# American Institute of Mathematical Sciences

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

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