Partial Differential Equation Methods in Kinetic Theory and Their Applications

运动理论中的偏微分方程方法及其应用

• 批准号: 2106650
• 负责人: Yan Guo
• 金额: $ 36.77万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2106650&HistoricalAwards=false
• 关键词: Partial; Differential; Equation; Methods; Kinetic

The project considers boundary effects for several physical models widely used to describe a hot plasma. The questions studied are motivated by, and have implications for, a broad range of applications. As in the case of a nuclear fusion device, where it is important to control and understand the plasma-wall interaction, or in Einstein's theory for general relativity, where a fundamental open question, known as cosmic censorship conjecture, is whether a gravitational collapse of a star cannot (i.e., black holes) or can (i.e., naked singularity) be observed generically. This work will construct examples of exact gravitational collapse in Einstein's theory that can be observed, and will investigate the dynamics of contact lines as well as the effect of a Coriolis force in oceans. This project provides training opportunities for graduate students.Kinetic theory provides important models for describing a confined plasma in a device. Because of the severe mathematical difficulties caused by the presence of a grazing set at the boundary, questions of well-posedness for kinetic plasma models in the presence of magnetic effect remain open. Motivated by applications such as contact-line dynamics and the effects of a constant rotation or a constant magnetic field on a fluid, the investigator will pursue several lines of research. The problems investigated are the asymptotical stability of BGK waves; the rigorous analysis of numerical evidence that indicates the existence of a relativistic Larson-Penston self-similar gravitational collapse, leading to formation of a stable naked singularity and the violation of the cosmic censorship hypothesis; and the well-posedness of a hydrodynamic model describing contact line dynamics. The project will also develop a new mechanism to construct long time (global) smooth inviscid fluid flows based on the dispersive effect induced by a constant rotation or a magnetic field.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.

该项目考虑了广泛用于描述热等离子体的几种物理模型的边界效应。研究的问题受到广泛应用的启发，并对其产生影响。就像在核聚变装置的情况下，控制和理解等离子体壁相互作用很重要，或者在爱因斯坦的广义相对论中，一个基本的开放问题，被称为宇宙审查猜想，是恒星的引力塌缩是否不能（即黑洞）或可以（即裸奇点）被普遍观察到。这项工作将构建爱因斯坦理论中可观测到的精确引力塌缩的例子，并将研究接触线的动力学以及科里奥利力在海洋中的影响。该项目为研究生提供了培训机会。动力学理论为描述设备中的受限等离子体提供了重要模型。由于边界处存在放牧区会造成严重的数学困难，因此在存在磁效应的情况下动态等离子体模型的适定性问题仍然悬而未决。在接触线动力学以及恒定旋转或恒定磁场对流体的影响等应用的推动下，研究人员将进行多项研究。研究的问题是BGK波的渐近稳定性；对数值证据的严格分析表明存在相对论的拉尔森-彭斯顿自相似引力塌缩，导致稳定的裸奇点的形成并违反了宇宙审查假说；以及描述接触线动力学的流体动力学模型的适定性。该项目还将开发一种新机制，基于恒定旋转或磁场引起的色散效应来构造长期（全局）平滑的无粘性流体流动。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Global axisymmetric Euler flows with rotation

• DOI: 10.1007/s00222-022-01145-6
• 发表时间: 2021-09
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Yan Guo;B. Pausader;Klaus Widmayer

Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

• DOI: 10.1007/s00205-022-01827-8
• 发表时间: 2022-11-16
• 期刊: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
• 影响因子: 2.5
• 作者: Guo, Yan;Hadzic, Mahir;Schrecker, Matthew
• 通讯作者: Schrecker, Matthew