Multiscale Homogenization with Bounded Ratios and Anomalous Slow Diffusion

Communications on Pure and Applied Mathematics

Volumn 56, Issue 1, 2003, Pages 80-113

  Arous, Gérard Ben ^a,b   Owhadi, Houman ^a,c  

^a EPFL   (Switzerland)

^b NEW YORK UNIVERSITY   (United States)

^c UNIVERSITÉ DE PROVENCE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037211751     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.10053     Document Type: Article

References (43)

1. Allaire, G.; Briane, M. Multiscale convergence and reiterated homogenisation. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 2, 297-342.
2. Ancona, A. Some results and examples about the behavior of harmonic functions and Green's functions with respect to second order elliptic operators. Nagoya Math. J. 165 (2002), 123-158.
3. Avellaneda, M. Iterated homogenization, differential effective medium theory and applications. Comm. Pure Appl. Math. 40 (1987), no. 5, 527-554.
4. Avellaneda, M. Homogenization and renormalization: the mathematics of multi-scale random media and turbulent diffusion. Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994), 251-268. Lectures in Applied Mathematics, 31. American Mathematical Society, Providence, R.I., 1996.
5. Avellaneda, M.; Majda, A. J. Mathematical models with exact renormalization for turbulent transport. Comm. Math. Phys. 131 (1990), no. 2, 381-429.
6. Avellaneda, M.; Majda, A. J. Homogenization and renormalization of multiple-scattering expansions for Green functions in turbulent transport. Composite media and homogenization theory (Trieste, 1990), 13-35. Progress in Nonlinear Differential Equations and Their Applications, 5. BirkhA̧ä user, Boston, 1991.
7. Barlow, M. T. Diffusions on fractals. Lectures on probability theory and statistics (Saint-Flour, 1995), 1-121. Lecture Notes in Mathematics, 1690. Springer, Berlin, 1998.
8. Barlow, M. T.; Bass, R. F. Local times for Brownian motion on the Sierpiński carpet. Probab. Theory Related Fields 85 (1990), no. 1, 91-104.
9. Barlow, M. T.; Bass, R. F. Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51 (1999), no. 4, 673-744.
10. Ben Arous, G.; Owhadi, H. Super-diffusivity in a shear flow model from perpetual homogenization. Comm. Math. Phys. 227 (2002), no. 2, 281-302.
11. Bensoussan, A.; Lions, J.-L.; Papanicolaou, G. Asymptotic analysis for periodic structures. Studies in Mathematics and Its Applications, 5. North-Holland, Amsterdam-New York, 1978.
12. Bhattacharya, R. Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media. Ann. Appl. Probab. 9 (1999), no. 4, 951-1020.
13. Bhattacharya, R.; Denker, M.; Goswami, A. Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales. Stochastic Process. Appl. 80 (1999), no. 1, 55-86.
14. Boucher, S. Modules effectifs de materiaux composites quasi homogenes et quasi isotropes, constitues d'une matrice elastique et d'inclusions elastiques. II. Cas des concentrations finies en inclusions. Rev. Metall. 22 (1976), 31-36.
15. Bruggeman, D. A. G. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. Ann. Phys. (Leipzig) 24 (1935), 636-679.
16. Cleary, M. P.; Chen, I. W.; Lee, S. M. Self-consistent techniques for heterogeneous media. ASCE J. Eng. Mech. 106 (1980), 861-887.
17. Electrical transport and optical properties of inhomogeneous media (Ohio State University, 1977). AIP Conference Proceedings, 40. American Institute of Physics, New York, 1977.
18. Fannjiang, A. Anomalous diffusion in random flows. Mathematics of multiscale materials (Minneapolis, MN, 1995-1996), 81-99. IMA Volumes in Mathematics and Its Applications, 99. Springer, New York, 1998.
19. Fannjiang, A.; Komorowski, T. Fractional Brownian motions in a limit of turbulent transport. Ann. Appl. Probab. 10 (2000), no. 4, 1100-1120.
20. Fannjiang, A.; Papanicolaou, G. Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 (1994), no. 2, 333-408.
21. Fannjiang, A.; Papanicolaou, G. Diffusion in turbulence. Probab. Theory Related Fields 105 (1996), no. 3, 279-334.
22. Furtado, F.; Glimm, J.; Lindquist, B.; Pereira, F. Multiple length scale calculus of mixing length growth in tracer floods. Proceedings of the emerging technologies conference, Houston, TX, 251-259. Institute for Improved Oil Recovery, 1990.
23. Furtado, F.; Glimm, J.; Lindquist, B.; Pereira, F.; Zhang, Q. Time dependent anomalous diffusion for flow in multi-fractal porous media. Proceedings of the workshop on numerical methods for the simulation of multiphase and complex flow, 79-89. Lecture Notes in Physics, 398. Springer, New York, 1992.
24. Glimm, J.; Lindquist, B.; Pereira, F.; Peierls, R. The multi-fractal hypothesis and anomalous diffusion. Mat. Apl. Comput. 11 (1992), no. 2, 189-207.
25. Hambly, B. M.; Kumagai, T.; Kusuoka, S.; Zhou, X. Y. Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets. J. Math. Soc. Japan 52 (2000), no. 2, 373-408.
26. Isichenko, M. B.; Kalda, J. Statistical topography. II. Two-dimensional transport of a passive scalar. J. Nonlinear Sci. 1 (1991), no. 4, 375-396.
27. Jikov, V. V.; Kozlov, S. M. Multiscaled homogenization. Homogenization, 35-64. Series on Advances in Mathematics for Applied Sciences, 50. World Scientific, River Edge, N.J., 1999.
28. Keller, G. Equilibrium states in ergodic theory. London Mathematical Society Student Texts, 42. Cambridge University, Cambridge, 1998.
29. Komorowski, T.; Olla, S. On homogenization of time-dependent random flows. Theory Related Fields 121 (2001), no. 1, 98-116.
30. Kozlov, D. Multiscaled approach in homogenization. Proceedings of the second workshop on composite media and homogenization theory, International Centre for Theoretical Physics, Trieste, 1993, 217-229. World Scientific, Singapore-River Edge, N.J., 1995.
31. Li, Y. Y.; Vogelius, M. Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91-151.
32. McLaughlin, R. A. A study of the differential scheme for composite materials. Int. J. Eng. Sci. 15 (1997), 237-244.
33. Norris, A. N. A differential scheme for the effective moduli of composites. Mechanics of Materials 4 (1985), 1-16.
34. Osada, H. Isoperimetric constants and estimates of heat kernels of pre Sierpiński carpets. Probab. Theory Related Fields 86 (1990), no. 4, 469-490.
35. Owhadi, H. Anomalous diffusion and homogenization on an infinite number of scales. Doctoral dissertation, Swiss Federal Institute of Technology, 2001. Available at: http://www.cmi.univ-mrs.fr/~owhadi/.
36. Owhadi, H. Super-diffusion in turbulence: Kolmogorov Richardson laws versus Landau Lagrangian chaos. In preparation, 2001.
37. Owhadi, H. Anomalous slow diffusion from perpetual homogenization. Ann. Probab., in press. Available at: http://www.cmi.univ-mrs.fr/~owhadi/.
38. Ruelle, D. Thermodynamic formalis. The mathematical structures of classical equilibrium statistical mechanics. Encyclopedia of Mathematics and Its Applications, 5. Addison-Wesley, Reading, Mass., 1978.
39. Sen, P. N.; Scala, C.; Cohen, M. H. A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophys. 46 (1981), 781-795.
40. Stampacchia, G. Le probléme de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965), no. 1, 189-258.
41. Stampacchia, G. Equations elliptiques du second ordre à coefficients discontinus. Séminaire de mathématiques supérieures, SMS016. CRM Université de Montréal, 1971.
42. Zhang, Q. A multi-length scale theory of mixing length growth for tracer flow in heterogeneous porous media. J. Statist. Phys. 66 (1992), 485-501.
43. Zhang, Q.; Glimm, J. Inertial range scaling of laminar shear flow as a model of turbulent transport. Comm. Math. Phys. 146 (1992), no. 2, 217-229.