Mathematical Sciences: Analysis of Collisionless Plasmas

这项研究的目的将是分析无碰撞等离子体在其自感电磁场影响下运动的时间演化模型。这些模型采用偏微分方程组、泊松-弗拉索夫系统和麦克斯韦-弗拉索夫系统的形式。在每个模型中，未知的是等离子体的密度，麦克斯韦-弗拉索夫系统在某种程度上是两个系统中更完整的，表现出电磁效应和相对论效应。全球对称解于1952年首次建立，随后进行了多次扩展。然而，一般数据经典解的全局存在性问题对于这两个系统中的任何一个仍然是悬而未决的。最近的结果建立了某些解在有限时间内的分解。这项工作的一个目标是证明当初始数据近球对称时，整体解的存在是有保证的。第二个研究方向是对太阳风和地球磁场之间的相互作用进行建模。通过这一点，人们理解了带电粒子流的中性运动迅速远离太阳。它主要由质子和电子组成，当它们遇到地球磁场时，感应电流增加到总磁场中。由此产生的相互作用被认为是造成北极光和地磁风暴的原因。这种相互作用的另一个特征是弓形激波，等离子体的行为在弓形激波中突然改变。无碰撞等离子体中冲击波的确切性质仍不清楚。提出了一个混合初边值问题来模拟上述相互作用。最初要做的工作将是建立解的全局存在性结果，并确定解随着时间的增加而趋于稳定状态的条件。物理科学的应用是显而易见的。

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.