Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media

Pages: 159 - 185, Volume 24, Issue 1, May 2009      doi:10.3934/dcds.2009.24.159

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Stéphane Mischler - Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris cedex 16, France (email)
Clément Mouhot - CNRS & Université Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775, Paris cedex 16, France (email)

Abstract: We consider a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing (in the framework of so-called constant normal restitution coefficients $\alpha \in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_$∗,1) for some constructive $\alpha_$∗, $\in$ [0,1)) we prove uniqueness of the stationary solution for given values of the restitution coefficient $\alpha \in [\alpha_$∗,1), the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.

Keywords:  Inelastic Boltzmann equation, granular gases, random forcing, hard spheres, stationary solution, uniqueness, stability, small inelasticity, elastic limit, degenerated perturbation, spectrum.
Mathematics Subject Classification:  Primary: 76P05; Secondary: 76T25.

Received: November 2007;      Revised: July 2008;      Available Online: January 2009.