Mathematical Problems in Collisionless Kinetic Theory

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。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，可以控制碰撞效应。高温、低密度无碰撞等离子体的运动由Vlasov-Maxwell方程组描述，该方程组是一个非线性的双曲型偏微分方程组。在这种设置中，碰撞被忽略，而电荷和电流密度（驱动麦克斯韦系统）是确定在一个自洽的方式从速度时刻的解决方案的弗拉索夫方程。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，奇点会从光滑的初始值发展出来吗？在某些情况下，例如在较低维的相对论公式（例如，一个空间和两个速度变量），已知存在光滑的全局解。 另一个要研究的问题是系统中电荷和电流密度以及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时间衰减，或者是否存在足够的相互作用，以便即使在时间接近无穷大时也能维持它们的强度？动力学理论包括等离子体的运动和性质的研究。等离子体通常被称为物质的第四种状态（固体，液体和气体之后），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，因此是优良的电导体。例如，“等离子发动机”已经由一些航天机构开发，最近被用于为一些NASA航天器提供动力。此外，通过核聚变使用等离子体作为清洁能源目前具有巨大的科学兴趣。 无碰撞等离子体的典型例子包括太阳风、地球电离层、银河系星云、低密度聚变反应堆和彗星尾。等离子体的运动由物理学规定的许多复杂方程描述。数学家的目标之一是证明这些方程有解（在适当的条件下），确定它们的定性行为，并在数值上近似它们（以便人们可以确定地预测未来情况下的行为）。证明弗拉索夫-麦克斯韦系统有一个“好”的解也将证实方程组是描述等离子体相关现象的“正确”的方程组。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).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 approaches 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. For 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. The motion of a plasma is described by a number of complicated equations dictated by physics. Among the mathematician's goals are to show that these equations possess solutions (under appropriate conditions), determine their qualitative behavior, and approximate them numerically (so that one can predict behavior in future situations with certainty). A proof that the Vlasov-Maxwell system has a "nice" solution would also confirm that the system of equations is the "right" one to describe plasma-related phenomena.