Time-periodic solutions of the Boltzmann equation with soft potentials in ${\mathbb{R}}^3$

We establish the existence of time-periodic solutions to the Boltzmann equation for the full range of soft potentials and prove their global stability. The proof relies on refined energy estimates combining Besov and Sobolev regularity with velocity-weighted energy methods. The analysis is complicated by the lack of a spectral gap in the soft potential regime together with the absence of time integrability of the periodic forcing over ${\mathbb{R}}^{+}$. This provides a framework for the study of time-periodic behavior for the Boltzmann equation.