Asymptotics of solutions for dispersive quasilinear problems

这个项目旨在研究广义的非线性色散问题中的几个现象。这些问题来自不同的物理学领域，例如广义相对论中时空结构和物质结构之间的相互作用，地形变化对海浪传播的影响，或者仅仅是等离子体物理和激光传播中的一些玩具模型，这些模型是为了理解较小的色散通道中可能的能量积累而设计的。这些情况是不同的，但都是几种不同“传播模式”之间复杂相互作用的结果，它们结合在一起创造了新的动态，不能简单地通过孤立地查看系统的每个组件来预测。更好地理解这些影响可以开发更好、更简单的模型，准确地描述考虑了很长一段时间的系统的行为。这个项目的目的是详细描述这些效应相关的例子，并突出潜在的共同数学结构，同时分离出每个特定现象的特定特征。本项目的主要主题是研究三个不同的非线性双曲和色散系统在散射不成立且非线性效应对解的定性行为有强烈影响的情况下的长期行为。主要研究人员将考虑下列问题：(I)广义相对论问题，即存在质量标量场时爱因斯坦方程的Minkowski空间的稳定性；(Ii)涉及较小体积区域上Sobolev范数可能增长的模型(NLS)问题；(Iii)试图证明海底地形的突然和局部变化如何影响孤立波传播的水波理论模型。在每一种情况下，首席研究人员都将专注于使用傅立叶分析和调和分析、双线性估计、常微分方程组和几何方法(矢量场、规范形式、哈密顿动力学工具)以及自适应函数空间的组合来控制全局解。其目标是推导出控制渐近行为的改进的“有效”动力学。

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.