Analytical and Numerical Methods in Collisionless Kinetic Theory

该项目开发了分析和计算方法来回答等离子体动力学动力学理论中的数学问题。等离子体是带电气体，通常被称为物质的第四种状态，占宇宙中所有物质的99%以上，由于它们是优良的导电体，因此具有重要的实际意义。世界各地的航天机构都在开发等离子发动机，最近已被用于为选定的航天器提供动力。人们还在探索在核聚变反应堆中使用等离子体作为一种清洁能源。其他在自然现象中出现等离子体的著名例子有太阳风、地球电离层、银河星云和彗星尾巴。对太阳风动力学的全面了解具有重要的实际意义，因为它决定了空间天气的强度，而空间天气通常会对绕地球运行的卫星造成昂贵的损害。等离子体的运动可以用一个偏微分方程系统来建模，这个项目的目标之一是证明这些方程具有现实的解，确定它们的定性行为，计算它们对模型参数（如质量、电荷和温度）的敏感性，并通过计算近似它们来精确预测未来的行为。该项目提供研究生研究培训机会。等离子体是一种完全电离的气体，其中的电磁力通常很强，足以控制碰撞效应。用非线性双曲偏微分方程组弗拉索夫-麦克斯韦方程组描述了高温低密度无碰撞等离子体的运动。在这种情况下，碰撞被忽略，驱动麦克斯韦系统的电荷和电流密度以自一致的方式由系统中离子分布的速度平均值确定，满足弗拉索夫方程。该项目解决的一个主要问题是，在无碰撞等离子体中是否存在冲击，也就是说，奇点是否可以随着时间的推移从平滑的初始数据发展而来。在某些情况下，例如在低维相对论公式中，已知存在光滑的全局解。其他问题涉及粒子位置和动量、电荷和电流密度、电磁场和系统内粒子分布函数的大时间限制行为。更具体地说，该项目确定方程中的色散效应是否会导致这些量随时间衰减，或者是否有足够的相互作用来维持它们的强度，即使在时间渐近极限中。最后，利用全局灵敏度指标、相应的色散关系和粒子池模拟计算了场和密度相对于模型输入参数的灵敏度。该奖项反映了美国国家科学基金会的法定使命，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。

This project develops analytic and computational methods to answer mathematical questions in the kinetic theory of plasma dynamics. Plasmas are charged gases, often referred to as the fourth state of matter, that account for more than 99% of all material in the universe and are of significant practical interest since they serve as excellent conductors of electricity. Plasma engines have been developed by space agencies around the world and were recently employed to power selected spacecraft. The use of plasmas within nuclear fusion reactors is also being explored as a source of clean energy. Other notable examples of plasmas occurring in natural phenomena are the solar wind, the Earth's ionosphere, galactic nebulae, and the tails of comets. A full understanding of the dynamics of the solar wind is of practical importance, as it dictates the intensity of space weather, which is often responsible for expensive damage to satellites orbiting the Earth. The motion of a plasma can be modeled by a system of partial differential equations, and among the goals of this project are to demonstrate that these equations possess realistic solutions, to determine their qualitative behavior, to compute their sensitivity with respect to model parameters, such as masses, charges, and temperature, and to approximate them computationally to predict future behavior with precision. The project provides graduate research training opportunities. A plasma is a fully ionized gas in which electromagnetic forces are often strong enough to dominate collisional effects. The motion of a high temperature, low density collisionless plasma is described by the Vlasov-Maxwell equations, a nonlinear system of hyperbolic partial differential equations. In this setting, collisions are neglected while the charge and current densities, which drive the Maxwell system, are determined in a self-consistent manner from velocity averages of the distribution of ions in the system, which satisfies the Vlasov equation. A major question this project addresses is if there are shocks in a collisionless plasma, that is, if a singularity can develop from smooth initial data as time progresses. In some cases, such as in lower dimensional relativistic formulations, smooth global solutions are known to exist. Additional questions concern the large time limiting behavior of the particle positions and momenta, charge and current densities, electromagnetic fields, and particle distribution functions within the system. More specifically, the project determines whether dispersive effects in the equations cause these quantities to decay over time, or if there is sufficient interaction to sustain their strength even in the time asymptotic limit. Finally, the sensitivity of the fields and densities with respect to model input parameters are computed using global sensitivity metrics, corresponding dispersion relations, and particle-in-cell simulations.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.