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.

非线性色散部分微分方程方程是一种非常宽的方程式，用于描述各种现象，从水面的波到血浆物理学和量子力学。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分散方程的理解，该方法是由主要研究者及其合作者Nader Masmoudi和Jalal Shatah最近提出的。时空共振的方法结合了谐振的标准概念，众所周知，至少自庞加雷（Poincare）时代以来，研究了解决方案的空间定位。其目的是了解平衡的稳定性。它涉及精致的多线性谐波分析问题，主要研究者计划系统地研究该问题。该项目的第二个目的是加深我们对自相似爆炸的理解。这是在非线性分散方程解决方案解决方案中形成奇点的一种机制，这是对此类方程式研究的基本重要性的主题。如此称呼的非线性分散偏微分方程是描述非常多样化的物理现象的一类方程式：地表水波，表面水波，负责液体的物理学（质量效应），或量效应。应用的范围非常广泛：水浪的物理学包括对海啸的研究；至于血浆物理学，这不仅对我们对宇宙的理解非常重要，而且对于许多工业应用也至关重要。首席研究者的目的是加深我们对基本层面上这一非常一般的方程式类别的理解。他打算特别研究两个问题。首先，这些方程式的平衡何时稳定？其次，这些方程如何发展出奇异性，或者以不同的方式陈述，这些方程的“平滑”解决方案如何变得越来越“锯齿状”？例如，对这些问题的答案将有助于了解海啸波如何传播或激光光束如何聚焦。