Aspects of Nonlinear Hamiltonian PDE

[摘要]非线性Hamiltonian PDE的几个方面。PI打算继续研究非线性哈密顿偏微分方程的几个方面。研究的第一行是PDE的kam理论（在有界域上-假设周期边界条件），涉及时间周期和准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正则性数据。第一个研究方向与光滑动力系统理论（这里是无限维相空间）和晶格薛定谔算子的局域化理论密切相关。近年来有了实质性的技术进步，也导致了更高空间维度（d2）的结果，例如在二维NLS的时间周期解和准周期解的构建中。但是，这些发展仍处于初级阶段，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。第二个方向是利用调和分析的最新进展及其与守恒定律和先验不等式的相互作用，以解决一些悬而未决的问题，如某些非线性薛定谔方程的柯西问题，例如，临界非线性。一个典型的问题是大数据散焦情况下的全局适位性问题。2. 非线性哈密顿偏微分方程如非线性波动方程和非线性薛定谔方程是物理和工程中许多问题的数学模型。它们由许多不同的科学家小组研究，重点不同，通常使用不同的（数学）工具。我对这个提议感兴趣的是纯粹严谨的分析调查。从这个角度来看，NLW （resp）、NLS （resp）是有限（resp）的典型模型方程。无限速度传播。在考虑全局动力学问题时，应该区分有界和无界空间域的情况，其中不同的特征应该是预期的，“自然”问题（通常未解决）也是不同的。当然，要问的第一个问题是，解决方案是否一直存在，并且在平滑数据的情况下保持平滑。接下来是长时间行为的问题，包括单个数据和不同相空间中的动力学。在有界域的情况下，我们对概周期行为特别感兴趣。在线路上，由于色散，散射经常发生，我们打算在某些临界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.