Partial Differential Equation Methods in Kinetic Theory and Their Applications

等离子体是稀释的、快速移动的带电粒子的集合。据信，宇宙中95%以上的物质以等离子体（第四种状态）的形式存在。这项研究的主要动机是促进核聚变装置（如托卡马克）的设计，在这个装置中，一个受限的等离子体被高速加速以产生高能量。本项目主要进行与托卡马克等离子体研究相关的基础数学研究。研究课题包括等离子体-壁相互作用、等离子体结构的动态稳定性以及等离子体中的激波。所有这些问题都是具有潜在应用前景的核聚变装置等离子体控制的理论基础。更具体地说，该项目旨在建立正则性，以及在凸域上玻尔兹曼理论的边界层扩展的几何修正的有效性。关键的方法是基于首席研究员以前对中子输运方程类似边界层展开的规律性的研究。另一个要追求的方向是在有界区域内求解Vlasov-Landau方程的适定性。这一方程组是建立等离子体碰撞模型的基础，而边值问题是等离子体-壁相互作用的模型。研究的其他问题包括：Vlasov-Poisson系统中BGK波的朗道阻尼和长时间稳定性；欧拉-麦克斯韦系统在描述等离子体中基本双流体模型的非零涡度存在下的长期行为；流体中移动接触线的动力学。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。

A plasma is a collection of dilute, fast moving charged particles. It is believed that more than 95% of the matter in the universe is in the form of a plasma (the fourth state). The main motivation of this research is to promote design of nuclear fusion devices (such as a tokamak), in which a confined plasma is accelerated at high speed to produce high energy. This project focuses on basic mathematical research related to the study of a plasma in a tokamak. Research topics include study of plasma-wall interaction, dynamic stability of plasma configurations, as well as shock waves in plasma. All of such topics are fundamental theoretically in the plasma control for a nuclear fusion device with potential applications.More specifically, the project is aimed at establishing regularity, and the validity of boundary layer expansion with geometric corrections for the Boltzmann theory in a convex domain. The key methodology is based on the principal investigator's previous work on regularity of similar boundary layer expansions for the neutron transport equations. Another direction to be pursued is well-posedness for the Vlasov-Landau equations in a bounded domain. This system of equations is fundamental in modeling a collisional plasma, and the boundary value problem models the plasma-wall interaction. Among other problems to be investigated are: The Landau damping and long-time stability for the BGK waves in a Vlasov-Poisson system; the long-time behavior of Euler-Maxwell system in the presence of a non-zero vorticity that describes the fundamental two-fluid models in a plasma; the dynamics of moving contact lines in fluids.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.