Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions

Journal of Statistical Physics

Volumn 116, Issue 5-6, 2004, Pages 1651-1682

  Bobylev, A V ^a   Gamba, I M ^b   Panferov, V A ^b,c  

^a KARLSTAD UNIVERSITY   (Sweden)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c UNIVERSITY OF VICTORIA   (Canada)

Author keywords

Boltzmanu equations; Granular flows; Heat bath; High energy tails; Inelastic collisions; Povzner inequalities; Shear flow

Indexed keywords

EID: 7244221021     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:JOSS.0000041751.11664.ea     Document Type: Article

References (36)

1. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984. Reprint of the 1972 edition.
2. A. Baldassarri, U. M. B. Marconi, and A. Puglisi, Influence of Correlations on the Velocity Statistics of Scalar Granular Gases. Europhys. Lett. 58:14-20 (2002).
3. C. Bizon, M. D. Shattuck, J. B. Swift, and H. L. Swinney, Transport Coefficients for Granular Media from Molecular Dynamics Simulations, Phys. Rev. E, 60 (1999), pp. 4340-4351, ArXiv:cond-mat/9904132.
4. A. V. Bobylev, Moment Inequalities for the Boltzmann Equation and Applications to Spatially Homogeneous Problems. J. Statist. Phys. 88:1183-1214 (1997).
5. A. V. Bobylev, J. A. Carrillo, and I. M. Gamba, On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions. J. Statist. Phys. 98:743-773 (2000).
6. A. V. Bobylev and C. Cercignani, Moment Equations for a Granular Material in a Thermal Bath. J. Stat. Phys. 106:547-567 (2002).
7. A. V. Bobylev and C. Cercignani, Self-similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions. J. Statist. Phys. 110: 333-375 (2003)
8. A. V. Bobylev, C. Cercignani, and G. Toscani, Proof of an Asymptotic. Property of Self-similar Solutions of the Boltzmann Equation for Granular Materials. J. Statist. Phys. 111:403-417 (2003).
9. A. V. Bobylev, M. Groppi, and G. Spiga, Approximate Solutions to the Problem of Stationary Shear Flow of Smooth Granular Materials. Eur. J. Mech. B Fluids 21:91-103 (2002).
10. C. Cercignani, Recent Developments in the Mechanics of Granular Materials, in fisica matematica e ingegneria delle strutture, Pitagora Editrice, Bologna, 1995, pp. 119-132.
11. C. Cercignani, Shear Flow of a Granular Material. J. Statist. Phys. 102:1407-1415 (2001).
12. L. Desvillettes, Some Applications of the Method of Moments for the Homogeneous Boltzmann and Kac Equations. Arch. Rational Mech. Anal. 123:387-404 (1993).
13. T. Elmroth, Global Boundedness of Moments of Solutions of the Boltzmann Equation for Forces of Infinite Range. Arch. Rational Mech. Anal. 82:1-12 (1983).
14. M. H. Ernst and R. Brito, Velocity Tails for Inelastic Maxwell Models, cond-mat/0111093, (2001).
15. M. H. Ernst and R. Brito, Driven Inelastic Maxwell Models with High Energy Tails. Phys. Rev. E 65:040301(1-4) (2002).
16. M. H. Ernst and R. Brito, Scaling Solutions of Inelastic Boltzmann Equations with Over-Populated High Energy Tails, J. Stat. Phys. to appear (2002), ArXiv:cond-mat/0112417.
17. S. E. Esipov and T. Pöschel, The Granular Phase Diagram. J. Stat. Phys. 86:1385 (1997).
18. I. M. Gamba, V. Panferov, and C. Villani, work in progress.
19. I. M. Gamba, V. Panferov, and C. Villani, On the Boltzmann Equations for Diffusively Excited Granular Media. Comm. Math. Phys. 246: 503-541 (2004).
20. I. M. Gamba, S. Rjasanow, and W. Wagner, Direct Simulation of the Uniformly Heated Granular Boltzmann Equation. Submitted, (2003).
21. I. Goldhirsch, Scales and Kinetics of Granular Flows. Chaos, 9: 659-672 (1999).
22. I. Goldhirsch, Rapid Granular flows: Kinetics and Hydrodynamics. In Modeling in applied sciences: A kinetic theory approach, Birkhäuser Boston, Boston, MA, 2000, pp. 21-79.
23. P-properties for the Spatially Homogeneous Boltzmann Equation, Arch. Rational Mech. Anal. 103:1-38 (1988).
24. P. L. Krapivsky and E. Ben-Naim, Multiscaling in Infinite Dimensional Collision Processes. Phys. Rev. E, 61:R5 (2000) ArXiv:cond-mat/9909176.
25. P. L. Krapivsky and E. Ben-Naim, Nontrivial Velocity Distributions in Inelastic Gases. J. Phys. A 35:p. L147 (2002) ArXiv:cond-mat/011104.
26. P. L. Krapivsky and E. Ben-Naim, The Inelastic. Maxwell Model, cond-mat/0301238, (2003).
27. X. Lu, Conservation of Energy, Entropy Identity, and Local Stability for the Spatially Homogeneous Boltzmann equation, J. Statist. Phys. 96:765-796 (1999).
28. S. Mischler and B. Wennberg, On the Spatially Homogeneous Boltzmann Equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16:467-501 (1999).
29. J. M. Montanero and A. Santos, Computer Simulation of Uniformly Heated Granular Fluids. Gran. Matt. 2:53-64 (2000) ArXiv:cond-mat/0002323.
30. S. J. Moon, M. D. Shattuck, and J. B. Swift, Velocity Distributions and Correlations in Homogeneously Heated Granular Media, Phys. Rev. E 64:031303(1-10) (2001) ArXiv:cond-mat/0105322.
31. A. Y. Povzner, On the Boltzmann Equation in the Kinetic Theory of Gases. Mat. Sb. (N.S.) 58(100):65-86 (1962).
32. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1997.
33. T. P. C. van Noije and M. H. Ernst, Velocity Distributions in Homogeneously Cooling and Heated Granular Fluids. Gran. Matt. 1:57 (1998) ArXiv:cond-mat/9803042.
34. C. Villani, A Survey of Mathematical Topics in Collisional Kinetic Theory. In Handbook of Mathematical Fluid Mechanics, S. Friedlander, and D. Serre, eds., vol. 1, Elsevier, 2002, ch. 2.
35. B. Wennberg, Entropy Dissipation and Moment Production for the Boltzmann Equation. J. Statist. Phys. 86:1053-1066 (1997).
36. D. R. M. Williams and F. C. MacKintosh, Driven Dranular Media in One Dimension: Correlations and Equation of State. Phys. Rev. E 54:9-12 (1996).