  In this paper, we present an energy method for the system of Boltzmann
equations in the multicomponent mixture case, based on a micro-macro
decomposition. More precisely, the perturbation of a solution to the Boltzmann
equation around a global equilibrium is decomposed into the sum of a
macroscopic and a microscopic part, for which we obtain a priori estimates at
both lower and higher orders. These estimates are obtained under a suitable
smallness assumption. The assumption can be justified a posteriori in the
higher-order case, leading to the closure of the corresponding estimate.
