On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions
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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.
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Journal of Statistical Physics
Volume 98, Issue 3-4 , pp 743-773
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- homogeneous inelastic Boltzmann
- large-time asymptotics
- self-similar solutions
- Industry Sectors
- Author Affiliations
- 1. M. V. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia
- 2. Department of Mathematics, University of Texas at Austin, 78712, Austin, Texas
- 3. Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain
- 4. Department of Mathematics, University of Texas at Austin, 78712, Austin, Texas, USA