Local and global dynamics for nonlinear dispersive equations

非线性色散方程的局部和全局动力学

• 批准号: 0801261
• 负责人: Daniel Tataru
• 金额: $ 83.29万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801261&HistoricalAwards=false
• 关键词: Local; global; dynamics; nonlinear; dispersive

The proposed work is devoted to a broad range of problems within the field of nonlinear dispersive equations. The primary goal is to achieve a good understanding of the interplay between linear dispersive dynamics and nonlinear effects. On the linear side, the project will study the global-in-time dispersive properties for wave and Schroedinger equations evolving on curved backgrounds. This is also in part directed toward the conjectured stability of the Schwarzchild and Kerr black hole solutions of the Einstein equations in general relativity. On the nonlinear side, a key topic to be explored is that of multiscale analysis. This corresponds to nonlinear dispersive equations that evolve in regimes where linear and nonlinear effects are strongly interlaced. Thus while one might see coherent linear dynamics on a very short, possibly frequency dependent time scale, these are lost at larger times. The challenge is then to uncover larger patterns that remain coherent on longer and longer time scales.Broadly speaking, dispersive equations are equations whose solutions can be thought of as superpositions of waves travelling in different directions. Nonlinear effects allow these waves to interact, which may lead to modified propagation patterns (creating new waves) or possibly to the phenomenon described by mathematicians as "blow-up." Many of the equations considered in this project have their origins in physical theories such as general relativity, many-body quantum field theory, surface wave propagation, and plasma physics. For this reason, it is hoped that the results of the research may shed some light on the corresponding physical phenomena. The project involves graduate students and postdocs, as well as several domestic and international collaborators.

所提出的工作专门用于非线性分散方程领域内的广泛问题。主要目标是对线性色散动力学和非线性效应之间的相互作用有良好的了解。在线性方面，该项目将研究在弯曲背景上演变的波和施罗丁格方程的全局分散性能。这也部分针对爱因斯坦方程的施瓦兹柴尔德和克尔黑洞溶液的猜想稳定性。在非线性方面，要探讨的关键主题是多尺度分析。这对应于在线性和非线性效应强烈交织的状态下进化的非线性色散方程。因此，尽管人们可能会在非常短的，可能依赖于频率的时间尺度上看到连贯的线性动力学，但它们在较大的时间丢失。然后，挑战是要揭示在越来越长的时间尺度上保持连贯性的较大模式。在路线上，分散方程是方程式的方程式，可以将其视为朝着不同方向传播的波的叠加。非线性效应允许这些波相互作用，这可能会导致修改的传播模式（创建新的波浪），或者可能导致数学家描述为“爆炸”的现象。该项目中考虑的许多方程都起源于物理理论，例如一般相对论，多体量子场理论，表面波传播和等离子体物理学。 因此，希望研究结果能够阐明相应的物理现象。该项目涉及研究生和博士后，以及几位国内和国际合作者。