Abstract
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.
Similar content being viewed by others
REFERENCES
N. Bellomo, M. Esteban, and M. Lachowicz, Nonlinear kinetic equations with dissipative collisions, Appl. Math. Letter 8:47–52 (1995).
D. Benedetto and E. Caglioti, The collapse phenomenon in one dimensional inelastic point particle system, to appear in Physica D.
D. Benedetto, E. Caglioti, J. A. Carrillo, and M. Pulvirenti, A non-maxwellian steady distribution for one-dimensional granular media, J. Stat. Phys. 91:979–990 (1998).
D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media, Math. Mod. and Num. An. 31:615–641 (1997).
C. Bizon, J. B. Shattuck, M. D. Swift, W. D. McCormick, H. L. Swinney, Patterns in 3D vertically oscillated granular layers: Simulation and experiment, Phys. Rev. Letters 80:57–60 (1998).
A. V. Bobylev, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a maxwellian gas, Teoret. Mat. Fiz. 60:280–310 (1984).
A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Rev. C Math. Phys. 7:111–233 (1988).
J. J. Brey, J. W. Dufty, A. Santos, Dissipative dynamics for hard spheres, J. Stat. Phys. 87:1051–1068 (1997).
C. Cercignani, Recent developments in the mechanics of granular materials, Fisica matematica e ingegneria delle strutture, pp. 119–132, Pitagora Editrice, Bologna (1995).
C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Series in Applied Mathematical Sciences, Vol. 106 (Springer-Verlag, 1994).
P. Constantin, E. Grossman, and M. Mungan, Inelastic collisions of three particles on a line as a two dimensional billiard, Physica D 83:409–420 (1995).
M. J. Esteban and B. Perthame, On the modified Enskog equation for elastic and inelastic collisions. Models with spin, Ann. Inst. H. Poincaré Anal. Non Linaire 8:289–308 (1991).
D. Gidaspow, Multiphase Flow and Fluidization, Continuum and Kinetic Theory Descriptions (Academic Press, 1994).
D. Goldman, M. D. Shattuck, C. Bizon, W. D. McCormick, J. B. Swift, H. L. Swinney, Absence of inelastic collapse in a realistic three ball model, Phys. Rev. E 57:4831–4833 (1998).
A. Goldshtein and M. Shapiro, Mechanics of collisional motion of granular materials. Part I. General hydrodynamic equations, J. Fluid Mech. 282:75–114 (1995).
J. T. Jenkins and M. W. Richman, Grad's 13-moment system for a dense gas of inelastic spheres, Arch. Rat. Mech. Anal. 87:355–377 (1985).
J. Mathews and R. Walker, Mathematical Methods of Physics (Addison-Wesley, 1970).
N. Sela and I. Goldhirsch, Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order, J. Fluid Mech. 361:41–74 (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bobylev, A.V., Carrillo, J.A. & Gamba, I.M. On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions. Journal of Statistical Physics 98, 743–773 (2000). https://doi.org/10.1023/A:1018627625800
Issue Date:
DOI: https://doi.org/10.1023/A:1018627625800