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”系统开始被合理地理解。但是，从对微观对象的理解到适度复杂的配置仍然是一个具有挑战性的问题。该项目旨在了解略带复杂系统的简单但基本的问题，重点是与宏观等离子体物理有关的情况和模型。 等离子体是许多带电颗粒的集合，由于它们的远距离相互作用表现出一种集体宏观行为，该行为很复杂，而且仍然很熟悉。反过来，这限制了一些潜在的主要应用程序，受控融合也许是最著名的。因此，进一步了解这种“小规模到大规模”的信息可以具有主要的应用程序。该提议的重点是大型问题，例如“血浆宏观可以自发地发展出巨大的行为，例如高浓度或速度，或自发形成真空？”。另一个相关的问题是，要找到远离安静的中性平衡的等离子体的大型行为的可能场景。研究这些问题涉及对由于集体基本简单相互作用而导致能量转移在等离子体内部不同部分的深层研究。该项目旨在开发对物理学的两个基本方程的研究。首先，我们考虑了两个维度的2流体Euler-Maxwell方程的稳定性问题。这是一个基本方程，对等离子体的动力学特性进行了建模。目的是证明在某些条件下，平衡的小扰动不会产生冲击，实际上，即使在没有耗散的情况下，等离子体也会恢复平衡。这样的结果将具有很大的物理和数学重要性，因为在没有自洽的电磁场的情况下，由于可压缩的Euler方程而言是错误的。在第二部分中，我们在弯曲的背景下研究Schroedinger方程。 该方程是在许多时间可逆方程中出现的通用模型，特别是在某些血浆模型中。当这个方程式摆在非构造背景上时，欧几里得理论中的许多经典工具都会崩溃，并且人们期望由于几何形状的影响而导致各种新现象的出现。我们特别研究了体积生长对溶液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分散剂受到限制的情况下了解解决方案的渐近动力学。另一个统一的主题是尝试找到更简单的极限方程，并了解其对完整模型的意义。最后一个统一的主题涉及分散系统的特异性和相关性。我们计划使用和开发的一些工具是浓度紧凑度方法，研究空间和时间共振，对分散系统的研究以及具有单数倍增器的伪造估计。