Survival Threshold for Collective Plasma Oscillations

该项目的主要目标是研究由足够多的相互作用粒子（如等离子体物理学中的等离子体和量子力学中的凝聚体）组成的物质的最终状态问题。该研究将促进对等离子体物理学和量子力学中湍流的理解，为解决物理学中未解决的问题提供基础数学，并推动当前数学技术的边界。该研究将为偏微分方程理论、数学物理、动力系统和应用数学提供新的技术。该项目包括旨在培训研究生和年轻研究人员的活动。该项目将证明长期以来关于等离子体物理学和量子力学中使用的数学平均场模型解的大时间行为的猜想。正在研究的主要数学模型包括相对论的弗拉索夫-麦克斯韦系统和用于模拟无限多个相互作用粒子的非线性集体效应的哈特里方程。该研究将严格验证包括等离子体振荡、相变、相位混合、朗道阻尼和相干结构形成在内的非线性物理现象。建立了Vlasov方程和Hartree方程在非平凡平移不变平衡附近的散射理论和周期结构的形成。该项目的工作涉及光谱理论、解析分析、傅立叶分析、色散偏微分方程、概率和统计物理等数学技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main objective of this project is to investigate questions about the final states of matter consisting of a sufficiently large number of interacting particles such as plasmas in plasma physics and condensates in quantum mechanics. The research will advance the understanding of turbulence in plasma physics and quantum mechanics, provide foundational mathematics to tackle unsolved problems in physics, and push the boundaries of current mathematical techniques. The research will contribute new techniques to the theory of partial differential equations, mathematical physics, dynamical systems, and applied mathematics. The project includes activities aimed at training graduate students and young researchers.The project will prove longstanding conjectures concerning the large time behavior of solutions to the mathematical mean field models that are used in plasma physics and quantum mechanics. The primary mathematical models under investigation include the relativistic Vlasov-Maxwell system and the Hartree equations used to model the nonlinear collective effects of infinitely many interacting particles. The research will rigorously validate nonlinear physical phenomena including plasma oscillations, phase transition, phase mixing, Landau damping, and the formation of coherent structures. The scattering theory as well as the formation of periodic structures for the Vlasov and Hartree equations near nontrivial translation-invariant equilibria will be established. The work of the project involves mathematical techniques from spectral theory, resolvent analysis, Fourier analysis, dispersive PDEs, probability, and statistical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.