Research on Dispersive Partial Differential Equations

9800879吉奥拉·斯塔菲拉尼研究项目摘要我的研究涉及到一些色散偏微分方程组的研究。我的工作集中在两个特殊的色散方程的柯西问题：非线性薛定谔方程和广义的Korteweg-de Vries方程。在我的项目中我要解决的第一个问题是：给定一个色散方程，人们必须假定初始轮廓(初始数据条件)有多大的规律性，才能确保以后波解的存在和唯一性？当一个人还施加边界条件时，这样的问题就变得更加复杂。目前，可用的技术(调和分析和傅立叶分析)仅适用于周期边界条件的情况。我要解决的第二个问题是，一旦假定波在所有时间都“活着”并且是光滑的，如何在时间上保持波解的某些正则性。更准确地说，在这些条件下，我研究了在考虑可微阶的情况下测得的波大小的渐近行为。这是一个与上面描述的问题非常不同的问题。然而，我用来回答这两个问题的技巧在本质上是非常相似的。色散偏微分方程用来模拟自然界中的许多波动现象。例如，信号在光纤中的传播，磁场中等离子体中的非线性离子声波和等离子体中的长波。在我的工作中，我改进和开发了一些抽象的数学工具，以便从波的初始轮廓的一般性质中获得关于波本身演化的信息。这些信息涉及波的存在时间、某些性质的持久性、奇点附近的行为以及波携带的能量。在某些情况下，这些信息的一部分可以通过在实验室中进行实验来恢复。相反，我得到的结果为某些经验观察提供了数学证明。

DMS-9800879 Gigiola Staffilani Abstract of the Research Project My research involves the study of certain Dispersive Partial Differential Equations. I concentrate my work on the Cauchy problem for two particular dispersive equations: the nonlinear Schr\"odinger equation and the generalized Korteweg-de Vries equation. The first question I address in my project is the following: given a dispersive equation, how much regularity does one have to assume for the initial profile (initial data condition) in order to be able to insure existence and uniqueness of the wave solution at later times? Such a question becomes more complicated to answer when one also imposes boundary conditions. At the moment, the techniques available (Harmonic and Fourier Analysis) only apply to the case of periodic boundary conditions. The second question I address is how certain regularity properties of the wave solution are preserved in time, once it is assumed that the wave ``lives'' for all times and it is smooth. More precisely, under these conditions, I study the asymptotic behavior of the size of the wave measured taking into account the order of differentiability. This is a very different problem than the one described above. Nevertheless the techniques I rely upon to answer both questions are very similar in their nature. Dispersive partial differential equations are introduced to model many wave phenomena that occur in nature. Some examples are, for instance, propagation of signals in optic fibers, nonlinear ionic-sonic waves in plasma in a magnetic field and long waves in plasma. In my work I improve and develop some abstract mathematical tools in order to obtain from general properties of the initial profile of the wave information about the evolution of the wave itself. This information regards the time of existence of the wave, the persistence of certain properties, the behavior near singular points, and the energy carried by the wave. In some cases part of this information can be recovered by conducting experiments in laboratories. Conversely, the results I obtain provide mathematical justifications for certain empirical observations.