zbMATH — the first resource for mathematics

Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres. (English) Zbl 1178.82056
The spatially homogeneous Boltzmann equation is considered for hard spheres undergoing inelastic collisions with a constant normal restitution coefficient in the range [0,1]. One needs to know rudiments of a physical theory of granular gases and a mathematical introduction to this theory, set by the present authors [part I with M. Rodriguez Ricard, J. Stat. Phys. 124, No. 2–4, 655–702 (2006; Zbl 1135.82325), part II with C. Mouhot, ibid., 703–746 (2006; Zbl 1135.82030)]. A brief review is given of why simplified Boltzmann models for inelastic Maxwell molecules or pseudo inealstic hard spheres do not capture some crucial physical features of the cooling process of granular gases. Basic mathematical works on spatially homogeneous inelastic hard spheres Boltzmann models are mentioned as a departure point of the analysis.
In the present paper, the self-similarity properties of the Boltzmann equation for inelastic hard spheres are studied. The uniqueness and attractivity of self-similar solutions are established, thus giving a complete answer to the Ernst-Brito conjecture [M. H. Ernst and R. Brito, J. Stat. Phys. 109, No. 3–4, 407–432 (2002; Zbl 1015.82030)] for inelastic hard spheres with small inelasticity. The proofs are based on adopting a suitable elastic limit rescaling and the construction of a smooth path of self-similar profiles. Suitable tools from perturbative theory of linear operators are employed as well.

82C22 Interacting particle systems in time-dependent statistical mechanics
76T25 Granular flows
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C70 Transport processes in time-dependent statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35F25 Initial value problems for nonlinear first-order PDEs
PDF BibTeX Cite
Full Text: DOI arXiv
[1] Abrahamsson F.: Strong L 1 convergence to equilibrium without entropy conditions for the Boltzmann equation. Comm. Part. Diff. Eqs. 24, 1501–1535 (1999) · Zbl 1059.35014
[2] Bisi M., Carrillo J.A., Toscani G.: Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria. J. Stat. Phys. 118(1–2), 301–331 (2005) · Zbl 1085.82008
[3] Bisi M., Carrillo J.A., Toscani G.: Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model. J. Stat. Phys. 124(2–4), 625–653 (2006) · Zbl 1135.82028
[4] Blake M.D.: A spectral bound for asymptotically norm-continuous semigroups. J. Op. Th. 45, 111–130 (2001) · Zbl 0994.47039
[5] Bobylev A.V., Carillo J.A., Gamba I.: On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Stat. Phys. 98, 743–773 (2000) · Zbl 1056.76071
[6] Baranger C., Mouhot C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Matem. Iberoam. 21, 819–841 (2005) · Zbl 1092.76057
[7] Bobylev A.V., Cercignani C., Toscani G.: Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Stat. Phys. 111, 403–417 (2003) · Zbl 1119.82318
[8] Bobylev A.V., Gamba I., Panferov V.: Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions. J. Stat. Phys. 116, 1651–1682 (2004) · Zbl 1097.82021
[9] Brilliantov N.V., Pöschel T.: Kinetic Theory of Granular Gases. Oxford Graduate Texts. Oxford University Press, Oxford (2004)
[10] Caglioti E., Villani C.: Homogeneous cooling states are not always good approximations to granular flows. Arch. Rat. Mech. Anal. 163, 329–343 (2002) · Zbl 1053.74012
[11] Carleman T.: Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60, 91–146 (1932) · Zbl 0006.40002
[12] Carleman, T.: Problèmes mathématiques dans la théorie cinétique des gaz. Uppsala: Almqvist and Wiksells Boktryckeri Ab, 1957 · Zbl 0077.23401
[13] Cercignani, C.: Recent developments in the mechanics of granular materials. In: Fisica matematica e ingegneria delle strutture, Bologna: Pitagora Editrice, 1995, pp. 119–132
[14] Csiszár I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, 85–108 (1963) · Zbl 0124.08703
[15] Ernst M.H., Brito R.: Driven inelastic Maxwell molecules with high energy tails. Phys. Rev. E 65, 85–108 (2002)
[16] Ernst M.H., Brito R.: Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails. J. Stat. Phys. 109, 407–432 (2002) · Zbl 1015.82030
[17] Gamba I., Panferov V., Villani C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246, 503–541 (2004) · Zbl 1106.82031
[18] Grad, H.: Asymptotic theory of the Boltzmann equation. II. In: Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962), Vol. I, New York: Academic Press, 1963, pp 26–59
[19] Haff P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430 (1983) · Zbl 0537.76005
[20] Hilbert, D.: Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. Math. Ann. 72, (1912), New York: Chelsea Publ., 1953 · JFM 43.0423.01
[21] Kato T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1995) · Zbl 0836.47009
[22] Kullback S.: Information Theory and Statistics. John Wiley, New York (1959) · Zbl 0088.10406
[23] Mischler S., Mouhot C., Rodriguez Ricard M.: Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem. J. Stat. Phys. 124, 655–702 (2006) · Zbl 1135.82325
[24] Mischler S., Mouhot C.: Cooling process for inelastic Boltzmann equations for hard spheres, Part I:Self-similar solutions and tail behavior. J. Stat. Phys. 124, 703–746 (2006) · Zbl 1135.82030
[25] Mischler, S., Mouhot, C.: Work in progress
[26] Mischler S., Wennberg B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. Henri Poincaré, Analyse non linéaire 16, 467–501 (1999) · Zbl 0946.35075
[27] Mouhot C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 261, 629–672 (2006) · Zbl 1113.82062
[28] Mouhot C., Villani C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rat. Mech. Anal. 173, 169–212 (2004) · Zbl 1063.76086
[29] Nirenberg, L.: Topics in Nonlinear Functional Analysis. With a chapter by E. Zehnder. Notes by R. A. Artino. Lecture Notes, 1973–1974. New York: Courant Institute of Mathematical Sciences, New York University, 1974 · Zbl 1198.35202
[30] Pulvirenti A., Wennberg B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 183, 145–160 (1997) · Zbl 0866.76077
[31] Villani C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234, 455–490 (2003) · Zbl 1041.82018
[32] Yao P.F.: On the inversion of the Laplace transform of C 0 semigroups and its applications. SIAM J. Math. Anal. 26, 1331–1341 (1995) · Zbl 0845.47032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.