Aspects of Nonlinear Hamiltonian PDE

DMS-9801013 Jean Bourgain摘要 非线性哈密顿偏微分方程的若干问题 J. Bourgain 1. PI打算继续研究非线性哈密顿偏微分方程的几个方面。第一条调查线是KAM理论的PDE的 (on有界域-假设周期性边界条件，例如），有关 时间周期解和拟周期解的存在性和持久性。 第二个主题是初值问题的适定性理论， 特别强调临界非线性和最小规律性数据。 第一个研究方向与光滑动力系统（这里是在无限维相空间中）和局部化理论密切相关 格点薛定谔算子理论 在极大程度 近年来的技术进步也导致了更高的 空间维度（D2），例如在时间周期的构造中 解和拟周期解。 但这些发展是 仍处于初始阶段，需要完成许多具有挑战性的工作 主要涉及所谓的“小维度”问题。 第二个方向旨在利用调和分析的最新进展及其与守恒律和先验不等式的相互作用，以便 关于柯西问题的一些未决问题的进展， 某些非线性薛定谔方程，例如，与临界 非线性 一个典型的问题是全局适定性问题 大数据的散焦情况。 2.非线性哈密顿偏微分方程，如非线性波动方程 (NLW)和非线性薛定谔方程（NLS）是一个数学模型， 物理学和工程学中的一些问题。 他们被许多人研究 不同的科学家群体，有不同的重点，往往不同 （数学）工具。我对这个提议的兴趣纯粹是分析性的 严格的调查。从这个角度来看，NLW（或NLS）是 典型的模型方程有限（分别）无限）速度传播。 在考虑全球动力学问题时，应该区分 有界和无界空间域的情况下，不同的功能应该 这两个“自然”问题（通常未解决）也是截然不同的。 当然，首先要问的问题是，解决方案是否永远存在 并且在平滑数据的情况下保持平滑。 接下来是时间长的问题 行为，无论是个人数据和动态在各种相空间。 在有界域的情况下，我们特别感兴趣的几乎 周期性行为 在线路上，由于色散，经常发生散射 我们打算在某些关键情况下研究这种现象。

DMS-9801013 Jean Bourgain ABSTRACT Aspects of nonlinear Hamiltonian PDE's J. Bourgain 1. The PI intend to continue research on several aspects of nonlinear Hamiltonian PDE's. A first line of investigation is the KAM-theory for PDE's (on bounded domains - assuming periodic boundary conditions say), concerned with existence and persistency of time periodic and quasi-periodic solutions. A second theme is the wellposedness theory for the initial value problem with special emphasis on critical nonlinearity and data of minimal regularity. The first research direction is closely related to the theory of smooth dynamical systems (here in infinite dimensional phase space) and localization theory for lattice Schrodinger operators. There has been a substantial technological progress over the recent years leading also to results in higher space dimension (D 2), for instance in the construction of time-periodic solutions and quasi-periodic solutions for 2D NLS. But these developments are still in an initial stage and a lot of challenging work needs to be done involving mainly so-called "small-dimension" questions. The second direction aims to exploit recent advances in Harmonic Analysis And its interplay with conservation laws and apriori inequalities, in order to progress on some of the remaining open problems on the Cauchy problem for certain nonlinear Schrodinger equations for instance, with critical nonlinearity. A typical issue are the global wellposedness questions in the defocusing case for large data. 2. Nonlinear Hamiltonian PDE's such as the nonlinear wave equations (NLW) and nonlinear Schrodinger equations (NLS) are mathematical models in a number of issues in physics and engineering. They are studied by many distinct groups of scientists, with different emphasis and often different (mathematical) tools. My interest in this proposal is an analytically purely rigorous inv estigation. From this point of view NLW (resp, NLS) are the typical model equations for finite (resp. infinite) speed propagation. When considering questions of global dynamics, one should distinguish the case of bounded and unbounded spatial domains where different features should be expected and the "natural" problems (often unsolved) are also distinct. Of course, a first question to ask is whether the solution exists for all time and remains smooth in case of smooth data. Next is the question of long time behaviour, both for individual data and the dynamics in various phase space. In the case of bounded domains, we are particularly interested in almost periodic behaviour. On the line, due to dispersion, scattering often occurs and we intend to investigate this phenomenon in certain critical cases.