Asymptotic dynamics for nonlinear dispersive systems

非线性色散系统的渐近动力学

• 批准号: 1362940
• 负责人: Benoit Pausader
• 金额: $ 17.85万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362940&HistoricalAwards=false
• 关键词: Asymptotic; dynamics; nonlinear; dispersive; systems

At present, some simple "elementary" systems start to be reasonably well understood. However, passing from the understanding of microscopic objects to moderately complex configurations remains a challenging problem. This projects aims at understanding simple but basic questions for slightly complicated systems, focusing on situations and models related to macroscopic plasma physics. A plasma is a collection of many charged particles that, because of their long range interactions exhibit a collective macroscopic behavior which is complex and remains poorly understood. This in turns limits some potential major applications, controlled fusion being perhaps the most well-known. Thus, furthering the understanding of this "small scale to large scale" cascade of information can have major possible applications. The emphasis of this proposal is on large-time questions such as "can a plasma macroscopically at rest spontaneously develop dramatic behavior such as high concentration or velocities, or spontaneously form a vacuum?". Another related question is to find possible scenarios for the large-time behavior of a plasma that remains away from the quiet neutral equilibrium. Investigating these questions involve a fine study of the deep mechanisms of transfer of energy inside different parts of the plasma as a result of collective elementary simple interactions.This projects aims at developing the study of two fundamental equations from physics. First we consider stability issues for the 2-fluid Euler-Maxwell equation in two dimensions. This is a fundamental equation modeling the dynamical properties of a plasma. The goal is to prove that under certain conditions, small perturbations of an equilibrium will not develop shocks and that actually, the plasma will get back to equilibrium, even in the absence of dissipation. Such a result would be of great physical and mathematical importance as it is known to be false for the compressible Euler equation in the absence of a self-consistent electromagnetic field. In a second part, we study the Schroedinger equation on a curved background. This equation is a universal model appearing in many time reversible equations, notably in some plasma models. When this equation is posed on a nonconstant background, many classical tools from the Euclidian theory break down and one expects the appearance of various new phenomena due to the influence of the geometry. We study in particular the effect of the growth of the volume on the global behavior of the solutions and on the possibility to obtain asymptotic dynamics different from scattering. A unifying theme is to understand the asymptotic dynamics of solutions in a context where the dispersion is limited. Another unifying theme is to try to find simpler limit equations and understand their significance for the full model. A last unifying theme concerns the specificity and relevance of dispersive systems. Some tools we plan to use and develop are concentration compactness methods, study of the space and time resonances, study of dispersive systems and pseudo-products estimates with singular multipliers.

目前，一些简单的“基本”系统开始被很好地理解。然而，从对微观物体的理解过渡到中等复杂的结构仍然是一个具有挑战性的问题。该项目旨在理解稍微复杂的系统的简单但基本的问题，重点关注与宏观等离子体物理相关的情况和模型。 等离子体是许多带电粒子的集合，由于它们的长程相互作用，它们表现出一种复杂且仍知之甚少的集体宏观行为。这反过来又限制了一些潜在的主要应用，受控聚变可能是最著名的。因此，进一步理解这种“小规模到大规模”级联的信息可能具有重大的潜在应用。该提案的重点是一些重大问题，例如“宏观静止的等离子体能否自发地发展出戏剧性的行为，例如高浓度或速度，或自发地形成真空？”。另一个相关问题是寻找远离安静中性平衡的等离子体长时间行为的可能场景。研究这些问题涉及对等离子体不同部分内部能量转移的深层机制（由于集体基本简单相互作用的结果）的深入研究。该项目旨在发展对物理学中两个基本方程的研究。首先，我们考虑二维 2 流体欧拉-麦克斯韦方程的稳定性问题。这是模拟等离子体动力学特性的基本方程。目标是证明在某些条件下，平衡的小扰动不会产生冲击，并且实际上，即使没有耗散，等离子体也会恢复平衡。这样的结果将具有重大的物理和数学重要性，因为众所周知，在缺乏自洽电磁场的情况下，可压缩欧拉方程是错误的。在第二部分中，我们研究弯曲背景上的薛定谔方程。 该方程是出现在许多时间可逆方程中的通用模型，特别是在一些等离子体模型中。当这个方程在非恒定背景下提出时，欧几里德理论中的许多经典工具就会失效，并且由于几何的影响，人们预计会出现各种新现象。我们特别研究体积增长对解的全局行为的影响以及获得不同于散射的渐近动力学的可能性。一个统一的主题是在色散有限的情况下理解解的渐近动力学。另一个统一的主题是尝试找到更简单的极限方程并理解它们对整个模型的意义。最后一个统一的主题涉及分散系统的特殊性和相关性。我们计划使用和开发的一些工具包括浓度紧致度方法、空间和时间共振研究、色散系统研究以及奇异乘数的伪乘积估计。