Asymptotic dynamics for nonlinear dispersive systems

目前，一些简单的“初级”系统开始得到相当好的理解。然而，从对微观物体的理解过渡到适度复杂的构型仍然是一个具有挑战性的问题。这个项目的目的是理解简单但基本的问题，稍微复杂的系统，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粒子的集合，由于它们的长距离相互作用，表现出一种复杂的集体宏观行为，并且仍然鲜为人知。这反过来又限制了一些潜在的主要应用，受控融合可能是最广为人知的。因此，进一步了解这种“从小规模到大规模”的信息级联可能具有重大的应用价值。这项建议的重点是大时间的问题，如“宏观上处于静止状态的等离子体是否会自发产生高浓度或高速度等戏剧性行为，或自发形成真空？”另一个相关的问题是为远离安静中性平衡的等离子体的长时间行为找到可能的情景。对这些问题的研究包括深入研究作为集体基本简单相互作用结果的等离子体不同部分之间的能量传递的深层机制。该项目旨在从物理学的角度发展对两个基本方程的研究。首先，我们考虑了二维二维Euler-Maxwell方程的稳定性问题。这是一个模拟等离子体动力学性质的基本方程。我们的目标是证明，在某些条件下，平衡的微小扰动不会产生激波，实际上，即使没有耗散，等离子体也会回到平衡。这样的结果将具有重要的物理和数学意义，因为众所周知，在没有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，我们研究了弯曲背景下的薛定谔方程。这个方程是出现在许多时间可逆方程中的普遍模型，特别是在一些等离子体模型中。当这个方程在非常数背景下提出时，欧几里得理论中的许多经典工具都失效了，人们预计会因为几何的影响而出现各种新的现象。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试图找到更简单的极限方程，并理解它们对完整模型的意义。最后一个统一的主题涉及分散系统的特殊性和相关性。我们计划使用和开发的工具包括集中紧致性方法，空间和时间共振的研究，色散系统的研究和带有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.