Mathematical Sciences: Nonlinear Partial Differential Equations in Plasma Physics

数学科学：等离子体物理中的非线性偏微分方程

• 批准号: 8721721
• 负责人: Robert Glassey
• 金额: $ 9.96万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-05-01 至 1991-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8721721&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Partial; Differential

This investigation will focus on the analysis of several specific nonlinear hyperbolic partial differential equations occurring in plasma physics. Questions of global solutions to the Cauchy problem, stability, possible development of shocks and collisional effects will be studied for the Vlasov-Maxwell system of equations. Collisionless plasmas are dilute ionized gases. In them, the interaction of particles takes place through the electromagnetic field which is generated by the particles themselves. Their trajectories are defined by certain differential equations involving relativistically high speeds. The principal investigator has shown recently that the Cauchy problem for the simpler Vlasov equations has a unique global smooth solution provided the momenta of the individual particles are uniformly bounded. Work will now be done on models in which the effects of collisions are taken into account. Heretofore, the electromagnetic effects were allowed to dominate the collisional. No sufficient condition for classical solvability is known at this time. A second line of investigation involves equations with rotational symmetry and the question of stability of solutions. Certain cases have affirmative answers (i.e. stability is confirmed); the applied literature goes further, stating - without proof - that stability is present under certain general conditions. What first must be done is a clarification of the distinctions, if any, between asymptotic stability and normal stability. Work will be done towards this end. Studies into the open question of the existence of shocks in collisionless plasma will be carried out. The work will begin with two dimensional problems where fields and densities remain bounded for all time. What is needed now are estimates on the derivatives of the fields and densities involved. There lacks a Huygens' Principle in two dimensions and it is unclear whether this is a real physical barrier (i.e. a shock really occur) or is simply a shortcoming of current methods. In addition to its contributions to the theory of partial differential equations, this work has important potential applications to mathematical physics and statistical mechanics.

本次调查将重点分析几个 特殊非线性双曲型偏微分方程 发生在等离子体物理学中。 全球解决办法的问题 柯西问题，稳定性，可能的发展冲击和 我们将研究Vlasov-Maxwell系统的碰撞效应 方程的。 无碰撞等离子体是稀释的电离气体。 在他们身上， 粒子之间的相互作用是通过 粒子产生的电磁场 自己 它们的轨迹是由某些 涉及相对论高速的微分方程。 首席研究员最近表明，柯西 简单的弗拉索夫方程的问题有一个独特的全球 光滑的溶液提供了单个颗粒的动量 是一致有界的 现在将对模型进行研究， 考虑了碰撞的影响。 在此， 电磁效应被允许主导 碰撞。 经典可解性的不充分条件 在这个时候是已知的。 第二条调查路线涉及方程， 旋转对称性和解的稳定性问题。 某些情况下有肯定的答案（即稳定性是 （注：此为文学作品），《文学》更进一步， 在没有证据的情况下，稳定性在某些一般情况下是存在的。 条件 首先必须做的是澄清 渐近稳定性和正态性之间的区别，如果有的话， 稳定 将为此开展工作。 研究了在不确定性中存在冲击的未决问题， 将进行无碰撞等离子体。 工作将开始 二维问题中的场和密度保持不变 永远有界。 现在需要的是对 所涉及的场和密度的导数。 缺少一个 惠更斯原理在二维空间中， 这是一个真实的物理屏障（即，确实发生了电击），或者 这只是现有方法的一个缺点。 除了它对局部理论的贡献外， 微分方程，这项工作具有重要的潜力 应用于数学物理和统计力学。