Asymptotics of solutions for dispersive quasilinear problems

色散拟线性问题解的渐近性

基本信息

批准号：
1700282
负责人：
Benoit Pausader
金额：
$ 22.8万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700282&HistoricalAwards=false
关键词：
Asymptotics solutions dispersive quasilinear problems

项目摘要

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.

该项目旨在研究广泛的非线性色散问题中的几种现象。这些问题来自不同的物理领域，例如广义相对论中时空结构与物质之间的相互作用、地形变化对海浪传播的影响，或者只是等离子体物理学和激光中的一些玩具模型传播旨在了解较小色散通道中可能的能量积累。这些情况各不相同，但都是由几种不同“传播模式”之间复杂的相互作用造成的，这些相互作用结合起来创造了新的动态，而这些动态不能简单地通过孤立地观察系统的每个组件来预测。更好地理解这些影响可以让我们开发出更好、更简单的模型，准确地描述长期考虑的系统行为。该项目的目标是详细描述此类效应相关的示例，并突出潜在的共同数学结构，同时隔离负责每个特定现象学的具体特征。该项目的主题是研究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，三种不同非线性双曲和色散系统的长期行为。首席研究员将考虑以下内容：（i）广义相对论的问题，即存在大标量场时爱因斯坦方程的闵可夫斯基空间的稳定性； (ii) 模型（NLS）问题，涉及较小体积域上 Sobolev 范数的可能增长； (iii) 水波理论模型，旨在证明底部地形的突然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里叶和调和分析、双线性估计、常微分方程和几何方法（矢量场、范式、哈密顿动力学工具）和适应函数空间的组合来全局控制解决方案。目标是导出控制渐近行为的改进的“有效”动力学。