Dispersive Phenomena in Linear and Nonlinear Partial Differential Equations

所提出的工作的主要目的是调查色散现象在偏微分方程中的作用。虽然人们对研究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（这可能会导致爆破）的解决方案的空间浓度的区域更强。因此，分散降低了爆破的可能性。 然而，如果爆发发生，其模式应在很大程度上取决于分散所允许的最差集中类型。 另一方面，非线性相互作用可以影响色散。 因此，人们从线性色散的研究，双线性估计，并进一步分析完全非线性相互作用。近年来，这条线的攻击已被证明是非常成功的研究非线性色散方程。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。描述色散的一种简单方法是说波（例如声波，弹性波，水波，电磁波等）不能在空间上集中很长一段时间;相反，它们必须扩散和衰减。在线性问题中，不同的波相互交叉而没有相互作用。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，也取决于它们的交叉模式。这些非线性相互作用在各种物理系统的短时行为和长时行为的研究中都起着重要的作用。例子包括固体中的弹性波，广义相对论中的引力波，以及许多其他的。所提出的工作的目标是有助于理解的动力学的非线性波相互作用的背景下，物理激励的色散系统。

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.