Mathematical Sciences: Analysis of Collisionless Plasmas

本研究的目的将是分析模型的 无碰撞等离子体在影响下运动的时间演化 它们自身感应的电磁场。 这些模型需要 偏微分方程组的形式，泊松- Vlasov系统和Maxwell-Vlasov系统。 每个人心中的未知 模型是等离子体的密度，麦克斯韦-弗拉索夫系统 是两个系统中比较完整的一个， 磁效应和相对论效应。 全局对称解最早是在1952年建立的， 随后又有许多扩展。 然而，全球问题 一般数据的经典解的存在性仍然是开放的 对于任何一个系统。 最近的研究结果表明， 在有限时间内的解决方案。 这项工作的一个目标是 为了表明当初始数据接近球形时 对称的，整体存在的解决方案是有保证的。 第二 调查路线建议建立模型， 太阳风和地球的磁场。 被这一 理解带电粒子的中性流快速移动 远离太阳 它主要由质子和电子组成 当它们遇到地球磁场时， 加到总磁场中的电流。 所得 相互作用被认为是北极光的原因， 地磁暴 这种相互作用的另一个特征是弓 等离子体的行为发生变化的冲击波 突然 无碰撞中激波的精确性质 等离子体仍不清楚。 混合初值-边值 提出了一个问题来模拟上述相互作用。 初步工作 要做的将是建立全球存在的结果， 解决方案，并确定条件下，解决方案往往 随着时间的增加而稳定。 在物理科学中的应用是明确的。

The purpose of this research will be to analyze models of the time evolution of collisionless plasmas moving under the influence of their self-induced electromagnetic fields. These models take the form of systems of partial differential equations, the Poisson- Vlasov system and the Maxwell-Vlasov system. The unknown in each model is the density of the plasma, and the Maxwell-Vlasov system is somewhat more complete of the two systems, exhibiting electrical and magnetic effects as well as relativistic. Globally symmetric solutions were first established in 1952, and numerous extensions followed. However, the problem of global existence of classical solutions for general data is still open for either system. Recent results establish the breakdown of certain solutions in finite time. One goal of this work will be to show that when the initial data is nearly spherically symmetric, global existence of solutions is guaranteed. A second line of investigation proposes to model the interaction between the solar wind and the earth's magnetic field. By this one understands the neutral stream of charged particles moving rapidly away from the sun. It consists mainly of protons and electrons which, when they encounter the earth's magnetic field, induce currents adding to the total magnetic field. The resulting interaction is believed to account for the aurora borealis and geomagnetic storms. Another feature of this interaction is a bow shock wave across which the behavior of the plasma changes abruptly. The precise nature of shock waves in collisionless plasmas is still unclear. A mixed initial value-boundary value problem is proposed to model the above interaction. Initial work to be done will be in establishing global existence results for solutions and to determine conditions under which solutions tend to a steady state as time increases. Applications to the physical sciences are clear.