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

Mouhot, Clément

Mouhot, Clément (2006), Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Communications in Mathematical Physics, 261, p. 629-672. http://dx.doi.org/10.1007/s00220-005-1455-x

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00076709/en/

Date

2006

Journal name

Communications in Mathematical Physics

Volume

261

Publisher

Springer

Pages

629-672

Abstract (EN)

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.

Subjects / Keywords

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