Deterministic and probabilistic well-posedness results for nonlinear dispersive and wave equations

非线性色散方程和波动方程的确定性和概率适定性结果

• 批准号: 1748083
• 负责人: Aynur Bulut
• 金额: $ 4.46万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1748083&HistoricalAwards=false
• 关键词: Deterministic; probabilistic; well; posedness; results

The research supported in this proposal concerns the study of nonlinear dispersive and wave equations. These arise as fundamental models of a wide variety of physical systems, including the propagation of waves and the study of nonlinear optics, and are closely related to models of fluids, as well as a number of aspects of statistical and quantum mechanics. The development of mathematical tools to understand important issues such as existence and uniqueness of solutions, as well the behavior of the corresponding evolutions on a qualitative and quantitative level, is therefore an issue of fundamental scientific importance. As a particular example, several of the questions to be investigated in this research involve studying long-time properties of solutions to the nonlinear Schrodinger and nonlinear wave equations evolving from "generic" randomly chosen initial data. In addition to being of substantial mathematical interest, investigation into these questions addresses the broader scientific issue of whether possible singularities arising in the mathematical formulation can occur in physically relevant settings. Moreover, the topics to be studied are closely connected to a wide range of issues in partial differential equations, probability, and harmonic analysis, and the ideas and techniques developed will provide important contributions to the availability of mathematical tools for future research on related questions.The particular scope of this research project is to investigate several problems concerning local and global well-posedness properties for nonlinear dispersive and wave equations, focusing on the nonlinear Schrodinger, nonlinear wave, and Korteweg-de Vries equations. The research encompasses four broad directions, which are often interrelated, and which each contribute to the wider theme of understanding the precise dynamical behavior of solutions both locally and globally in time. The first two directions of interest concern global in time existence of solutions, in particular focusing first on the development of global well-posedness results adapted to the energy-supercritical setting, and second on important issues surrounding probabilistic global well-posedness results in endpoint and limiting situations (in this probabilistic framework, initial data for the problem is chosen as a random Fourier series, and results are obtained by excluding sets occurring with small probability). The third direction of interest turns to the study of local in time stability properties of solutions, focusing in particular on initial data of low-regularity. The fourth direction of study investigates dispersive models with higher-order nonlocal terms, in which a key ingredient will be the adaptation of recent developments in the study of nonlinear elliptic and parabolic partial differential equations with similar nonlocal features. In each of these settings, the PI and collaborators will incorporate techniques from harmonic analysis, probability, and spectral theory to analyze the dynamical features involved in the evolution.

该提案支持的研究涉及非线性色散和波动方程的研究。 这些模型是各种物理系统的基本模型，包括波的传播和非线性光学的研究，并且与流体模型以及统计和量子力学的许多方面密切相关。 因此，开发数学工具来理解重要问题，例如解决方案的存在性和唯一性，以及定性和定量层面上相应演化的行为，是一个具有基础科学重要性的问题。 作为一个特定的例子，本研究要研究的几个问题涉及研究从“通用”随机选择的初始数据演变而来的非线性薛定谔和非线性波动方程解的长期特性。 除了具有重大的数学意义外，对这些问题的研究还解决了更广泛的科学问题，即数学公式中可能出现的奇点是否会出现在物理相关的环境中。 此外，要研究的主题与偏微分方程、概率和调和分析中的广泛问题密切相关，所开发的思想和技术将为相关问题的未来研究提供数学工具的可用性做出重要贡献。该研究项目的具体范围是研究有关非线性色散方程和波动方程的局部和全局适定性性质的几个问题，重点是非线性薛定谔方程、非线性波和科特韦格-德弗里斯方程。 该研究涵盖四个广泛的方向，这些方向通常是相互关联的，并且每个方向都有助于理解局部和全局解决方案的精确动态行为这一更广泛的主题。 前两个兴趣方向涉及解决方案的全球时间存在性，特别是首先关注适应能量超临界环境的全球适定性结果的发展，其次关注围绕端点和中的概率性全球适定性结果的重要问题。限制情况（在这个概率框架中，问题的初始数据被选择为随机傅立叶级数，并且通过排除小概率出现的集合来获得结果）。 第三个兴趣方向转向解的局部时间稳定性特性的研究，特别关注低正则性的初始数据。 第四个研究方向研究具有高阶非局部项的色散模型，其中一个关键因素是适应具有类似非局部特征的非线性椭圆和抛物型偏微分方程研究的最新进展。 在每种设置中，PI 和合作者都将结合谐波分析、概率和谱理论的技术来分析演化中涉及的动态特征。