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

Stéphane Mischler; Clément Mouhot

doi:10.3934/dcds.2009.24.159

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2009, Volume 24, Issue 1: 159-185. Doi: 10.3934/dcds.2009.24.159

This issue Previous Article Boltzmann equation, boundary effects Next Article Energy estimates for electro-reaction-diffusion systems with partly fast kinetics

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
2.
CNRS & Université Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775, Paris cedex 16

Received: October 31, 2007

Revised: June 30, 2008

Published: December 2008

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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.

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Mathematics Subject Classification: Primary: 76P05; Secondary: 76T25.

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

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