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January  2009, 24(1): 159-185. doi: 10.3934/dcds.2009.24.159

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

Stéphane Mischler 1, and  Clément Mouhot 2,

1. 

Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris cedex 16, France
2. 

CNRS & Université Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775, Paris cedex 16, France

Received  November 2007 Revised  July 2008 Published  January 2009

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: spectrum., hard spheres, uniqueness, granular gases, small inelasticity, stability, elastic limit, degenerated perturbation, Inelastic Boltzmann equation, random forcing, stationary solution.
Mathematics Subject Classification: Primary: 76P05; Secondary: 76T2.
Citation: Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159
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