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

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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

Abstract
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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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