Space-Time Resonances and Asymptotics; Stability of Self-Similar Solutions

时空共振和渐近；

• 批准号: 1101269
• 负责人: Pierre Germain
• 金额: $ 16.79万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101269&HistoricalAwards=false
• 关键词: Space; Time; Resonances; Asymptotics; Stability

Nonlinear dispersive partial differential equations equations are a very wide class of equations that are used to describe varied phenomena, ranging from waves at the surface of water to plasma physics and quantum mechanics. The first aim of this project is to extend our understanding of nonlinear dispersive equations by using the method of space-time resonances, which was introduced recently by the principal investigator and his collaborators, Nader Masmoudi and Jalal Shatah. The method of space-time resonances combines the standard notion of resonances, well known since at least Poincare's time, with a study of the spatial localization of solutions. Its aim is to understand the stability of equilibria. It involves delicate multilinear harmonic analysis questions, which the principal investigator plans to study systematically. The second aim of this project is to deepen our understanding of self-similar blow-up. This is a mechanism for the formation of singularities in the solutions of nonlinear dispersive equations, a topic of fundamental importance in the study of such equations.So-called nonlinear dispersive partial differential equations are a class of equations describing extremely varied physical phenomena: surface water waves, the physics of charged fluids (plasmas), or quantum mechanical effects, to name but three. The range of applications is very broad: the physics of water waves includes the study of tsunamis; as for plasma physics, it is not only very important for our understanding of the universe but is also crucial for many industrial applications. The aim of the principal investigator is to deepen our understanding of this very general class of equations at a fundamental level. He intends to study two questions in particular. First, when are the equilibria of these equations stable? Second, how do these equations develop singularities or, stated differently, how can "smooth" solutions of these equations become increasingly "jagged"? Answers to these questions would, for instance, help understand how tsunami waves propagate or how laser beams focus.

非线性色散偏微分方程是一类非常广泛的方程，用于描述各种现象，从水面的波浪到等离子体物理和量子力学。该项目的第一个目的是通过使用时空共振方法来扩展我们对非线性色散方程的理解，该方法最近由首席研究员及其合作者Nader Masmoudi和Jalal Shatah引入。时空共振的方法结合了共振的标准概念，众所周知，因为至少庞加莱的时间，与研究的空间定位的解决方案。它的目的是了解平衡点的稳定性。它涉及微妙的多线性谐波分析问题，首席研究员计划系统地研究。本项目的第二个目的是加深我们对自相似爆破的理解。非线性色散偏微分方程（英语：Nonlinear dispersion partial differential equations）是一类描述各种物理现象的方程，如表面水波、带电流体（等离子体）物理、量子力学效应等。非线性色散偏微分方程（英语：Nonlinear dispersion partial differential equations）是一类描述各种物理现象的方程。应用范围非常广泛：水波物理学包括海啸的研究;至于等离子体物理学，它不仅对我们了解宇宙非常重要，而且对许多工业应用也至关重要。主要研究者的目的是加深我们对这类非常一般的方程的基本水平的理解。他打算特别研究两个问题。首先，这些方程的平衡点何时稳定？第二，这些方程如何发展奇点，或者换句话说，这些方程的“光滑”解如何变得越来越“锯齿状”？例如，这些问题的答案将有助于理解海啸波如何传播或激光束如何聚焦。