Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials

Clément Mouhot (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

For the spatially homogeneous Boltzmann equation with hard po- tentials and Grad's cutoff (e.g. hard spheres), we give quantitative estimates of exponential convergence to equilibrium, and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, on which we provide a lower bound. Our approach is based on establishing spectral gap-like estimates valid near the equilibrium, and then connecting the latter to the quantitative nonlinear theory. This leads us to an explicit study of the linearized Boltzmann collision operator in functional spaces larger than the usual linearization setting.

Mots clés: Boltzmann equation; spatially homogeneous; linearized Boltzmann collision operator; spectrum; spectral gap; explicit; trend to equilibrium; rate of convergence; entropy production.; entropy production

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Clément Mouhot. Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. 2006. ⟨hal-00076709v1⟩