Asymptotic dynamics for nonlinear dispersive systems

目前，一些简单的“初级”系统开始被很好地理解。然而，从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，由于它们的远距离相互作用，表现出一种集体的宏观行为，这种行为是复杂的，至今仍知之甚少。这反过来又限制了一些潜在的主要应用，控制聚变可能是最著名的。因此，进一步理解这种“从小规模到大规模”的信息级联可能具有重大的应用价值。这个提议的重点是大时间的问题，如“宏观上静止的等离子体能否自发地发展出戏剧性的行为，如高浓度或高速度，或自发地形成真空？”另一个相关的问题是寻找远离安静的中性平衡的等离子体的大时间行为的可能情况。要研究这些问题，需要对等离子体不同部分内部能量传递的深层机制进行细致的研究，这是集体基本简单相互作用的结果。本项目旨在发展物理学中两个基本方程的研究。首先，我们考虑二维流体欧拉-麦克斯韦方程的稳定性问题。这是一个模拟等离子体动力学特性的基本方程。目的是证明在一定条件下，平衡的小扰动不会产生激波，实际上，等离子体会回到平衡状态，即使没有耗散。这样的结果将具有重大的物理和数学意义，因为已知在没有自洽电磁场的情况下，可压缩欧拉方程是不成立的。在第二部分中，我们研究了弯曲背景下的薛定谔方程。该方程是许多时间可逆方程中普遍存在的模型，特别是在一些等离子体模型中。当该方程在非常数背景下提出时，由于几何的影响，欧几里得理论中的许多经典工具失效，人们期望出现各种新现象。我们特别研究了体积的增长对解的整体行为的影响，以及对获得不同于散射的渐近动力学的可能性的影响。一个统一的主题是理解解的渐近动力学在一个背景下，色散是有限的。另一个统一的主题是试图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是色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集中紧致方法，研究空间和时间共振，研究色散系统和奇异乘子的伪积估计。

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.