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.

这项研究将集中于对血浆物理学中发生的几种特定非线性双曲线偏微分方程的分析。 对于弗拉索夫·马克斯韦尔方程式系统，将研究全球解决问题，稳定性，可能发生冲击和碰撞影响的问题的问题。 无碰撞等离子体是稀的电离气体。 在其中，颗粒的相互作用是通过电磁场进行的，该电磁场是由颗粒本身产生的。 它们的轨迹由涉及相对高速的某些微分方程定义。 主要研究者最近表明，较简单的vlasov方程的库奇问题具有独特的全局平滑解决方案，只要单个颗粒的力量均匀地界限。 现在将对碰撞效果的模型进行工作。 迄今为止，电磁效应被允许占主导地位。 目前尚无足够的经典溶解性条件。 第二条调查涉及旋转对称性的方程式和解决方案的稳定性问题。 某些案件具有肯定的答案（即确认稳定性）；应用文献进一步说明 - 没有证据表明在某些一般条件下存在稳定性。 必须首先要做的是阐明渐近稳定性和正常稳定性之间的区别（如果有）。 工作将在这一目标上完成。 将对无碰撞血浆中存在冲击的开放问题进行研究。 这项工作将从两个维度问题开始，在这些问题中，场和密度一直始终存在。 现在需要的是对所涉及的田间和密度的衍生产品的估计。 在两个维度上缺乏huygens的原理，目前尚不清楚这是否是真正的物理障碍（即真正发生冲击），还是仅仅是当前方法的缺点。 除了对部分微分方程理论的贡献外，这项工作还对数学物理学和统计力学也具有重要的潜在应用。