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

For the spatially homogeneous Boltzmann equation with hard potentials 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 gaplike estimates valid near the equilibrium, and then connecting the latter to the quantitative non-linear theory. This leads us to an explicit study of the linearized Boltzmann collision operator in functional spaces larger than the usual linearization setting.