Kinetic Equations in Bounded Domains

有界域中的动力学方程

• 批准号: 2104775
• 负责人: Lei Wu
• 金额: $ 22.6万
• 依托单位: Lehigh University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104775&HistoricalAwards=false
• 关键词: Kinetic; Equations; Bounded; Domains

Kinetic theory concerns the dynamics of a large number of particles, such as flows of air or water, plasmas, neutron transport, and radiation transfer. In classical mechanics, such systems can be described under different scales. In the microscopic scale, Newton's laws track the position and velocity of each particle. In the macroscopic scale, fluid mechanics and thermodynamics provide effective tools to predict the behaviors of averaged statistical properties like pressure and temperature. Kinetic theory forms a bridge between these two approaches and utilizes probabilistic tools in the position-velocity space, the so-called phase space, to obtain a mesoscopic description. The probability density of particles present in the phase space satisfies the Boltzmann equation or the Landau equation, which are evolutionary partial differential equations. This project focuses on the kinetic equations in bounded domains, where the particles may be reflected or absorbed by the physical boundary. The purpose is to develop novel mathematical tools to characterize the multi-scale behaviors of these particle systems in applications such as medical imaging, fluid mechanics, and nuclear fusion. The project provides opportunities for research training of graduate students.This project concentrates on the theory of hydrodynamic limits, a key step to tackle the so-called "Hilbert's sixth Problem" to treat physics in an axiomatic manner. The aim is to study the asymptotic behavior of kinetic equations when the Knudsen number, which measures the relative distance a particle can travel between two collisions, shrinks to zero. The focus is in justifying the validity of asymptotic approximations of kinetic equations in the presence of boundary layer effects, where the geometric correction of the boundary is non-negligible. The investigator will develop new boundary layer decomposition, regularization, and reflection extension techniques.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.

动力学理论涉及大量颗粒的动力学，例如空气或水流，等离子体，中子传输和辐射转移。在经典的力学中，可以在不同的尺度上描述这样的系统。在微观量表中，牛顿的定律跟踪每个粒子的位置和速度。在宏观量表中，流体力学和热力学提供了有效的工具来预测压力和温度等平均统计特性的行为。动力学理论在这两种方法之间形成了桥梁，并利用位置空间（所谓的相位空间）中的概率工具，以获得介绍性描述。相位空间中存在的粒子的概率密度满足玻尔兹曼方程或兰道方程，这些方程是进化的偏微分方程。该项目的重点是有界域中的动力学方程，其中粒子可以被物理边界反映或吸收。目的是开发新型的数学工具来表征这些粒子系统在医学成像，流体力学和核融合等应用中的多尺度行为。该项目为研究生的研究培训提供了机会。该项目集中在流体动力学限制理论上，这是解决所谓的“希尔伯特的第六个问题”的关键步骤，以方便治疗物理学。目的是研究动力学方程的渐近行为，当knudsen数量测量粒子可以在两次碰撞之间传播的相对距离，缩小到零。重点是在存在边界层效应的情况下证明动力学方程渐近近似的有效性，其中边界的几何校正是不可忽略的。研究人员将开发新的边界层分解，正则化和反射扩展技术。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来支持的。

项目成果

Hydrodynamic Limit of 3dimensional Evolutionary Boltzmann Equation in Convex Domains

凸域中三维演化玻尔兹曼方程的流体力学极限

• DOI: 10.1137/20m1375735
• 发表时间: 2022
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Wu, Lei;Ouyang, Zhimeng
• 通讯作者: Ouyang, Zhimeng

On the quantum Boltzmann equation near Maxwellian and vacuum

• DOI: 10.1016/j.jde.2022.01.056
• 发表时间: 2021-02
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Zhimeng Ouyang;Lei Wu