Time-periodic solutions of the Boltzmann equation

The Boltzmann equation with a time-periodic inhomogeneous term is solved on the existence of a time-periodic solution that is close to an absolute Maxwellian and has the same period as the inhomogeneous term, under some smallness assumption on the inhomogeneous term and for the spatial dimension $n\ge 5$, and also for the case $n=3$ and $ 4$ with an additional assumption that the spatial integral of the macroscopic component of the inhomogeneous term vanishes. This solution is a unique time-periodic solution near the relevant Maxwellian and asymptotically stable in time. Similar results are established also with the space-periodic boundary condition. As a special case, our results cover the case where the inhomogeneous term is time-independent, proving the unique existence and asymptotic stability of stationary solutions. The proof is based on a combination of the contraction mapping principle and time-decay estimates of solutions to the linearized Boltzmann equation.