Abstract
We consider a question related to the kinetic theory of granular materials. The model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of colliding particles. Our main result is that, in the space-homogeneous case, a self-similar asymptotics holds, as conjectured by Ernst–Brito. The proof holds for any initial distribution function with a finite moment of some order greater than two.
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REFERENCES
A. V. Bobylev and C. Cercignani, Exact eternal solutions of the Boltzmann equation, J. Stat. Phys. 106:1019-1038 (2002).
A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications, J. Stat. Phys. 106:1039-1071 (2002).
M. H. Ernst and R. Brito, High energy tails for inelastic Maxwell models, preprint (submitted to Europhys. Lett., 2001).
M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, preprint (submitted to J. Stat. Phys., 2002).
A. Baldassarri, U. Marini Bettolo Marconi, and A. Puglisi, Influence of correlations on the velocity statistics of scalar granular gases, Europhys. Lett. 58:14-20 (2002).
P. L. Krapivski and E. Ben-Naim, Nontrivial velocity distributions in inelastic gases, J. Phys. A: Math. Gen. 35:L147-L152 (2002).
A. V. Bobylev, J. A. Carrillo, and I. A. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interaction, J. Stat. Phys. 98:743-773 (2000).
A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys. 110:333-375 (2003).
E. Gabetta, G. Toscani, and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Stat. Phys. 81:901-934 (1995).
A. V. Bobylev, The Fourier transform methods in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR 225:1041-1044 (1975) [In Russian].
A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci Rev. C7:111-233 (1988).
W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1 (Wiley,N.Y., 1966).
A. V. Bobylev and C. Cercignani, Moment equations for a granular material in a thermal bath, J. Stat. Phys. 106:547-567 (2002).
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Bobylev, A.V., Cercignani, C. & Toscani, G. Proof of an Asymptotic Property of Self-Similar Solutions of the Boltzmann Equation for Granular Materials. Journal of Statistical Physics 111, 403–417 (2003). https://doi.org/10.1023/A:1022273528296
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DOI: https://doi.org/10.1023/A:1022273528296