Dispersive Phenomena in Linear and Nonlinear Partial Differential Equations

提出的工作的主要目的是研究色散现象在偏微分方程中的作用。虽然人们对线性偏微分方程的色散现象有一定的兴趣，但主要的动机来自非线性偏微分方程。事实上，非线性效应（可能导致爆炸）在溶液的空间集中区域更强。因此，分散降低了爆炸的可能性。尽管如此，如果发生爆炸，其模式应在很大程度上取决于扩散所允许的最坏浓度类型。另一方面，非线性相互作用会影响色散。因此，从线性色散的研究到双线性估计，再到完全非线性相互作用的分析。近年来，这种方法在非线性色散方程的研究中被证明是非常成功的。然而，还有许多工作要做，人们有一种感觉，我们只看到了冰山一角。描述色散的一个简单方法是，波（如声波、弹性波、水波、电磁波等）不能长时间保持空间集中；相反，它们必须扩散和腐烂。在线性问题中，不同的波相互交叉而不相互作用。然而，在非线性现象中，波总是相互作用的。这种相互作用的强度取决于每个波的强度，也取决于它们的相交模式。这些非线性相互作用在研究各种物理系统的短时行为和长时行为中都起着重要作用。例子包括固体中的弹性波，广义相对论中的引力波等等。所提出的工作的目标是有助于理解在物理驱动色散系统的背景下非线性波相互作用的动力学。

The primary aim of the proposed work is to investigate the role of dispersive phenomena in partial differential equations. While there certainly is some interest in studying dispersive phenomena for linear partial differential equations, the main motivation comes from nonlinear partial differential equations. Indeed, nonlinear effects (which can possibly lead to blowup) are stronger in regions of spatial concentration of the solutions. Hence the dispersion reduces the potential for blow-up. If nevertheless blow-up occurs, its pattern should largely be determined by the worst type of concentration allowed by the dispersion. On the other hand, nonlinear interactions can affect the dispersion. Thus one is led from the study of linear dispersion to bilinear estimates and further to the analysis of fully nonlinear interactions. In recent years this line of attack has proved to be highly successful in the study of nonlinear dispersive equations. Yet much more remains to bedone, and one has the feeling that we have only seen the tip of the iceberg. A simple way to describe dispersion is to say that waves (e.g. sound waves, elastic waves, water waves, electromagnetic waves, etc) cannot stay spatially concentrated for a long period of time; instead they must spread out and decay. In linear problems different waves cross each other without interaction. However, in nonlinear phenomena waves will always interact. The strength of this interaction depends on the strength of each wave but also on their intersection pattern. These nonlinear interactions play an essential role in both the study of the short time behavior and of the long time behavior of various physical systems. Examples include elastic waves in solids, gravitational waves in general relativity, and many others. The goal of the proposed work is to contribute to the understanding of the dynamics of nonlinear wave interactions in the context of physically motivated dispersive systems.