Aspects of Nonlinear Hamiltonian PDE

非线性哈密顿量偏微分方程的各个方面

• 批准号: 9801013
• 负责人: Jean Bourgain
• 金额: $ 13万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-05-15 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801013&HistoricalAwards=false
• 关键词: Aspects; Nonlinear; Hamiltonian; PDE

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.

DMS-9801013 Jean Bourain是非线性哈密顿偏微分方程解的抽象方面1。PI打算继续研究非线性哈密顿偏微分方程解的几个方面。一线研究是关于有界域上的偏微分方程解的KAM理论，涉及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正则性数据。第一个研究方向与光滑动力系统理论(这里是在无限维相空间中)和格型薛定谔算子的局部化理论密切相关。近年来，技术有了很大的进步，也导致了更高的空间维度(D2)，例如，在构造二维NLS的时间周期解和准周期解方面。但这些发展仍处于初级阶段，需要做大量具有挑战性的工作，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律和先验不等式的相互作用，以便在某些具有临界非线性的非线性薛定谔方程的柯西问题的一些遗留问题上取得进展。一个典型的问题是大数据散焦情况下的全球适宜性问题。2.非线性波动方程(NLW)和非线性薛定谔方程(NLS)等非线性哈密顿偏微分方程组是许多物理和工程问题中的数学模型。它们由许多不同的科学家群体研究，他们有不同的侧重点，通常也有不同的(数学)工具。我对这项提议的兴趣是一项分析性的纯粹严格的调查。从这个角度来看，NLW(RESP，NLS)是有限(REP，NLS)的典型模型方程。无限)速度传播。在考虑全球动力学问题时，人们应该区分有界和无界空间域的情况，在这种情况下，应该预期不同的特征，并且“自然”问题(通常是未解决的)也是截然不同的。当然，首先要问的问题是，解决方案是否一直存在，并在数据平稳的情况下保持平稳。接下来是长时间行为的问题，包括个人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，经常发生散射，我们打算在某些关键情况下研究这种现象。