Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models
Lods, Bertrand; Mouhot, Clément; Toscani, Giuseppe (2008), Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models, Kinetic and related models, 1, 2, p. 223-248
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00220504/en/
Journal nameKinetic and related models
American Institute of Mathematical Sciences
MetadataShow full item record
Abstract (EN)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.
Subjects / Keywordslinear Boltzmann equation; entropy production; spectral gap; diffusion approximation; Fick's law; diffusive coefficient; Granular gas dynamics
Showing items related by title and author.
Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior Mouhot, Clément; Mischler, Stéphane (2006) Article accepté pour publication ou publié
Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres Mouhot, Clément; Mischler, Stéphane (2009) Article accepté pour publication ou publié
Rodriguez Ricard, Mariano; Mouhot, Clément; Mischler, Stéphane (2006) Article accepté pour publication ou publié