Partial differential equation methods in kinetic theory and their applications

动力学理论中的偏微分方程方法及其应用

• 批准号: 1611695
• 负责人: Yan Guo
• 金额: $ 26.11万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611695&HistoricalAwards=false
• 关键词: Partial; differential; equation; methods; kinetic

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.

GuoDMS-1611695 许多重要的科学应用需要了解粒子的稀释气体的动力学，例如，航天飞机重返大气层，气流经过飞机，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气体或等离子体的动力学。 受真实的世界中各种科学应用问题的启发，研究者的目标是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，无碰撞等离子体的长时间行为的表征，流体中接触线的动力学（例如，咖啡杯侧面的咖啡边缘），以及边界层理论对高雷诺数流动的有效性。 研究生参与了该项目。 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起着重要的作用，但由于其具有挑战性的特性，尚未对其进行深入研究。 作为他的主要目标之一，研究者和他的合作者继续研究这样的边值问题。 更具体地说，研究者进一步研究了存在外力时凸域中玻尔兹曼解的规律性，有界域中朗道方程的适定性，动力学理论中边界层理论几何校正的一般理论，作为玻尔兹曼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，和最近Ren-E模型的接触线动力学的适定性。 他还试图验证高雷诺数流动的稳定普朗特边界层展开的有效性。 研究生参与了该项目。