Time-periodic solutions of the Boltzmann equation

Abstract

The Boltzmann equationwith a time-periodic inhomogeneous termis solved on the existence ofa time-periodic solution that is close to an absolute Maxwellian and hasthe same period as the inhomogeneous term, under some smallness assumption onthe inhomogeneous term and for the spatial dimension $n\ge 5$, and alsofor the case $n=3$ and $ 4$ with an additionalassumption that the spatial integral ofthe macroscopic component of the inhomogeneous termvanishes.This solution is a uniquetime-periodic solution near the relevant Maxwellianand asymptotically stable in time. Similar results are established alsowith the space-periodicboundary condition. As a special case,our results cover the casewhere the inhomogeneous term is time-independent,proving the uniqueexistence and asymptotic stability of stationary solutions.The proof is based on a combination ofthe contraction mapping principle and time-decay estimates ofsolutions to the linearized Boltzmann equation.