Partial Differential Equation Methods in Kinetic Theory and Their Applications

该项目考虑边界效应的几个物理模型广泛用于描述热等离子体。所研究的问题的动机，并有影响，广泛的应用。就像在核聚变装置的情况下，控制和理解等离子体壁的相互作用是很重要的，或者在爱因斯坦的广义相对论中，一个基本的悬而未决的问题，被称为宇宙审查猜想，是一个星星的引力坍缩是否不能（即，黑洞）或可以（即，裸奇点（naked singularity）。这项工作将构建爱因斯坦理论中可以观察到的精确引力坍缩的例子，并将研究接触线的动力学以及海洋中科里奥利力的影响。本计画为研究生提供训练机会。动力学理论为描述装置中的受限电浆提供重要的模型。由于在边界处存在掠集所造成的严重的数学困难，存在磁效应的动力学等离子体模型的适定性问题仍然是开放的。受到诸如接触线动力学和恒定旋转或恒定磁场对流体的影响等应用的启发，研究人员将进行几项研究。研究的问题是BGK波的渐近稳定性;数值证据的严格分析表明存在相对论性Larson-Penston自相似引力坍缩，导致形成稳定的裸奇点和违反宇宙审查假设;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的机制，以构建基于恒定旋转或磁场引起的分散效应的长时间（全球）平滑无粘流体流。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.