Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 901084, 12 pages
http://dx.doi.org/10.1155/2011/901084
Research Article

-Distributions: An Extension of -Measures to an Setting

1Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička c. 30, 10 000 Zagreb, Croatia
2Faculty of Mathematics, University of Montenegro, Cetinjski put bb, 81 000 Podgorica, Montenegro

Received 13 March 2011; Revised 22 May 2011; Accepted 20 June 2011

Academic Editor: P. J. Y. Wong

Copyright © 2011 Nenad Antonić and Darko Mitrović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, vol. 74, American Mathematical Society, 1990.
  2. L. Tartar, “H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 115, no. 3-4, pp. 193–230, 1990. View at Google Scholar · View at Zentralblatt MATH
  3. P. Gérard, “Microlocal defect measures,” Communications in Partial Differential Equations, vol. 16, no. 11, pp. 1761–1794, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J. Aleksić, D. Mitrović, and S. Pilipović, “Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media,” Journal of Evolution Equations, vol. 9, no. 4, pp. 809–828, 2009. View at Publisher · View at Google Scholar
  5. J. Aleksić and D. Mitrović, “On the compactness for two dimensional scalar conservation law with discontinuous flux,” Communications in Mathematical Sciences, vol. 7, no. 4, pp. 963–971, 2009. View at Google Scholar · View at Zentralblatt MATH
  6. N. Antonić and M. Lazar, “Parabolic variant of H-measures in homogenisation of a model problem based on Navier-Stokes equation,” Nonlinear Analysis. Real World Applications, vol. 11, no. 6, pp. 4500–4512, 2010. View at Publisher · View at Google Scholar
  7. N. Antonić and Marko Vrdoljak, “Parabolic H-convergence and small-amplitude homogenization,” Applicable Analysis, vol. 88, no. 10-11, pp. 1493–1508, 2009. View at Publisher · View at Google Scholar
  8. R. V. Kohn, “The relaxation of a double-well energy,” Continuum Mechanics and Thermodynamics, vol. 3, no. 3, pp. 193–236, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. Lewandowski, “Vorticities in a LES model for 3D periodic turbulent flows,” Journal of Mathematical Fluid Mechanics, vol. 8, no. 3, pp. 398–422, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. G. Métivier and S. Schochet, “Trilinear resonant interactions of semilinear hyperbolic waves,” Duke Mathematical Journal, vol. 95, no. 2, pp. 241–304, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. Mielke, “Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner-Husimi transforms,” Archive for Rational Mechanics and Analysis, vol. 181, no. 3, pp. 401–448, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. D. Mitrović and I. Ivec, “A generalization of H-measures and application on purely fractional scalar conservation laws,” Communications on Pure and Applied Analysis, vol. 10, p. 11, 2011. View at Google Scholar
  13. E. Y. Panov, “Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,” Archive for Rational Mechanics and Analysis, vol. 195, no. 2, pp. 643–673, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. N. Antonić, “H-measures applied to symmetric systems,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 126, no. 6, pp. 1133–1155, 1996. View at Google Scholar · View at Zentralblatt MATH
  15. N. Antonić and M. Lazar, “A parabolic variant of H-measures,” Annali dell'Universitá di Ferrara, vol. 54, no. 2, pp. 183–201, 2008. View at Publisher · View at Google Scholar
  16. L. Tartar, The General Theory of Homogenization, vol. 7, Springer, Berlin, Germany, 2009. View at Publisher · View at Google Scholar
  17. G. O. Okikiolu, Aspects of the Theory of Bounded Integral Operators in Lp-Spaces, Academic Press, London, UK, 1971.
  18. L. Grafakos, Classical Fourier Analysis, vol. 249, Springer, New York, NY, USA, 2nd edition, 2008.
  19. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970.
  20. N. Antonić and M. Lazar, “H-measures and variants applied to parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 207–225, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. E. Y. Panov, “Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property,” Journal of Mathematical Sciences, vol. 159, no. 2, pp. 180–228, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. S. A. Sazhenkov, “The genuinely nonlinear Graetz-Nusselt ultraparabolic equation,” Rossiĭskaya Akademiya Nauk, vol. 47, no. 2, pp. 431–454, 2006 (Russian), Translation in Siberian Mathematical Journal, vol. 47, no. 2. pp. 355–375, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. L. Hörmander, The Analysis of Linear Partial Differential Operators I–IV, Springer, 1985.
  24. F. Murat, “A survey on compensated compactness,” in Contributions to Modern Calculus of Variations, vol. 148, pp. 145–183, Longman Sci. Tech., Harlow, UK, 1987. View at Google Scholar
  25. F. Murat, “Compacité par compensation,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, vol. 5, no. 3, pp. 489–507, 1978. View at Google Scholar · View at Zentralblatt MATH