# American Institute of Mathematical Sciences

February  2015, 8(1): 77-90. doi: 10.3934/dcdss.2015.8.77

## Multi-scales H-measures

 1 University Professor of Mathematics emeritus, Carnegie Mellon University, Pittsburgh, PA 15213-3890, United States

Received  February 2013 Revised  July 2013 Published  July 2014

This paper introduces a new tool so called Multi-scales H-measures to analyse the effect of heterogeneities occurring at several scales. In a first place, it recalls the course that brought the introduction of new tools for homogenization and it recalls what are H-Measures. Then the paper gives the definition and the framework of Semi-Classical Measures, presents their capability, and illustrates some of their limitations. Finally, it introduces the concept of Multi-Scale H-measures.
Citation: Luc Tartar. Multi-scales H-measures. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 77-90. doi: 10.3934/dcdss.2015.8.77
##### References:
 [1] Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 397-417. [2] Y. Amirat, K. Hamdache and A. Ziani, Étude d'une équation de transport à mémoire, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 685-688. [3] P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822. [4] P. Gérard, Mesures semi-classiques et ondes de Bloch, in Séminaire sur les Équations aux Dérivées Partielles, 1990-1991, Exposé XVI, École Polytechnique, Palaiseau, 19 pp. [5] P.-L. Lions and T. Paul, Sur les mesures de Wigner, Revista Matemática Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143. [6] L. Tartar, Approximations of H-measures, Research Report 97-204, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890. [7] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 193-230. doi: 10.1017/S0308210500020606. [8] L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-05195-1.

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##### References:
 [1] Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 397-417. [2] Y. Amirat, K. Hamdache and A. Ziani, Étude d'une équation de transport à mémoire, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 685-688. [3] P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822. [4] P. Gérard, Mesures semi-classiques et ondes de Bloch, in Séminaire sur les Équations aux Dérivées Partielles, 1990-1991, Exposé XVI, École Polytechnique, Palaiseau, 19 pp. [5] P.-L. Lions and T. Paul, Sur les mesures de Wigner, Revista Matemática Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143. [6] L. Tartar, Approximations of H-measures, Research Report 97-204, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890. [7] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 193-230. doi: 10.1017/S0308210500020606. [8] L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-05195-1.
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