Boundary Problems of Kinetic Theory

稀薄粒子的运动学理论在理解气体动力学、等离子体物理和流体方程等问题上发挥了重要作用。在许多物理情况下(例如，托卡马克聚变反应堆中的等离子体)，粒子与边界相互作用，并且它们的相互作用显著影响全球动力学。边界附近粒子分布的奇异行为是数学理解和实验中的主要困难。这个项目旨在开发新的数学工具来处理这些问题，并回答几个基本问题。本研究项目研究了动力学模型和流体方程中出现的几个重要的边界问题。特别是，PI计划研究具有边界条件的Vlasov-Maxwell-Boltzmann系统的正则性。该方程是非线性、非局部、退化的耗散耦合双曲型方程(波动方程)。由于边界条件的特殊性，解一般都是奇异的。对该提议的成功研究将标志着奇点偏微分方程组理论的重大发展。该项目的成果有望提高对其他科学家和工程师感兴趣的广泛问题的建模能力。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The kinetic theory on rarefied particles have played an important role in the understanding of problems in gas dynamics, plasma physics, and fluid equations. In many physical situations (for example, plasma in tokamak fusion reactor), the particles interact with boundaries and their interaction affects global dynamics significantly. Singular behavior of particle distribution near the boundary is major difficulty in both mathematical understanding and the experiment. This project aims to develop new mathematical tools to handle these problems and answer several fundamental questions. This research project studies several important boundary problems arising in kinetic models and fluid equations. In particular, the PI plans to study regularity property of the Vlasov-Maxwell-Boltzmann system with boundary conditions. The equation is nonlinear, nonlocal, and degenerated dissipative coupled with hyperbolic equation (wave equation). Due to the characteristics boundary condition, the solutions are singular in general. Successful investigations for the proposal would mark a significant development in the PDE theory with singularity. Results of the project are expected to improve modeling capabilities for a wide range of problems of interest to other scientists and engineers.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.