Journal of Statistical Physics

, Volume 105, Issue 1–2, pp 337–352 | Cite as

On Diffusive Equilibria in Generalized Kinetic Theory

  • Carlo Cercignani
  • Reinhard Illner
  • Cristina Stoica


We investigate the solvability of equations

Q(f,f)+∈ 2 Δf=0

in term of nonnegative integrable densities fL1+(R3). Here, Q(ff) is a generalized collision operator. If Q is the Boltzmann operator, the only solution is 0. In contrast, we show that if Q is the pseudo-Maxwellian collision operator for granular flow, then there are non -trivial weak solutions of (★).

diffusive equilibria kinetic granular flow 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. J. DiPerna and P. L. Lions, On Fokker–Planck–Boltzmann equations, Commun. Math. Phys. (1988).Google Scholar
  2. 2.
    A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic, I. Modeling, and II. Numerical Investigations, SIAM J. Appl. Math. 59:983–1001 (1999).Google Scholar
  3. 3.
    M. Günther, A. Klar, Th. Materne, and R. Wegener, An Explicit Solvable Kinetic Model for Vehicular Traffic and Associated Macroscopic Equations, Math. Models and Methods in the Appl. Sci., to appear.Google Scholar
  4. 4.
    R. Illner and C. Stoica, Kinetic Equilibria in Traffic flow Models, preprint.Google Scholar
  5. 5.
    N. Sela and I. Goldhirsch, Hydrodynamic equations for rapid flows of smooth inelastic spheres to Burnett order, J. Fluid Mech. 361:41–74 (1998).Google Scholar
  6. 6.
    A. V. Bobylev, J. Carrillo, and I. A. Gamba, On some kinetic properties and hydrodynamics equations for inelastic interactions, J. of Stat. Phys. 98(3/4):743–773 (2000).Google Scholar
  7. 7.
    C. Cercignani, Shear flow of a granular material, J. Stat. Phys., to appear.Google Scholar
  8. 8.
    N. Dunford and L. Schwartz, Linear Operators I, Interscience Publ. (1967), Section IV.8.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Carlo Cercignani
    • 1
  • Reinhard Illner
    • 2
  • Cristina Stoica
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

Personalised recommendations