Nonlinear Partial Differential Equations in Kinetic Theory

运动理论中的非线性偏微分方程

• 批准号: 0204227
• 负责人: Robert Glassey
• 金额: $ 12.75万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204227&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Kinetic

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.

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。 高温、低密度无碰撞等离子体的运动由 Vlasov-Maxwell 方程（双曲偏微分方程的非线性系统）描述。 在此设置中，碰撞被忽略，而电荷和电流密度（驱动麦克斯韦系统）以自洽的方式从 Vlasov 方程解的速度矩确定。 要研究的主要问题是：三维无碰撞等离子体中是否存在激波？ 也就是说，随着时间的推移，奇点是否可以从平滑规定的初始值发展而来？众所周知，平滑全局解存在于低维情况（例如，两个空间和速度变量）以及数据“较小”时。动力学理论包括对等离子体的运动和特性的研究。 等离子体通常被称为物质的第四态（继固体、液体和气体之后）；它们几乎涵盖了宇宙中的所有物质。 等离子体是带电气体，因此是优良的电导体。 “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、银河星云和彗尾。 等离子体的运动由物理学规定的许多复杂方程来描述。数学家的目标之一是证明这些方程有解（在适当的条件下）并对其进行数值近似（以便可以预测未来情况下的某些行为）。 要表明弗拉索夫-麦克斯韦系统有一个“好的”解决方案，至少可以部分确认该系统是描述这些现象的“正确”系统。