Global Behaviour of Critical Nonlinear PDE

临界非线性偏微分方程的全局行为

• 批准号: 0649473
• 负责人: Terence Tao
• 金额: $ 106.22万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0649473&HistoricalAwards=false
• 关键词: Global; Behaviour; Critical; Nonlinear; PDE

Global Behavior of Critical Nonlinear PDEAbstract of Proposed ResearchTerence TaoThis project is to study (and hopefully solve) the global regularity question for the Cauchy problem for several well-known, critical, nonlinear, dispersive and wave equations. They include the two-dimensional wave maps equation into hyperbolic space, the mass-critical defocusing nonlinear Schrodinger equation and the mass-critical generalized Korteweg-de Vries equation. The very recent breakthroughs on this area, including Bourgain's induction-on-energy argument and the successes of concentration-compactness methods, as well as the recently completely resolution of the global regularity problem for the energy-critical nonlinear Schrodinger equation suggest that the resolution of these problems are now within reach.Many wave phenomena in physics (e.g. light, water, sound, gravity, etc.) are described using nonlinear partial differential equations. These equations often encode a struggle between dispersion, which acts to spread out the wave and make it decay over time, and nonlinearity, which can instead cause the wave to concentrate and even to develop singularities (or "blow up") in relatively short periods of time. An important class of equations are the "critical" equations, in which the dispersion and the nonlinearity are in some sense exactly balanced against each other. It is generally believed that if the nonlinearity has a "defocusing" nature then the dispersion should eventually "win", and no singularities will form, whereas the converse should be true in the "focusing" case. Until very recently, this intuition was only confirmed for a handful of critical equations but, in the last few years, some powerful new technical tools have been developed which should now allow us to prove results about a much larger range of critical equations. This project will pursue such issues for some specific and well-known equations arising in physics.

临界非线性偏微分方程的全局性态研究建议摘要Terence Tao这个项目是研究（并希望解决）几个著名的临界非线性色散波动方程的柯西问题的全局正则性问题。它们包括双曲空间中的二维波映射方程、质量临界散焦非线性Schrodinger方程和质量临界广义Korteweg-de弗里斯方程。 最近在这一领域的突破，包括Bourgain的能量归纳论证和浓度紧性方法的成功，以及最近能量临界非线性薛定谔方程的整体正则性问题的完全解决，表明这些问题的解决现在是触手可及的。物理学中的许多波动现象（如光、水、声音、引力等），都是由波的性质决定的。用非线性偏微分方程描述。这些方程通常编码了色散和非线性之间的斗争，色散的作用是分散波并使其随时间衰减，而非线性则会导致波集中，甚至在相对较短的时间内发展出奇点（或“爆炸”）。一类重要的方程是“临界”方程，其中色散和非线性在某种意义上是相互精确平衡的。 一般认为，如果非线性具有“散焦”性质，那么色散最终应该“获胜”，并且不会形成奇点，而在“聚焦”情况下，匡威的情况应该是正确的。 直到最近，这种直觉只被证实为少数关键方程，但在过去的几年里，一些强大的新技术工具已经开发出来，现在应该允许我们证明有关更大范围的关键方程的结果。 本项目将针对物理学中出现的一些特定和众所周知的方程来探讨这些问题。