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On some properties of kinetic and hydrodynamic equations for inelastic interactions. (English) Zbl 1056.76071

Summary: 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 of 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.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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