Exponential Decay for Soft Potentials near Maxwellian

We consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches zero at the rate of e−λt p for some λ > 0 and 0 < p < 1. Our method is based on an unified energy estimate with appropriate exponential velocity weight. Our results extend the classical result Caflisch of [2] to the case of very soft potential and Coulomb interactions, and also improve the recent “almost exponential” decay results by [5, 14].