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。 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偏微分方程组。 要解决的主要问题是：在无碰撞的等离子体中是否存在激波？ 也就是说，奇异性是否能随着时间的推移从平稳的初始值发展而来？ 此外，还将研究“感应加热”。 在这一领域，人们研究如何利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种状态（在固体，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、银河系星云和彗星尾巴。 等离子体的运动由物理学规定的许多复杂方程描述。 数学家的目标是证明这些方程有解，至少在某些条件下。 这是提案的主旨。 一个完全不同的应用被称为“感应加热”。“在这里，人们试图确定如何通过电磁波穿过各种材料来加热它们。 微波加热食品就是一个很好的例子。

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.