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.

所提出的工作致力于解决非线性色散方程领域内的广泛问题。主要目标是很好地理解线性色散动力学和非线性效应之间的相互作用。在线性方面，该项目将研究在弯曲背景上演化的波和薛定谔方程的全局时间色散特性。这在一定程度上也是针对广义相对论中爱因斯坦方程的史瓦西和克尔黑洞解的推测稳定性。在非线性方面，需要探索的一个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情况下演化的非线性色散方程。因此，虽然人们可能会在非常短的、可能与频率相关的时间尺度上看到相干的线性动力学，但这些在较长的时间内就会丢失。接下来的挑战是发现在越来越长的时间尺度上保持相干的更大模式。广义上讲，色散方程是其解可以被视为沿不同方向传播的波的叠加的方程。非线性效应使这些波相互作用，这可能会导致传播模式改变（产生新的波），或者可能导致数学家描述的“爆炸”现象。该项目中考虑的许多方程都起源于物理理论，例如广义相对论、多体量子场论、表面波传播和等离子体物理学。 因此，希望研究结果能够为相应的物理现象提供一些线索。该项目涉及研究生和博士后，以及一些国内和国际合作者。