Time-periodic solutions of the Vlasov-Poisson-Boltzmann system with a general external force in $\mathbb{R}^3$

Abstract

In this paper, we study the time-periodic problem for the Vlasov-Poisson-Boltzmann (VPB) system with a given time-periodic external force in the whole space $\mathbb{R}^3$. The force is allowed to be non-potential. Around the global Maxwellian, we prove the global existence of small solutions in a hybrid function space that combines the low-frequency Besov framework for the forced Boltzmann equation with a corresponding control of the self-consistent electric field. The main novelty lies in the treatment of the nonlinear Vlasov force $-\nabla_xϕ\cdot \nabla_vf + \frac{1}{2}(v \cdot \nabla_xϕ)f$ at low frequencies. Rather than treating it as a generic source term, we exploit the Poisson equation and macroscopic balance laws to recover the structural cancellation required for the VPB semi-group estimates, which combined with high-frequency energy estimates and weighted microscopic propagation, yields a closed global well-posedness theory. We further prove the asymptotic stability of small solutions driven by the same force. When the external force is time-periodic, Serrin's method yields a unique time-periodic solution with the same period, together with its stability. As a direct consequence, our result also gives the existence and stability of stationary solutions when the external force is time-independent.