A one dimensional Boltzmann equation with inelastic collisions

  • D. Benedetto
  • E. Caglioti
  • M. Pulvirenti


We consider the Boltzmann equation for inelastic particles on the line and prove some preliminary results on existence and uniqueness of the solutions. We also discuss some connections with another kinetic equation investigated by the same authors.


Boltzmann Equation Particle System Granular Medium Inelastic Collision BBGKY Hierarchy 
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  1. [1]
    ARKERYD L., “Existence theorems for certain kinetic equations and large data”, Arch. Rational Mech. Anal.103, no. 2,(1988), 139–149MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    BENEDETTO D., CAGLIOTI E., “The Collapse Phenomenon in One Dimensional Inelastic Point Particle Systems”, to appear on Physica DGoogle Scholar
  3. [3]
    BENEDETTO D., CAGLIOTI E., CARRILLO J., and PULVIRENTI, M., “A non Maxwellian equilibrium distribution for one-dimensional granular media”, Jour. Stat. Phys.91 no. 5/6 (1998), 979–990MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    BENEDETTO D., CAGLIOTI E., and PULVIRENTI M., “A Kinetic Equation for Granular Media”, Math. Mod. and Num. An.31 (n. 5),(1997)615–641MATHMathSciNetGoogle Scholar
  5. [5]
    BERNU B., MAZIGHI R., “One-dimensional bounce of inelastically colliding marbles on a wall”, Jour. of Phys. A. Math. Gen.23,(1990),5745–5754MATHCrossRefGoogle Scholar
  6. [6]
    BONY M., “Solutions globales bornees pour le modeles discrets de l'equation de Boltzmann en dimension 1 d'espace”, Act. Jour. E.D.P. St. Jean de Monts (1987)Google Scholar
  7. [7]
    CAMPBELL S., “Rapid Granular Flows”, Ann. Rev. of Fluid Mech.22,(1990), 57–92CrossRefGoogle Scholar
  8. [8]
    CERCIGNANI C., ILLNER R., PULVIRENTI M., “The Mathematical Theory of Dilute Gases”, Springer Verlag, Appl. Mat. Sci.106 (1994)Google Scholar
  9. [9]
    CONSTANTIN P., GROSSMAN E., MUNGAN M., “Inelastic collisions of three particles on a line as a two-dimensional billiard”, Physica D83,(1995), 409–420MATHMathSciNetGoogle Scholar
  10. [10]
    DU Y., LI H., and KADANOFF, P., “Breakdown of Hydrodynamics in a One-dimensional System of Inelastic Particles”, Phys. Rev. Lett.74 (8),(1995), 12668–1271CrossRefGoogle Scholar
  11. [11]
    ESIPOV S.E., PÖSCHEL T., “Boltzmann Equation and Granular Hydrodynamics”, preprint (1995)Google Scholar
  12. [12]
    GOLDHIRSCH I., ZANETTI G., “Clustering Instability in Dissipative Gases”, Phys. Rev. Lett.70, (1993),1619–1622CrossRefGoogle Scholar
  13. [13]
    HAFF P.K., “Grain flow as fluid mechanic phenomenon”, J. Fluid Mech.134,(1983),1619–1622CrossRefGoogle Scholar
  14. [14]
    MAC NAMARA S., and YOUNG W.R., “Inelastic collapse and clumping in a one-dimensional granular medium”, Phys. of Fluids A4 (3),(1992),496–504CrossRefGoogle Scholar
  15. [15]
    MAC NAMARA S., and YOUNG W.R., “Kinetic of a one-dimensional granular in the quasi elastic limit medium”, Phys. of Fluids A 5 (1),(1993),34–45CrossRefGoogle Scholar
  16. [16]
    PUGLIESI A., LORETO V., MARINI BETTOLO MARCONI U., PETRI A., VULPIANI A., “Clustering and non-gaussian behavior in granular matter”, preprint (1997)Google Scholar
  17. [17]
    SELA N., GOLDHIRSCH I., “Hydrodynamics of a one-dimensional granular medium”, Phys. of Fluids7(3), (1995), 34–45CrossRefGoogle Scholar

Copyright information

© Birkhäuser-Verlag 1997

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversitá di Roma La SapienzaRoma

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