Global Behaviour of Critical Nonlinear PDE

关键非线性偏微分方程的全局行为拟研究摘要terence tao本项目旨在研究（并有望解决）几种著名的临界非线性色散方程和波动方程的柯西问题的全局正则性问题。它们包括双曲空间的二维波映射方程、质量临界离焦非线性薛定谔方程和质量临界广义Korteweg-de Vries方程。最近在这一领域的突破，包括布尔甘的能量感应论证和集中紧性方法的成功，以及最近对能量临界非线性薛定谔方程的全局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、重力等）都是用非线性偏微分方程来描述的。这些方程通常在色散和非线性之间进行斗争，色散的作用是扩散波并使其随时间衰减，而非线性则会导致波在相对较短的时间内集中甚至产生奇点（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平衡的。一般认为，如果非线性具有“散焦”性质，则色散最终会“获胜”，不会形成奇点，而在“聚焦”情况下则相反。直到最近，这种直觉只在少数关键方程中得到证实，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更大范围的关键方程的结果。本项目将对物理学中出现的一些具体而著名的方程进行此类问题的研究。

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.