Well-posedness and Behavior of Solutions to Kinetic Equations

这个项目将开发新的分析和计算方法来解决等离子体动力学动力学理论中的各种数学问题。等离子体通常被称为物质的第四种状态(仅次于固体、液体和气体)，占宇宙中所有物质的99.99%。由于等离子体是带电气体，它们是优良的导电体，因此具有很大的实用价值。例如，许多航天机构已经开发了等离子发动机，最近还用来为NASA的一些航天器提供动力。此外，在核聚变中使用等离子体作为清洁能源目前引起了极大的科学兴趣。其他值得注意的等离子体例子包括太阳风、地球电离层、银河星云和彗星尾巴。对太阳风的全面了解也将特别有用，因为这一自然现象决定了“空间天气”的强度，而“空间天气”往往是造成地球轨道卫星代价高昂的损害的原因。一般说来，等离子体的运动由一系列由物理决定的复杂的偏微分方程式来描述。当前项目的目标之一是证明这些方程具有实际解，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)的敏感度，并通过计算对它们进行近似，以便人们能够准确地预测未来情况的结果。等离子体是一种完全电离的气体，其中电磁力通常足够强，足以主导碰撞效应。用Vlasov-Maxwell方程描述高温、低密度无碰撞等离子体的运动，这是一个非线性的双曲型偏微分方程组。在这种情况下，碰撞被忽略，而驱动麦克斯韦系统的电荷和电流密度以一种自洽的方式从Vlasov方程的解的速度平均中确定。这个项目将研究的一个主要问题是：在无碰撞的等离子体中是否存在冲击波？也就是说，随着时间的推移，奇点会从平稳规定的初值发展出来吗？在某些情况下，例如在低维的相对论公式中(例如，一个空间和两个速度变量)，已知存在光滑的整体解。类似地，我们将研究关联的Fokker-Planck系统，以进一步阐明碰撞对粒子分布的平滑影响。另一个需要研究的问题是系统中电荷密度和电磁场的长时间行为。更具体地说，方程中的色散效应是否会导致这些量随着时间的推移而衰减，或者是否存在足够的相互作用，以便在时间趋于无穷大的情况下保持它们的强度？最后，将使用全局灵敏度度量和单胞粒子模拟来计算场和密度相对于模型输入参数的灵敏度。

This project will develop new analytic and computational methods to solve a variety of mathematical problems in the kinetic theory of plasma dynamics. Plasmas are often referred to as the fourth state of matter (after solids, liquids, and gases) and account for 99.99% of all material in the universe. Since plasmas are charged gases, they serve as excellent conductors of electricity, and thus are of great practical interest. As an example, plasma engines have been developed by a number of space agencies and recently used to power some NASA spacecraft. Additionally, the use of plasmas within nuclear fusion as a source of clean energy is currently of immense scientific interest. Other notable examples of plasmas include the solar wind, the Earth's ionosphere, galactic nebulae, and comet tails. A complete understanding of the solar wind would also be particularly useful, as this natural phenomenon dictates the intensity of "space weather", which is often responsible for expensive damage to satellites orbiting the Earth. In general, the motion of a plasma is described by a number of complicated partial differential equations dictated by physics. Among the goals of the current project are to demonstrate that these equations possess realistic solutions, determine their qualitative behavior, compute their sensitivity with respect to model parameters (such as masses, charges, and temperature), and computationally approximate them so that one can accurately predict outcomes in future situations.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 solutions to the Vlasov equation. One major question this project will study is: are there shocks in a collisionless plasma? That is, could a singularity develop from smoothly prescribed initial values as time progresses? In some cases, such as in lower dimensional, relativistic formulations (e.g., one space and two velocity variables), smooth global solutions are known to exist. Similarly, the associated Fokker-Planck system will be studied to further elucidate the smoothing influence of collisions on the particle distribution. Another problem to be investigated concerns the long-time behavior of the charge density and electromagnetic field in the system. More specifically, do dispersive effects in the equations cause these quantities to decay over time, or is there sufficient interaction so as to sustain their strength even as time tends to infinity? Finally, the sensitivity of fields and densities with respect to model input parameters will be computed using global sensitivity metrics and particle-in-cell simulations.