# American Institute of Mathematical Sciences

June  2008, 1(2): 223-248. doi: 10.3934/krm.2008.1.223

## Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models

 1 Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière Cedex, France 2 CNRS & Université Paris-Dauphine, UMR7534, F-75016 Paris, France 3 Department of Mathematics at the University of Pavia, via Ferrata 1, 27100 Pavia

Received  January 2008 Revised  January 2008 Published  May 2008

We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get exponential convergence to equilibrium with explicit rate. The proof is based on inequalities between the entropy production functional for hard spheres and Maxwell molecules. Mathematical proof of the convergence to some heat equation in the diffusive limit is also given. From the last two points we deduce the first explicit estimates on the diffusive coefficient in the Fick's law for (inelastic hard-spheres) dissipative gases.
Citation: Bertrand Lods, Clément Mouhot, Giuseppe Toscani. Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models. Kinetic and Related Models, 2008, 1 (2) : 223-248. doi: 10.3934/krm.2008.1.223
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