bensoussan, A., Lions, J.-L., Papanicoloau, G.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications 5. North-Holland Publishing, Amsterdam (1978). DOI 10.1016/s0168-2024(08)x7015-8 | MR 0503330 | Zbl 0404.35001
[6] Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications 17. Oxford University Press, New York (1999). MR 1765047 | Zbl 0939.35001
[7] Danielsson, T., Johnsen, P.: Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales. Math. Bohem. 146 (2021), 483-511. DOI 10.21136/MB.2021.0087-19 | MR 4336552 | Zbl 1499.35049
[8] Evans, L. C.: The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinb., ASect. A 111 (1989), 359-375. DOI 10.1017/S0308210500018631 | MR 1007533 | Zbl 0679.35001
[9] Evans, L. C.: Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinb., Sect. A 120 (1992), 245-265. DOI 10.1017/S0308210500032121 | MR 1159184 | Zbl 0796.35011
[10] Flodén, L., Holmbom, A., Olsson, M., Persson, J.: Very weak multiscale convergence. Appl. Math. Lett. 23 (2010), 1170-1173. DOI 10.1016/j.aml.2010.05.005 | MR 2665589 | Zbl 1198.35023
[11] Flodén, L., Holmbom, A., Lindberg, M. Olsson, Persson, J.: Two-scale convergence: Some remarks and extensions. Pure Appl. Math. Q. 9 (2013), 461-486. DOI 10.4310/PAMQ.2013.v9.n3.a4 | MR 3138471 | Zbl 1288.35041
[12] Flodén, L., Holmbom, A., Lindberg, M. Olsson, Persson, J.: 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 | MR 3176810 | Zbl 1406.35140
[13] Flodén, L., Olsson, M.: Reiterated homogenization of some linear and nonlinear monotone parabolic operators. Can. Appl. Math. Q. 14 (2006), 149-183. MR 2302654 | Zbl 1142.35331
[14] Flodén, L., Olsson, M.: Homogenization of some parabolic operators with several time scales. Appl. Math., Praha 52 (2007), 431-446. DOI 10.1007/s10492-007-0025-2 | MR 2342599 | Zbl 1164.35315
[15] Holmbom, A.: Homogenization of parabolic equations: An alternative approach and some corrector-type results. Appl. Math., Praha 42 (1997), 321-343. DOI 10.1023/A:1023049608047 | MR 1467553 | Zbl 0898.35008
[16] Kufner, A., John, O., Fučík, S.: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis 3. Noordhoff, Leyden (1977). MR 0482102 | Zbl 0364.46022
[17] Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35-86. MR 1912819 | Zbl 1061.35015
[18] Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608-623. DOI 10.1137/0520043 | MR 0990867 | Zbl 0688.35007
[19] Nguetseng, G., Woukeng, J. L.: Deterministic homogenization of parabolic monotone operators with time dependent coefficients. Electron. J. Differ. Equ. 2004 (2004), Article ID 82, 23 pages. MR 2075421 | Zbl 1058.35025
[20] Nguetseng, G., Woukeng, J. L.: $\Sigma$-convergence of nonlinear parabolic operators. Nonlinear Anal., Theory Methods Appl., Ser. A 66 (2007), 968-1004. DOI 10.1016/j.na.2005.12.035 | MR 2288445 | Zbl 1116.35011
[21] Persson, J.: Homogenization of monotone parabolic problems with several temporal scales. Appl. Math., Praha 57 (2012), 191-214. DOI 10.1007/s10492-012-0013-z | MR 2984600 | Zbl 1265.35018
[22] Persson, J.: Selected Topics in Homogenization: Doctoral Thesis. Mid Sweden University, Østersund (2012).
[23] Svanstedt, N.: $G$-convergence of parabolic operators. Nonlinear Anal., Theory Methods Appl. 36 (1999), 807-843. DOI 10.1016/S0362-546X(97)00532-4 | MR 1682689 | Zbl 0933.35020
[24] Svanstedt, N., Wellander, N., Wyller, J.: A numerical algorithm for nonlinear parabolic equations with highly oscillating coefficients. Numer. Methods Partial Differ. Equations 12 (1996), 423-440. DOI 10.1002/(SICI)1098-2426(199607)12:4<423::AID-NUM2>3.0.CO;2-O | MR 1396465 | Zbl 0859.65105
[25] Svanstedt, N., Woukeng, J. L.: Periodic homogenization of strongly nonlinear reactiondiffusion equations with large reaction terms. Appl. Anal. 92 (2013), 1357-1378. DOI 10.1080/00036811.2012.678334 | MR 3169106 | Zbl 1271.35006
[26] Woukeng, J. L.: Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales. Ann. Mat. Pura Appl. (4) 189 (2010), 357-379. DOI 10.1007/s10231-009-0112-y | MR 2657414 | Zbl 1213.35067
[27] Woukeng, J. L.: $\Sigma$-convergence and reiterated homogenization of nonlinear parabolic operators. Commun. Pure Appl. Anal. 9 (2010), 1753-1789. DOI 10.3934/cpaa.2010.9.1753 | MR 2684060 | Zbl 1213.35068
[28] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators. Springer, New York (1990). DOI 10.1007/978-1-4612-0981-2 | MR 1033498 | Zbl 0684.47029
[29] Zhikov, V. V.: On an extension of the method of two-scale convergence and its applications. Sb. Math. 191 (2000), 973-1014. DOI 10.1070/SM2000v191n07ABEH000491 | MR 1809928 | Zbl 0969.35048
Search
Partner of
