Three-dimensional time-periodic problem on the Boltzmann equation with external force

The time-periodic problem for the Boltzmann equation with a prescribed time-periodic external force in the three-dimensional whole space has remained open since it was first treated in [15], where the result was restricted to spatial dimensions not less than five. This paper gives an affirmative answer when the force is sufficiently small in $\mathcal{C}(\mathbb{R};{\dot{B}}_{2,\mathrm{\infty }}^{-3\mathrm{/}2}\cap {\dot{H}}^N)$ with $N\ge 4$. The proof follows Serrin’s method, through the global stability of the Cauchy problem for a time-periodic force. As a direct consequence, the same argument yields the existence and stability of stationary solutions in three dimensions for a time-independent force, which is not required to be a gradient and may therefore be rotational.