Kinetic Equations in Bounded Domains

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、中子输运和辐射传递。在经典力学中，这样的系统可以用不同的尺度来描述。在微观尺度上，牛顿定律追踪每个粒子的位置和速度。在宏观尺度上，流体力学和热力学为预测压力和温度等平均统计性质的行为提供了有效的工具。运动论在这两种方法之间架起了一座桥梁，并利用位置-速度空间（即所谓的相空间）中的概率工具来获得介观描述。粒子存在于相空间的概率密度满足演化偏微分方程玻尔兹曼方程或朗道方程。本项目主要研究粒子可能被物理边界反射或吸收的有界域中的动力学方程。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。该项目为研究生的研究训练提供了机会。这个项目集中于水动力极限理论，这是解决所谓的“希尔伯特第六问题”的关键一步，以一种公理化的方式对待物理。其目的是研究当克努森数（测量粒子在两次碰撞之间可以移动的相对距离）缩小到零时动力学方程的渐近行为。重点是在边界层效应存在的情况下证明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。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。

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.