Partial Differential Equation Methods in Kinetic Theory and Their Applications

等离子体是一种稀的、快速移动的带电粒子的集合。据信，宇宙中95%以上的物质是以等离子体（第四种状态）的形式存在的。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，其中限制等离子体以高速加速以产生高能量。该项目的重点是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与壁面相互作用、等离子体结构的动态稳定性以及等离子体中的冲击波。所有这些课题都是核聚变装置等离子体控制的基础理论，具有潜在的应用价值。更具体地说，该项目旨在建立规则性，以及凸域中Boltzmann理论的边界层展开与几何校正的有效性。关键的方法是基于主要研究者以前的工作规律类似的边界层膨胀的中子输运方程。另一个研究方向是有界域上的Vlasov-Landau方程的适定性。该方程组是模拟碰撞等离子体的基础，而边值问题模拟了等离子体与壁面的相互作用。其中包括：Vlasov-Poisson系统中BGK波的朗道阻尼和长时间稳定性，描述等离子体中基本双流体模型的非零涡量存在下Euler-Maxwell系统的长时间行为，等离子体中BGK波的稳定性，等离子体中BGK波的稳定该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。

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.