Collisions in Plasma: The Landau Equation and Related Models

等离子体中的碰撞：朗道方程和相关模型

• 批准号: 2206677
• 负责人: Maria Pia Gualdani
• 金额: $ 23万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206677&HistoricalAwards=false
• 关键词: Collisions; Plasma; Landau; Equation; Related

Phenomena in gas dynamics and plasma physics are driven by collisions and diffusion of particles. The mathematical modeling of these effects lies at the intersection of applied mathematics and statistical physics, and it involves the study of kinetic equations, a class of nonlinear partial differential equations that captures the behavior of large number of particles in terms of the particle density. This project concentrates on two specific kinetic equations: the first describes the time evolution of the particle distribution when particles interact through binary collisions that can occur at very large microscopical distances, the second arises in the statistical description of charged quantum particles. The intent is to validate the fidelity of these models to the physical systems by studying the qualitative properties of their solutions, to ensure that a temporary breakdown of the models does not occur. The project will provide training opportunities for graduate students and postdoctoral researchers. The project aims at expanding the mathematical understanding of the dynamics of collisions in dilute gases and plasma by analyzing two kinetic partial differential equations. The first is the Landau equation, which describes the time evolution of the particle distribution when particles interact through binary collisions of grazing type. The second is the Landau-Fermi-Dirac equation, which arises in the statistical description of charged quantum particles. Both equations present challenges due to nonlinear terms, nonlocal features, and degenerate coefficients. The first part of the project deals with the qualitative properties of solutions, such as global well-posedness and finite time blow-up, characterization of the long-time behavior, and connection with macroscopic fluid equations. The second part concerns the validity of the Landau-Fermi-Dirac approximation as a correction to the Boltzmann equation in the grazing regime. The techniques employed include a novel combination of classical kinetic theory and recent theories developed for nonlinear nonlocal integro-differential equations and degenerate nonlocal differential operators.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.

气体动力学和等离子体物理学中的现象是由碰撞和颗粒扩散驱动的。这些效果的数学模型在于应用数学和统计物理学的相交，它涉及动力学方程的研究，动力学方程是一类非线性偏微分方程，这些方程在粒子密度方面捕获了大量粒子的行为。该项目集中于两个特定的动力学方程：第一个描述粒子通过二进制碰撞相互作用时可能发生在非常大的显微镜距离时的时间演变，第二个出现在带电量子粒子的统计描述中。目的是通过研究其解决方案的定性特性来验证这些模型对物理系统的保真度，以确保不会发生暂时的模型分解。该项目将为研究生和博士后研究人员提供培训机会。该项目旨在通过分析两个动力学偏微分方程来扩展稀释气和等离子体中碰撞动力学的数学理解。第一个是Landau方程，它描述了粒子通过放牧类型的二进制碰撞相互作用时粒子分布的时间演变。第二个是Landau-Fermi-Dirac方程，它在带电量子颗粒的统计描述中产生。这两个方程都构成了由于非线性项，非本地特征和退化系数而引起的挑战。该项目的第一部分涉及解决方案的定性属性，例如全球适合度和有限的时间爆炸，长期行为的表征以及与宏观流体方程的联系。第二部分涉及Landau-Fermi-Dirac近似的有效性，以校正放牧方程中的Boltzmann方程。所采用的技术包括经典动力学理论和针对非线性非局部非局部间不同差异方程和退化非局部差异操作员开发的最新理论的新型组合。该奖项反映了NSF的法定任务，并被认为是通过该基金会的知识分子功能和广泛的影响来评估Criteria的智力优点和广泛的criteria，并被认为是值得的。