Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales

Tatiana Danielsson; Liselott Flodén; Pernilla Johnsen; Marianne Olsson Lindberg

Applications of Mathematics (2024)

  • Volume: 69, Issue: 1, page 1-24
  • ISSN: 0862-7940

Abstract

top
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 ε . The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems.

How to cite

top

Danielsson, Tatiana, et al. "Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales." Applications of Mathematics 69.1 (2024): 1-24. <http://eudml.org/doc/299205>.

@article{Danielsson2024,
abstract = {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.},
author = {Danielsson, Tatiana, Flodén, Liselott, Johnsen, Pernilla, Olsson Lindberg, Marianne},
journal = {Applications of Mathematics},
keywords = {homogenization; parabolic; monotone; two-scale convergence; multiscale convergence; very weak multiscale convergence},
language = {eng},
number = {1},
pages = {1-24},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales},
url = {http://eudml.org/doc/299205},
volume = {69},
year = {2024},
}

TY - JOUR
AU - Danielsson, Tatiana
AU - Flodén, Liselott
AU - Johnsen, Pernilla
AU - Olsson Lindberg, Marianne
TI - Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales
JO - Applications of Mathematics
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 1
EP - 24
AB - 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.
LA - eng
KW - homogenization; parabolic; monotone; two-scale convergence; multiscale convergence; very weak multiscale convergence
UR - http://eudml.org/doc/299205
ER -

References

top
  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. Amar, M., Andreucci, D., Bellaveglia, D., 10.4171/RLM/781, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 28 (2017), 663-700. (2017) Zbl1383.35015MR3729583DOI10.4171/RLM/781
  4. Amar, M., Andreucci, D., Gianni, R., Timofte, C., 10.1007/s00030-019-0592-4, NoDEA, Nonlinear Differ. Equ. Appl. 26 (2019), Article ID 52, 28 pages. (2019) Zbl1435.35035MR4029530DOI10.1007/s00030-019-0592-4
  5. Bensoussan, A., Lions, J.-L., Papanicoloau, G., 10.1016/s0168-2024(08)x7015-8, Studies in Mathematics and Its Applications 5. North-Holland Publishing, Amsterdam (1978). (1978) Zbl0404.35001MR0503330DOI10.1016/s0168-2024(08)x7015-8
  6. Cioranescu, D., Donato, P., An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications 17. Oxford University Press, New York (1999). (1999) Zbl0939.35001MR1765047
  7. Danielsson, T., Johnsen, P., 10.21136/MB.2021.0087-19, Math. Bohem. 146 (2021), 483-511. (2021) Zbl1499.35049MR4336552DOI10.21136/MB.2021.0087-19
  8. Evans, L. C., 10.1017/S0308210500018631, Proc. R. Soc. Edinb., ASect. A 111 (1989), 359-375. (1989) Zbl0679.35001MR1007533DOI10.1017/S0308210500018631
  9. Evans, L. C., 10.1017/S0308210500032121, Proc. R. Soc. Edinb., Sect. A 120 (1992), 245-265. (1992) Zbl0796.35011MR1159184DOI10.1017/S0308210500032121
  10. Flodén, L., Holmbom, A., Olsson, M., Persson, J., 10.1016/j.aml.2010.05.005, Appl. Math. Lett. 23 (2010), 1170-1173. (2010) Zbl1198.35023MR2665589DOI10.1016/j.aml.2010.05.005
  11. Flodén, L., Holmbom, A., Lindberg, M. Olsson, Persson, J., 10.4310/PAMQ.2013.v9.n3.a4, Pure Appl. Math. Q. 9 (2013), 461-486. (2013) Zbl1288.35041MR3138471DOI10.4310/PAMQ.2013.v9.n3.a4
  12. Flodén, L., Holmbom, A., Lindberg, M. Olsson, Persson, J., 10.1155/2014/101685, J. Appl. Math. 2014 (2014), Article ID 101685, 16 pages. (2014) Zbl1406.35140MR3176810DOI10.1155/2014/101685
  13. 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
  14. Flodén, L., Olsson, M., 10.1007/s10492-007-0025-2, Appl. Math., Praha 52 (2007), 431-446. (2007) Zbl1164.35315MR2342599DOI10.1007/s10492-007-0025-2
  15. Holmbom, A., 10.1023/A:1023049608047, Appl. Math., Praha 42 (1997), 321-343. (1997) Zbl0898.35008MR1467553DOI10.1023/A:1023049608047
  16. Kufner, A., John, O., Fučík, S., Function Spaces, Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis 3. Noordhoff, Leyden (1977). (1977) Zbl0364.46022MR0482102
  17. Lukkassen, D., Nguetseng, G., Wall, P., Two-scale convergence, Int. J. Pure Appl. Math. 2 (2002), 35-86. (2002) Zbl1061.35015MR1912819
  18. Nguetseng, G., 10.1137/0520043, SIAM J. Math. Anal. 20 (1989), 608-623. (1989) Zbl0688.35007MR0990867DOI10.1137/0520043
  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. (2004) Zbl1058.35025MR2075421
  20. Nguetseng, G., Woukeng, J. L., 10.1016/j.na.2005.12.035, Nonlinear Anal., Theory Methods Appl., Ser. A 66 (2007), 968-1004. (2007) Zbl1116.35011MR2288445DOI10.1016/j.na.2005.12.035
  21. Persson, J., 10.1007/s10492-012-0013-z, Appl. Math., Praha 57 (2012), 191-214. (2012) Zbl1265.35018MR2984600DOI10.1007/s10492-012-0013-z
  22. Persson, J., Selected Topics in Homogenization: Doctoral Thesis, Mid Sweden University, Østersund (2012). (2012) 
  23. Svanstedt, N., 10.1016/S0362-546X(97)00532-4, Nonlinear Anal., Theory Methods Appl. 36 (1999), 807-843. (1999) Zbl0933.35020MR1682689DOI10.1016/S0362-546X(97)00532-4
  24. Svanstedt, N., Wellander, N., Wyller, J., 10.1002/(SICI)1098-2426(199607)12:4<423::AID-NUM2>3.0.CO;2-O, Numer. Methods Partial Differ. Equations 12 (1996), 423-440. (1996) Zbl0859.65105MR1396465DOI10.1002/(SICI)1098-2426(199607)12:4<423::AID-NUM2>3.0.CO;2-O
  25. Svanstedt, N., Woukeng, J. L., 10.1080/00036811.2012.678334, Appl. Anal. 92 (2013), 1357-1378. (2013) Zbl1271.35006MR3169106DOI10.1080/00036811.2012.678334
  26. Woukeng, J. L., 10.1007/s10231-009-0112-y, Ann. Mat. Pura Appl. (4) 189 (2010), 357-379. (2010) Zbl1213.35067MR2657414DOI10.1007/s10231-009-0112-y
  27. Woukeng, J. L., 10.3934/cpaa.2010.9.1753, Commun. Pure Appl. Anal. 9 (2010), 1753-1789. (2010) Zbl1213.35068MR2684060DOI10.3934/cpaa.2010.9.1753
  28. Zeidler, E., 10.1007/978-1-4612-0981-2, Springer, New York (1990). (1990) Zbl0684.47029MR1033498DOI10.1007/978-1-4612-0981-2
  29. Zhikov, V. V., 10.1070/SM2000v191n07ABEH000491, Sb. Math. 191 (2000), 973-1014. (2000) Zbl0969.35048MR1809928DOI10.1070/SM2000v191n07ABEH000491

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.