METRICS FOR PROBABILITY-DISTRIBUTIONS AND THE TREND TO EQUILIBRIUM FOR SOLUTIONS OF THE BOLTZMANN-EQUATION

This paper deals with the trend to equilibrium of solutions to the spa ce-homogeneous Boltzmann equation for Maxwellian molecules with angula r cutoff as well as with infinite-range forces. The solutions are cons idered as densities of probability distributions. The Tanaka functiona l is a metric for the space of probability distributions, which has pr eviously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of orde r 2 + epsilon, then the convergence to equilibrium in his metric is ex ponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to t he Boltzmann equation, by proving that the Fourier-transformed solutio ns are at least as regular as the Fourier transform of the initial dat a. This is also used to prove that even if the initial data only posse ss a second moment, then integral(\upsilon\ > R) f(upsilon, t) \upsilo n\(2) d upsilon --> 0 as R --> infinity, and this convergence is unifo rm in time.