Partial differential equation methods in kinetic theory and their applications

GuoDMS-1611695许多重要的科学应用需要了解粒子稀薄气体的动力学，例如航天飞机在空中的再入、经过飞机的流动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薄气体或等离子体的动力学。受到现实世界中各种科学应用问题的启发，研究人员旨在从数学的角度理解以下基本而重要的现象和问题：稀薄气体与其周围壁面的相互作用，稀薄等离子体与其周围壁面的相互作用，中子束与周围壁面的相互作用，无碰撞等离子体的长期行为的表征，流体中接触线的动力学(例如，咖啡杯侧面的咖啡边缘)，以及高雷诺数流动的边界层理论的有效性。研究生参与了这个项目。动力学理论是多尺度模拟的核心，它将微观颗粒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的地位，但由于其具有挑战性的特点，至今尚未得到深入的研究。作为他的主要目标之一，这位研究员和他的合作者继续研究这样的边值问题。更具体地说，作者进一步研究了外力作用下凸域中Boltzmann解的正则性，有界域中Landau方程的适定性，运动论中边界层理论的几何修正的一般理论，作为Boltzmann理论扩散极限的通过障碍物的流动，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的Ren-E模型的适定性。他还试图验证普朗特定常边界层展开对于高雷诺数流动的有效性。研究生参与了这个项目。

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.