Homogenization of monotone parabolic problems with several temporal scales

Jens Persson

Applications of Mathematics (2012)

  • Volume: 57, Issue: 3, page 191-214
  • ISSN: 0862-7940

Abstract

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In this paper we homogenize monotone parabolic problems with two spatial scales and any number of temporal scales. Under the assumption that the spatial and temporal scales are well-separated in the sense explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the “slow” self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the ``rapid'' self-similar case), respectively.

How to cite

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Persson, Jens. "Homogenization of monotone parabolic problems with several temporal scales." Applications of Mathematics 57.3 (2012): 191-214. <http://eudml.org/doc/246651>.

@article{Persson2012,
abstract = {In this paper we homogenize monotone parabolic problems with two spatial scales and any number of temporal scales. Under the assumption that the spatial and temporal scales are well-separated in the sense explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the “slow” self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the ``rapid'' self-similar case), respectively.},
author = {Persson, Jens},
journal = {Applications of Mathematics},
keywords = {homogenization; $H$-convergence; multiscale convergence; parabolic; monotone; parabolic problem; homogenization; -convergence; multiscale convergence; parabolic problem},
language = {eng},
number = {3},
pages = {191-214},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homogenization of monotone parabolic problems with several temporal scales},
url = {http://eudml.org/doc/246651},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Persson, Jens
TI - Homogenization of monotone parabolic problems with several temporal scales
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 3
SP - 191
EP - 214
AB - In this paper we homogenize monotone parabolic problems with two spatial scales and any number of temporal scales. Under the assumption that the spatial and temporal scales are well-separated in the sense explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the “slow” self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the ``rapid'' self-similar case), respectively.
LA - eng
KW - homogenization; $H$-convergence; multiscale convergence; parabolic; monotone; parabolic problem; homogenization; -convergence; multiscale convergence; parabolic problem
UR - http://eudml.org/doc/246651
ER -

References

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  1. Allaire, G., 10.1137/0523084, SIAM J. Math. Anal. 23 (1992), 1482-1518. (1992) Zbl0770.35005MR1185639DOI10.1137/0523084
  2. Allaire, G., Briane, M., 10.1017/S0308210500022757, Proc. R. Soc. Edinb., Sect. A 126 (1996), 297-342. (1996) Zbl0866.35017MR1386865DOI10.1017/S0308210500022757
  3. Bensoussan, A., Lions, J.-L., Papanicolaou, G., Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications, Vol. 5, North-Holland Publishing Amsterdam-New York-Oxford (1978). (1978) MR0503330
  4. Evans, L. C., 10.1017/S0308210500018631, Proc. R. Soc. Edinb., Sect. A 111 (1989), 359-375. (1989) Zbl0679.35001MR1007533DOI10.1017/S0308210500018631
  5. Flodén, L., Olsson, M., Reiterated homogenization of some linear and nonlinear monotone parabolic operators, Can. Appl. Math. Q. 14 (2006), 149-183. (2006) Zbl1142.35331MR2302654
  6. Flodén, L., Olsson, M., 10.1007/s10492-007-0025-2, Appl. Math. 52 (2007), 431-446. (2007) Zbl1164.35315MR2342599DOI10.1007/s10492-007-0025-2
  7. Flodén, L., Holmbom, A., Olsson, M., Svanstedt, N., 10.1007/s11565-007-0014-0, Ann. Univ. Ferrara, Sez. VII Sci. Mat. 53 (2007), 217-232. (2007) Zbl1180.35076MR2358224DOI10.1007/s11565-007-0014-0
  8. Holmbom, A., Some modes of convergence and their application to homogenization and optimal composites design, Doctoral thesis 1996:208 D Department of Mathematics, Luleå University Luleå (1996). (1996) 
  9. Holmbom, A., 10.1023/A:1023049608047, Appl. Math. 42 (1997), 321-343. (1997) Zbl0898.35008MR1467553DOI10.1023/A:1023049608047
  10. Holmbom, A., Silfver, J., On the convergence of some sequences of oscillating functionals, WSEAS Trans. Math. 5 (2006), 951-956 2241788. (2006) MR2241788
  11. Holmbom, A., Svanstedt, N., Wellander, N., 10.1007/s10492-005-0009-z, Appl. Math. 50 (2005), 131-151. (2005) Zbl1099.35011MR2125155DOI10.1007/s10492-005-0009-z
  12. Kun'ch, R. N., Pankov, A. A., G-convergence of the monotone parabolic operators, Dokl. Akad. Nauk Ukr. SSR, Ser. A (1986), 8-10 Russian. (1986) Zbl0621.35005MR0897902
  13. Lions, J.-L., Lukkassen, D., Persson, L.-E., Wall, P., 10.1142/S0252959901000024, Chin. Ann. Math., Ser. B 22 (2001), 1-12. (2001) Zbl0979.35047MR1823125DOI10.1142/S0252959901000024
  14. Mascarenhas, M. L., Toader, A.-M., 10.1081/NFA-100103791, Numer. Funct. Anal. Optimization 22 (2001), 127-158. (2001) Zbl0995.49013MR1841866DOI10.1081/NFA-100103791
  15. Murat, F., H-convergence, Séminaire d'analyse fonctionelle et numérique de l'Université d'Alger (1978). (1978) 
  16. Nguetseng, G., 10.1137/0520043, SIAM J. Math. Anal. 20 (1989), 608-623. (1989) Zbl0688.35007MR0990867DOI10.1137/0520043
  17. Nguetseng, G., 10.4171/ZAA/1133, Z. Anal. Anwend. 22 (2003), 73-107. (2003) Zbl1045.46031MR1962077DOI10.4171/ZAA/1133
  18. Nguetseng, G., Woukeng, J. L., Deterministic homogenization of parabolic monotone operators with time dependent coefficients, Electron. J. Differ. Equ., paper No. 82 (2004), Electronic only. (2004) Zbl1058.35025MR2075421
  19. Nguetseng, G., Woukeng, J. L., 10.1016/j.na.2005.12.035, Nonlinear Anal., Theory Methods Appl. 66 (2007), 968-1004. (2007) Zbl1116.35011MR2288445DOI10.1016/j.na.2005.12.035
  20. Persson, J., Homogenisation of monotone parabolic problems with several temporal scales: The detailed arXiv e-print version, arXiv:1003.5523 [math.AP]. 
  21. Spagnolo, S., Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore, Ann. Sc. Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 21 (1967), 657-699 Italian. (1967) Zbl0153.42103MR0225015
  22. Svanstedt, N., G-convergence and homogenization of sequences of linear and nonlinear partial differential operators, Doctoral thesis 1992:105 D Department of Mathematics, Luleå University Luleå (1992). (1992) 
  23. Tartar, L., Cours peccot, Collège de France (1977), unpublished, partially written in []. (1977) 
  24. Tartar, L., Quelques remarques sur l'homogénéisation, In: Functional Analysis and Numerical Analysis, Proc. Japan-France Seminar 1976 M. Fujita Society for the Promotion of Science (1978), 468-482. (1978) 
  25. Woukeng, J. L., 10.1007/s10231-009-0112-y, Ann. Mat. Pura Appl. 189 (2010), 357-379. (2010) Zbl1213.35067MR2657414DOI10.1007/s10231-009-0112-y

Citations in EuDML Documents

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  1. Pernilla Johnsen, Tatiana Lobkova, Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales
  2. Tatiana Danielsson, Pernilla Johnsen, Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales
  3. Tatiana Danielsson, Liselott Flodén, Pernilla Johnsen, Marianne Olsson Lindberg, Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales

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