# 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 & 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. Google Scholar [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. Google Scholar [3] P. Gérard, Microlocal defect measures,, Comm. Partial Differential Equations, 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [4] P. Gérard, Mesures semi-classiques et ondes de Bloch,, in Séminaire sur les Équations aux Dérivées Partielles, (): 1990. Google Scholar [5] P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Revista Matemática Iberoamericana, 9 (1993), 553. doi: 10.4171/RMI/143. Google Scholar [6] L. Tartar, Approximations of H-measures,, Research Report 97-204, (1521), 97. Google Scholar [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. doi: 10.1017/S0308210500020606. Google Scholar [8] L. Tartar, The General Theory of Homogenization. A Personalized Introduction,, Lecture Notes of the Unione Matematica Italiana, (2009). doi: 10.1007/978-3-642-05195-1. Google Scholar

show all references

##### 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. Google Scholar [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. Google Scholar [3] P. Gérard, Microlocal defect measures,, Comm. Partial Differential Equations, 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [4] P. Gérard, Mesures semi-classiques et ondes de Bloch,, in Séminaire sur les Équations aux Dérivées Partielles, (): 1990. Google Scholar [5] P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Revista Matemática Iberoamericana, 9 (1993), 553. doi: 10.4171/RMI/143. Google Scholar [6] L. Tartar, Approximations of H-measures,, Research Report 97-204, (1521), 97. Google Scholar [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. doi: 10.1017/S0308210500020606. Google Scholar [8] L. Tartar, The General Theory of Homogenization. A Personalized Introduction,, Lecture Notes of the Unione Matematica Italiana, (2009). doi: 10.1007/978-3-642-05195-1. Google Scholar
 [1] Thomas Blanc, Mihaï Bostan. Multi-scale analysis for highly anisotropic parabolic problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 335-399. doi: 10.3934/dcdsb.2019186 [2] Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501 [3] Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200 [4] Eugene Kashdan, Svetlana Bunimovich-Mendrazitsky. Multi-scale model of bladder cancer development. Conference Publications, 2011, 2011 (Special) : 803-812. doi: 10.3934/proc.2011.2011.803 [5] Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013 [6] Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic & Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893 [7] Yuanhong Wei, Yong Li, Xue Yang. On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1095-1106. doi: 10.3934/dcdss.2017059 [8] Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic & Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505 [9] Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42. [10] Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449 [11] Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135 [12] Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451 [13] Jean-Philippe Bernard, Emmanuel Frénod, Antoine Rousseau. Modeling confinement in Étang de Thau: Numerical simulations and multi-scale aspects. Conference Publications, 2013, 2013 (special) : 69-76. doi: 10.3934/proc.2013.2013.69 [14] Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062 [15] Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234 [16] Samia Challal, Abdeslem Lyaghfouri. Hölder continuity of solutions to the $A$-Laplace equation involving measures. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1577-1583. doi: 10.3934/cpaa.2009.8.1577 [17] Darko Mitrovic, Ivan Ivec. A generalization of $H$-measures and application on purely fractional scalar conservation laws. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1617-1627. doi: 10.3934/cpaa.2011.10.1617 [18] Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313 [19] Markus Gahn. Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151 [20] Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012

2018 Impact Factor: 0.545