Cooling process for inelastic boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior

Journal of Statistical Physics

Volumn 124, Issue 2-4, 2006, Pages 703-746

  Mischler, S ^a   Mouhot, C ^a,b  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES   (France)

Author keywords

Boltzmann equation; Cooling process; Granular gas; Haff's law; Inelastic hard spheres; Regularity of the collision operator; Self similar solutions; Tail behavior

Indexed keywords

EID: 33751046314     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-006-9097-8     Document Type: Article

References (35)

1. 1 convergence to equilibrium without entropy conditions for the Boltzmann equation, Comm. Partial Differential Equations 24:1501-1535 (1999).
2. R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long range interactions, Arch. Ration. Mech. Anal. 152:327-355 (2000).
3. M. Balabane, Équations différentielles, cours de l'École des Ponts et Chaussées (1985).
4. D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media, Math. Mod. Numér. Anal. 31:615-641 (1997).
5. A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to the spatially homogeneous problems, J. Statist. Phys. 88:1183-1214 (1997).
6. A. V. Bobylev, J. A. Carillo, and I. Gamba, On some properties of kinetic and hydrodynamics equations for inelastic interactions, J. Statist. Phys. 98:743-773 (2000).
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, I. Gamba, and V. Panferov, Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions, J. Statist. Phys. 116:1651-1682 (2004).
10. F. Bouchut and L. Desvillettes, A proof of smoothing properties of the positive part of Boltzmann's kernel, Rev. Mat. Iberoamericana 14:47-61 (1998).
11. N. V. Brilliantov and T. Pöeschel, Kinetic Theory of Granular Gases (Oxford Graduate Texts. Oxford University Press, Oxford, 2004).
12. E. Caglioti and C. Villani, Homogeneous Cooling States are not always good approximations to granular flows, Arch. Ration. Mech Anal. 163:329-343 (2002).
13. T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math. 60: 91-146 (1933).
14. C. Cercignani, Recent developments in the mechanics of granular materials, in Fisica matematica e ingegneria delle structure (Pitagora Editrice, Bologna, 1995), pp. 119-132.
15. p estimates for the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22:127-142 (2005).
16. M. H. Ernst and R. Brito, Driven inelastic Maxwell molecules with high energy tails, Phys. Rev. E 65:1-4 (2002).
17. M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Statist. Phys. 109:407-432 (2002).
18. M. Escobedo, S. Mischler, and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire 22:99-125 (2005).
19. I. Gamba, V. Panferov, and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys. 246:503-541 (2004).
20. I. Gamba, V. Panferov, and C. Villani Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation. Work in progress.
21. p -estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal. 92:23-57 (1986).
22. p-properties for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal. 103:1-38 (1988).
23. H. Grad, Asymptotic theory of the Boltzmann equation. II Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de 1'UNESCO, Paris, 1962), Vol. I, pp 26-59, New York, 1963.
24. P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech. 134 (1983).
25. H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal 172:407-428 (2004).
26. P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications I, II, III, J. Math. Kyoto Univ. 34:391-427, 429-461, 539-584 (1994).
27. X. Lu, A direct method for the regularity of the gain term in the Boltzmann equation, J. Math. Anal. Appl. 228 (1998), 409-435.
28. S. Mischler, C. Mouhot, and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem, J. Statist. Phys., this volume.
29. S. Mischler and C. Mouhot, Convergence to self-similarity for a Boltzmann equation of dissipative hard spheres with small inelasticity (Work in progress).
30. S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 16:467-501 (1999).
31. C. Mouhot, Quantitative lower bound for the full Boltzmann equation, Part I: Periodic boundary conditions, Comm. Partial Differential Equations 30:881-917 (2005).
32. C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal. 173 (2004), 169-212.
33. A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys. 183:145-160 (1997).
34. C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, Vol. I, 71-305 (North-Holland, Amsterdam, 2002).
35. B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Partial Differential Equations 19:2057-2074 (1994).