Asymptotics of solutions for dispersive quasilinear problems

这个项目旨在研究非线性色散问题的广泛类别中的几种现象。这些问题来自物理学的不同领域，例如广义相对论中时空结构和物质之间的相互作用，地形变化对海浪传播的影响，或者仅仅是等离子体物理学和激光传播的一些玩具模型，旨在了解较小色散通道中可能的能量积累。这些情况是不同的，但都是由几种不同的“传播模式”之间的复杂相互作用产生的，它们结合在一起创造了新的动态，而这些动态不能简单地通过孤立地看待系统的每个组成部分来预测。更好地理解这些效应，可以开发出更好、更简单的模型，这些模型可以准确地描述系统在很长一段时间内的行为。本项目的目标是详细描述这些效应相关的例子，并突出基本的共同数学结构，同时又把每一种现象学的具体特征分离出来。这个项目的主题是研究长期以来的-三种不同的非线性双曲和色散系统的时间行为的情况下，散射不成立，非线性效应有很强的影响解决方案的定性行为。主要研究者将考虑以下问题：（i）广义相对论的问题，即在存在大量标量场的情况下爱因斯坦方程的闵可夫斯基空间的稳定性;（ii）涉及较小体积域上Sobolev范数可能增长的模型（NLS）问题;（iii）水波理论的模型，目的是显示海底地形的突然和局部变化如何影响孤立波的传播。在每种情况下，首席研究员将专注于使用傅立叶和调和分析，双线性估计，常微分方程和几何方法（向量场，范式，哈密顿动力学工具）和适应函数空间的组合来控制全局解决方案。我们的目标是得到改进的“有效”的动态控制渐近行为。

This project aims to study several phenomena in the broad category of nonlinear dispersive problems. These problems arise from varied domains of physics, such as the interaction between the structure of space-time and matter in general relativity, the influence of changes in the topography on the propagation of ocean waves, or simply some toy models from plasma physics and laser propagation devised to understand the possible buildup of energy in smaller dispersive channels. These situations are different and yet all result from complex interactions between several different "modes of propagation" combining to create new dynamics that cannot simply be predicted by looking at each component of the system in isolation. A better understanding of these effects allows one then to develop better and simpler models that accurately describe the behavior of the system under consideration for large time. The objective of this project is to describe in detail examples where such effects are relevant and to highlight the underlying common mathematical structure, while at the same time isolating the specific features responsible for each particular phenomenology.The main theme of this project is the study of the long-time behavior of three different nonlinear hyperbolic and dispersive systems in cases where scattering does not hold and nonlinear effects have a strong impact on the qualitative behavior of solutions. The principal investigator will consider the following: (i) a problem from general relativity, namely, the stability of Minkowski space for the Einstein equations in the presence of a massive scalar field; (ii) a model (NLS) problem that involves the possible growth of Sobolev norms on domains of smaller volume; (iii) a model from water-wave theory that seeks to demonstrate how a sudden and localized change in the bottom topography influences the propagation of solitary waves. In each case the principal investigator will focus on controlling solutions globally using a combination of Fourier and harmonic analysis, bilinear estimates, ordinary differential equations and geometric methods (vector fields, normal forms, tools from Hamiltonian dynamics), and adapted function spaces. The objective is to derive improved "effective" dynamics that control asymptotic behavior.