Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation |
| |
Authors: | G. Gabetta G. Toscani B. Wennberg |
| |
Affiliation: | (1) Dipartimento di Matematica, Università degli Studi di Pavia, 27100 Pavia, Italy;(2) Department of Mathematics, Chalmers University of Technology, 41296 Göteborg, Sweden |
| |
Abstract: | This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then v>Rf(v, t) v2dv0 asR, and this convergence is uniform in time. |
| |
Keywords: | Boltzmann equation Fourier transform probability measures weak convergence Prokhorov metric bivariate distributions with given marginals Tanaka functional |
本文献已被 SpringerLink 等数据库收录! |
|