发表机构
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay（巴黎萨克雷大学、法国国家科学研究中心、奥赛数学实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在解析空间中证明了Vlasov-Poisson系统势-密度映射的可逆性，并恢复了Runge-Gross型定理，解决了由给定密度分布反求电场的反问题。
AI 中文摘要
我们发展了由Manfredi [J. Plasma Phys., 86.2 (2020), 825860201] 引入的含时密度泛函理论在具有含时外部势的Vlasov方程框架下的数学分析。在此背景下，当外部势在空间变量上为（实）解析时，我们建立了$\n\mathbb{T}^d \times \mathbb{R}^d$上Vlasov-Poisson系统势-密度映射的可逆性。我们还为光滑（实）解析于时间的外部势恢复了Vlasov-Poisson方程的Runge-Gross型定理。势-密度映射的满射性是Vlasov-Poisson方程的一个反问题：我们寻找在给定时间区间上产生给定密度分布的电场。
英文摘要
We develop the mathematical analysis of the Time-Dependent Density Functional Theory introduced by Manfredi [J. Plasma Phys., 86.2 (2020), 825860201] in the setting of Vlasov equations with a time-dependent exterior potential. In this context, we establish the invertibility of the potential-density mapping for the Vlasov-Poisson system on $\mathbb{T}^d \times \mathbb{R}^d$ when the exterior potential is (real-)analytic in the space variable. We also recover a Runge-Gross type Theorem for the Vlasov-Poisson equation for smooth (real-)analytic in time exterior potential. The surjectivity of the potential-density mapping is an inverse problem for the Vlasov-Poisson equation: we find the electric field that generates a given density profile on a given time interval.
Comments21 pages
