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粒子的动力学，如空气或水的流动，等离子体，中子输运和辐射转移。在经典力学中，这样的系统可以在不同的尺度下描述。在微观尺度上，牛顿定律跟踪每个粒子的位置和速度。在宏观尺度上，流体力学和热力学提供了有效的工具来预测平均统计性质的行为，如压力和温度。动力学理论形成了这两种方法之间的桥梁，并利用概率工具的位置-速度空间，所谓的相空间，以获得介观的描述。存在于相空间中的粒子的概率密度满足作为演化偏微分方程的玻尔兹曼方程或朗道方程。这个项目的重点是在有界区域的动力学方程，其中粒子可能被反射或吸收的物理边界。其目的是开发新的数学工具来表征这些粒子系统在医学成像，流体力学和核聚变等应用中的多尺度行为。该项目为研究生提供了研究培训的机会。该项目集中于流体动力学极限理论，这是解决所谓的“希尔伯特第六问题”的关键一步，以公理化的方式处理物理学。其目的是研究当克努森数（Knudsen number）缩小到零时动力学方程的渐近行为，克努森数衡量粒子在两次碰撞之间可以移动的相对距离。重点是证明存在边界层效应时动力学方程的渐近近似的有效性，其中边界的几何校正是不可忽略的。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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