Nonlinear Partial Differential Equations in Kinetic Theory

运动理论中的非线性偏微分方程

基本信息

批准号：
9876820
负责人：
Robert Glassey
金额：
$ 11.55万
依托单位：
Indiana University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
1999
资助国家：
美国
起止时间：
1999-07-01 至 2002-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9876820&HistoricalAwards=false
关键词：
Nonlinear Partial Differential Equations Kinetic

项目摘要

A "collisionless" plasma is a fully ionized gas in whichelectromagnetic forces dominate collisional effects. The motion of ahigh temperature, low density collisionless plasma is described bythe Vlasov-Maxwell equations, a nonlinear system of partialdifferential equations. The major question to be addressed is this:are there shocks in a collisionless plasma? That is, could asingularity develop from smoothly prescribed initial values as timeprogresses? Additionally, "induction heating" will be studied. Inthis area one investigates how a conductive material is heated byusing electromagnetic waves. Microwave heating is a particularexample."Plasmas" are often called the fourth state of matter (after solids,liquids and gases). Plasmas account for a huge proportion of thematerial in the universe. Famous examples of plasmas include thesolar wind, the ionosphere, galactic nebulae and comet tails. Themotion of a plasma is described by a number of complicated equationsdictated by physics. The mathematician's goal is to show that theseequations have solutions, at least under certain conditions. This isthe main thrust of the proposal. A completely different applicationto be studied is called "induction heating." Here one tries todetermine how various materials can be heated by passing electromagnetic waves through them. The microwave heating of food is a particular example.

“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（一个非线性偏微分方程组）描述。要解决的主要问题是：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？此外，还将研究“感应加热”。在这一领域，人们研究如何使用电磁波加热导电材料。微波加热是一个特殊的例子。“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）。等离子体在宇宙物质中占很大比例。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。数学家的目标是证明方程组有解，至少在某些条件下是这样。这是该提案的主旨。有待研究的完全不同的应用称为“感应加热”。这里我们试图确定如何通过电磁波穿过各种材料来加热它们。食物的微波加热就是一个特例。