$\u211d^3$中带一般外力的Vlasov-Poisson-Boltzmann系统的时间周期解
Time-periodic solutions of the Vlasov-Poisson-Boltzmann system with a general external force in $\mathbb{R}^3$
AI总结：
本文研究$\u211d^3$中带一般非势时间周期外力的VPB系统，通过处理低频非线性Vlasov力的结构抵消，结合能量估计证明小解全局适定性、渐近稳定性及时间周期解的存在唯一性，也涵盖稳态解情形。
AI中文摘要：
本文研究全空间$\u211d^3$中带有给定时间周期外力的Vlasov-Poisson-Boltzmann（VPB）系统的时间周期问题，该外力允许为非势外力。在全局Maxwellian附近，我们在混合函数空间中证明了小解的全局存在性，该空间将受迫Boltzmann方程的低频Besov框架与自洽电场的相应控制相结合。\n主要创新点在于对低频处非线性Vlasov力$-\u2207_x\u03d5\cdot \u2207_vf + \frac{1}{2}(v \cdot \u2207_x\u03d5)f$的处理。我们并未将其视为一般源项，而是利用Poisson方程和宏观平衡律恢复VPB半群估计所需的结构抵消，结合高频能量估计和带权微观传播，得到了封闭的全局适定性理论。我们进一步证明了由相同外力驱动的小解的渐近稳定性。当外力为时间周期函数时，Serrin方法给出了具有相同周期的唯一时间周期解及其稳定性。作为直接推论，当外力不随时间变化时，我们的结果也给出了稳态解的存在性与稳定性。
英文摘要：
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.
