Journal of Statistical Physics

, Volume 98, Issue 3, pp 743–773

On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions


  • A. V. Bobylev
    • M. V. Keldysh Institute of Applied MathematicsRussian Academy of Sciences
  • J. A. Carrillo
    • Department of MathematicsUniversity of Texas at Austin
    • Departamento de Matemática Aplicada, Facultad de CienciasUniversidad de Granada
  • I. M. Gamba
    • Department of MathematicsUniversity of Texas at Austin

DOI: 10.1023/A:1018627625800

Cite this article as:
Bobylev, A.V., Carrillo, J.A. & Gamba, I.M. Journal of Statistical Physics (2000) 98: 743. doi:10.1023/A:1018627625800


We investigate a Boltzmann equation for inelastic scattering in which the relative velocity in the collision frequency is approximated by the thermal speed. The inelasticity is given by a velocity variable restitution coefficient. This equation is the analog to the Boltzmann classical equation for Maxwellian molecules. We study the homogeneous regime using Fourier analysis methods. We analyze the existence and uniqueness questions, the linearized operator around the Dirac delta function, self-similar solutions and moment equations. We clarify the conditions under which self-similar solutions describe the asymptotic behavior of the homogeneous equation. We obtain formally a hydrodynamic description for near elastic particles under the assumption of constant and variable restitution coefficient. We describe the linear long-wave stability/instability for homogeneous cooling states.

homogeneous inelastic Boltzmannlarge-time asymptoticsself-similar solutionshydrodynamics
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© Plenum Publishing Corporation 2000