Nonlinear Partial Differential Equations in Kinetic Theory

无碰撞等离子体是一种完全电离的气体，其中电磁力主导碰撞效应。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，而电荷和电流密度(驱动麦克斯韦系统)以一种自洽的方式从Vlasov方程的解的速度矩确定。要研究的主要问题是：三维无碰撞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？众所周知，光滑的整体解存在于低维情况下(例如，两个空间和速度变量)，并且当数据“小”时。动力学理论包括对等离子体运动和性质的研究。等离子体通常被称为物质的第四种状态(仅次于固体、液体和气体)；它们几乎占据了宇宙中所有的物质。等离子体是带电气体，因此是优良的导电体。“等离子发动机”最近被用来为美国宇航局的一些航天器提供动力。无碰撞等离子体的著名例子包括太阳风、电离层、银河星云和彗星尾巴。等离子体的运动由许多由物理决定的复杂方程来描述。数学家的目标之一是证明这些方程(在适当的条件下)是有解的，并用数字逼近它们(这样人们就可以预测未来情况下的某些行为)。为了证明Vlasov-Maxwell系统有一个“好”的解，至少可以部分地证实这个系统是描述这些现象的“正确”系统。

A collisionless plasma is a fully ionized gas in which electromagnetic forces 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 ignored 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 three dimensional collisionless plasma? That is, could a singularity develop from smoothly prescribed initial values as time progresses? Smooth global solutions are known to exist in lower dimensional situations (e.g., two space and velocity variables) and when the data are "small."Kinetic Theory includes the study of the motion and properties of plasmas. Plasmas are often called the fourth state of matter (after solids, liquids and gases); they account for practically all of the material in the universe. Plasmas are charged gases and hence are excellent conductors of electricity. "Plasma engines" have been used recently to power some NASA spacecraft. Notable examples of collisionless plasmas include the solar wind, the ionosphere, galactic nebulae 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 have solutions (under appropriate conditions) and to approximate them numerically (so that one can predict certain behavior in future situations). To show that the Vlasov-Maxwell system has a "nice" solution would at least partially confirm this system as the "right" one to describe these phenomena.