Two-scale div-curl lemma

Augusto Visintin

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 2, page 291-321
  • ISSN: 0391-173X

Abstract

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The div-curl lemma, one of the basic results of the theory of compensated compactness of Murat and Tartar, does not take over to the case in which the two factors two-scale converge in the sense of Nguetseng. A suitable modification of the differential operators however allows for this extension. The argument follows the lines of a well-known paper of F. Murat of 1978, and uses a two-scale extension of the Fourier transform. This result is also extended to time-dependent functions, and is applied to a two-scale formulation of the Maxwell system of electromagnetism, that accounts for the energy embedded in both coarse- and fine-scale oscillations.

How to cite

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Visintin, Augusto. "Two-scale div-curl lemma." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.2 (2007): 291-321. <http://eudml.org/doc/272264>.

@article{Visintin2007,
abstract = {The div-curl lemma, one of the basic results of the theory of compensated compactness of Murat and Tartar, does not take over to the case in which the two factors two-scale converge in the sense of Nguetseng. A suitable modification of the differential operators however allows for this extension. The argument follows the lines of a well-known paper of F. Murat of 1978, and uses a two-scale extension of the Fourier transform. This result is also extended to time-dependent functions, and is applied to a two-scale formulation of the Maxwell system of electromagnetism, that accounts for the energy embedded in both coarse- and fine-scale oscillations.},
author = {Visintin, Augusto},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {compensated compactness; two-scale Fourier transform},
language = {eng},
number = {2},
pages = {291-321},
publisher = {Scuola Normale Superiore, Pisa},
title = {Two-scale div-curl lemma},
url = {http://eudml.org/doc/272264},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Visintin, Augusto
TI - Two-scale div-curl lemma
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 2
SP - 291
EP - 321
AB - The div-curl lemma, one of the basic results of the theory of compensated compactness of Murat and Tartar, does not take over to the case in which the two factors two-scale converge in the sense of Nguetseng. A suitable modification of the differential operators however allows for this extension. The argument follows the lines of a well-known paper of F. Murat of 1978, and uses a two-scale extension of the Fourier transform. This result is also extended to time-dependent functions, and is applied to a two-scale formulation of the Maxwell system of electromagnetism, that accounts for the energy embedded in both coarse- and fine-scale oscillations.
LA - eng
KW - compensated compactness; two-scale Fourier transform
UR - http://eudml.org/doc/272264
ER -

References

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  1. [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal.23 (1992), 1482–1518. Zbl0770.35005MR1185639
  2. [2] G. Allaire, “Shape Optimization by the Homogenization Method”, Springer, New York, 2002. Zbl0990.35001MR1859696
  3. [3] T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal.21 (1990), 823–836. Zbl0698.76106MR1052874
  4. [4] J.-P. Aubin, Un théorème de compacité, C.R. Acad. Sci. Paris, Série I 256 (1963), 5042–5044. Zbl0195.13002MR152860
  5. [5] N. Bakhvalov and G. Panasenko, “Homogenisation: Averaging Processes in Periodic Media”, Kluwer, Dordrecht, 1989. Zbl0692.73012MR1112788
  6. [6] G. Bensoussan, J..L. Lions and G. Papanicolaou, “Asymptotic Analysis for Periodic Structures”, North-Holland, Amsterdam, 1978. Zbl0404.35001MR503330
  7. [7] A. Bourgeat, S. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal.27 (1996), 1520–1543. Zbl0866.35018MR1416507
  8. [8] H. Brezis, “Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert”, North-Holland, Amsterdam, 1973. Zbl0252.47055MR348562
  9. [9] M. Briane and J. Casado-Díaz, Lack of compactness in two-scale convergence, SIAM J. Math. Anal.37 (2005), 343–346. Zbl1112.35016MR2176106
  10. [10] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris, Sér. I 335 (2002), 99–104. Zbl1001.49016MR1921004
  11. [11] D. Cioranescu and P. Donato, “An Introduction to Homogenization”, Oxford Univ. Press, New York, 1999. Zbl0939.35001MR1765047
  12. [12] I. Ekeland and R. Temam, “Analyse Convexe et Problèmes Variationnelles”, Dunod Gauthier-Villars, Paris, 1974. Zbl0281.49001MR463993
  13. [13] J.-B. Hiriart-Urruty and C. Lemarechal, “Convex Analysis and Optimization Algorithms”, Springer, Berlin, 1993. Zbl0795.49001
  14. [14] J. D. Jackson, “Classical Electrodynamics”, Wiley, Chichester, 1962. Zbl0997.78500MR436782
  15. [15] V. V. Jikov, S. M. Kozlov and O.A. Oleinik, “Homogenization of Differential Operators and Integral Functionals”, Springer, Berlin, 1994. Zbl0801.35001MR1329546
  16. [16] M. Lenczner, Homogénéisation d’un circuit électrique, C.R. Acad. Sci. Paris, Ser. II 324 (1997), 537–542. Zbl0887.35016
  17. [17] J. L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires”, Dunod, Paris, 1969. Zbl0189.40603
  18. [18] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 489–507. Zbl0399.46022MR506997
  19. [19] F. Murat, Compacité par compensation. II, In: “Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis” held in Rome in 1978, E. De Giorgi and E. Magenes (eds.), Pitagora, Bologna, 1979, 245–256. Zbl0427.35008MR533170
  20. [20] F. Murat, A survey on compensated compactness, In: “Contributions to Modern Calculus of Variations”, Bologna, 1985, Pitman Res. Notes Math. Ser., Vol. 148, Longman, Harlow, 1987, 145–183. MR894077
  21. [21] F. Murat and L. Tartar, H-convergence, In: “Topics in the Mathematical Modelling of Composite Materials”, A. Cherkaev and R. Kohn (eds.), Birkhäuser, Boston 1997, 21–44. Zbl0920.35019MR1493039
  22. [22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal.20 (1989), 608–623. Zbl0688.35007MR990867
  23. [23] G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics, SIAM J. Math. Anal.21 (1990), 1394–1414. Zbl0723.73011MR1075584
  24. [24] E. Sanchez-Palencia, “Non-Homogeneous Media and Vibration Theory”, Springer, New York, 1980. Zbl0432.70002MR578345
  25. [25] J. Simon, Compact sets in the space L p ( 0 , T ; B ) , Ann. Mat. Pura Appl.146 (1987), 65–96. Zbl0629.46031MR916688
  26. [26] L. Tartar, “Course Peccot”, Collège de France, Paris 1977, unpublished, partially written in [21]. 
  27. [27] L. Tartar, Une nouvelle méthode de résolution d’équations aux dérivées partielles non linéaires, In: “Journées d’Analyse Non linéaire”, Springer, Berlin, 1978, 228–241. Zbl0414.35068MR519433
  28. [28] L. Tartar, Compensated compactness and applications to partial differential equations, In: “Nonlinear Analysis and Mechanics: Heriott-Watt Symposium”, Vol. IV, R. J. Knops (ed.), 1979, 136–212. Zbl0437.35004MR584398
  29. [29] L. Tartar, Mathematical tools for studying oscillations and concentrations: from Young measures to H-measures and their variants, In: “Multiscale Problems in Science and Technology”, N. Antonić, C. J. van Duijn, W. Jäger and A. Mikelić (eds.), Springer, Berlin, 2002, 1–84. Zbl1015.35001MR1998790
  30. [30] A. Visintin, Some properties of two-scale convergence, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15 (2004), 93–107. Zbl1225.35031MR2148538
  31. [31] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var.12 (2006), 371–397. Zbl1110.35009MR2224819
  32. [32] A. Visintin, Homogenization of doubly-nonlinear equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), 211–222. Zbl1223.35204MR2254068
  33. [33] A. Visintin, Two-scale convergence of first-order operators, Z. Anal. Anwendungen26 (2007), 133–164. Zbl1128.35018MR2314158
  34. [34] A. Visintin, Two-scale convergence of some integral functionals, Calc. Var. Partial Differential Equations29 (2007), 239–265. Zbl1129.35011MR2307775

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