Existence, Regularity, and Behavior of Solutions to Kinetic Equations

无碰撞等离子体是一种完全电离的气体，其中电磁力足够强大，可以控制碰撞效应。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，当驱动麦克斯韦系统的电荷和电流密度以自一致的方式由弗拉索夫方程解的速度矩确定时，碰撞被忽略。需要研究的主要问题是：在无碰撞的等离子体中是否存在激波？也就是说，随着时间的推移，奇点能否从平滑规定的初始值发展而来？在某些情况下，例如在较低维的相对论公式中（例如，一个空间和两个速度变量），已知存在光滑的全局解。另一个需要研究的问题涉及系统中电荷、电流密度和电磁场的长期行为。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时维持它们的强度？运动论包括对等离子体运动和性质的研究。等离子体通常被认为是物质的第四种状态（仅次于固体、液体和气体），占宇宙中所有物质的99.99%。它们具有很大的实际意义，因为它们是带电气体，因此是优良的导电体。例如，等离子发动机已经被许多太空机构开发出来，最近还被用于为美国宇航局的一些航天器提供动力。此外，通过核聚变利用等离子体作为一种清洁能源目前引起了巨大的科学兴趣。无碰撞等离子体的著名例子包括太阳风、地球电离层、银河星云、低密度聚变反应堆和彗星尾巴。对太阳风的全面了解也将非常有用，因为这种自然现象决定了“太空天气”的强度，而这通常是对地球轨道卫星造成昂贵损害的原因。等离子体的运动由物理学规定的许多复杂的微分方程来描述。当前项目的目标之一是表明这些方程具有解（在适当的条件下），确定它们的定性行为，并通过计算近似它们，以便人们可以确定地预测未来情况下的行为。

A collisionless plasma is a fully ionized gas in which electromagnetic forces are 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 moments of solutions to the Vlasov equation. The major question to be studied is this: 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. Another problem to be investigated concerns the long-time behavior of the charge and current densities and electromagnetic fields 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?Kinetic Theory includes the study of the motion and properties of plasma. 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. They are of great practical interest because they are charged gases, and thus serve as excellent conductors of electricity. 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 through nuclear fusion as a source of clean energy is currently of immense scientific interest. Notable examples of collisionless plasmas include the solar wind, the Earth's ionosphere, galactic nebulae, low-density fusion reactors, and comet tails. A complete understanding of the solar wind would also be extremely useful, as this natural phenomenon dictates the intensity of "space weather", which is often responsible for expensive damage to satellites orbiting the Earth. The motion of a plasma is described by a number of complicated differential equations dictated by physics. Among the goals of the current project are to show that these equations possess solutions (under appropriate conditions), determine their qualitative behavior, and approximate them computationally so that one can predict behavior in future situations with certainty.