Partial differential equation methods in kinetic theory and their applications

许多重要的科学应用需要了解稀粒子气体的动力学，例如，航天飞机的再入大气层，经过飞机的气流，或用于核聚变的托卡马克装置内的等离子体。动力学理论描述了稀释气体或等离子体的动力学。受现实世界中各种科学应用问题的激励，研究者旨在从数学的角度理解以下基本和重要的现象和问题：稀释气体与其周围壁的相互作用，稀释等离子体与其周围壁的相互作用，中子束与周围壁的相互作用，无碰撞等离子体长期行为的表征，流体中接触线的动力学（例如，咖啡杯侧面的咖啡边），以及边界层理论在高雷诺数流动中的有效性。研究生也参与了这个项目。动力学理论是多尺度建模的核心，它将微观粒子模型与宏观流体模型联系起来。尽管边界效应在动力学理论中占有重要地位，但由于其具有挑战性的特点，尚未得到深入的研究。作为他的主要目标之一，研究者和他的合作者继续研究这种边值问题。更具体地说，研究者进一步研究了外力存在下凸域Boltzmann解的规律性，有界域Landau方程的适定性，动力学中边界层理论的几何校正的一般理论，Boltzmann理论中通过障碍物的流动的扩散极限，等离子体无碰撞Vlasov理论中稳定BGK波周围的时间衰减，以及最近的接触线动力学Ren-E模型的适定性。他还试图验证稳定普朗特边界层展开对高雷诺数流动的有效性。研究生也参与了这个项目。

GuoDMS-1611695 Many important scientific applications require understanding dynamics of dilute gases of particles, e.g., re-entry of a space shuttle in the air, flows passing an airplane, or a plasma inside a tokamak device for nuclear fusion. Kinetic theory describes the dynamics of a dilute gas or plasma. Motivated by problems with various scientific applications in the real world, the investigator aims to understand the following basic and important phenomena and questions from a mathematical standpoint: the interaction of a dilute gas with its surrounding walls, interaction of a dilute plasma with its surrounding walls, interactions of neutron beams with surrounding walls, characterization of the long-time behavior of a collisionless plasma, dynamics of contact lines in fluids (e.g., the coffee edge on the side of a coffee cup), and the validity of boundary layer theory for flows with high Reynolds number. Graduate students are involved in the project. Kinetic theory is at the center of multi-scale modeling, which connects the microscopic particle models with macroscopic fluid models. Even though boundary effects play an important role in kinetic theory, they have not yet been studied deeply due to their challenging characteristic nature. As one of his main goals, the investigator and his collaborators continue to study such boundary value problems. More specifically, the investigator further studies regularity of Boltzmann solutions in convex domains in the presence of external forces, well-posedness for the Landau equation in a bounded domain, a general theory for geometric correction to the boundary layer theory in kinetic theory, flows passing an obstacle as a diffusive limit of Boltzmann theory, time decay around stable BGK waves in a collisionless Vlasov theory for a plasma, and the well-posedness of the recent Ren-E model for contact line dynamics. He also seeks to verify the validity of steady Prandtl boundary layer expansions for flows with high Reynolds number. Graduate students are involved in the project.