AI 中文总结
本文针对全空间和环面上的三维Vlasov-Poisson系统建立临界时空密度估计，解决能量守恒问题并闭合2<k≤3的速度矩传播区间。
AI 中文摘要
本文针对全空间和环面上的三维Vlasov-Poisson系统，建立了一个临界时空密度估计。该估计有两个结果：一是有界有限能量弱解类中所有具有有限初始动能的非负弱解均守恒总能量，在该类中对Lions和Perthame提出的能量守恒问题给出了肯定回答；二是在周期排斥情形下，它导出了所有阶k>2的速度矩的传播性，从而闭合了此前未解决的2<k≤3区间。
英文摘要
This paper establishes a critical space--time density estimate for the three-dimensional Vlasov--Poisson system in the whole space and on the torus. This estimate has two consequences. First, every nonnegative weak solution with finite initial kinetic energy in the bounded finite-energy weak-solution class conserves the total energy. This gives a positive answer, within this class, to the energy-conservation question raised by Lions and Perthame. Second, in the periodic repulsive case, it yields propagation of every velocity moment of order \(k>2\), thereby closing the previously unresolved interval \(2<k\leq3\).
