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

该建议支持的研究涉及非线性色散和波动方程的研究。 这些都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，发展数学工具来理解重要问题，如解的存在性和唯一性，以及相应演化在定性和定量水平上的行为，是一个具有根本科学意义的问题。 作为一个特殊的例子，在这项研究中要调查的几个问题涉及研究长期的非线性薛定谔方程和非线性波动方程的解决方案的性质，从“通用”随机选择的初始数据。 除了大量的数学兴趣，调查这些问题解决了更广泛的科学问题，是否可能出现的数学公式中的奇点可以发生在物理相关的设置。 此外，要研究的主题是密切相关的偏微分方程，概率和谐波分析，该研究项目的具体范围是研究有关局部和全局井的几个问题，非线性色散方程和波动方程的适定性，重点是非线性薛定谔方程、非线性波动方程和Korteweg-de弗里斯方程。 该研究包括四个广泛的方向，这些方向通常是相互关联的，每个方向都有助于理解局部和全球时间上解决方案的精确动力学行为。 感兴趣的前两个方向关注的全球时间存在的解决方案，特别是侧重于第一次发展的全球适定性的结果，以适应能源超临界设置，第二次的重要问题，周围的概率全球适定性的结果，在端点和限制的情况下（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，并且通过排除以小概率出现的集合来获得结果）。 感兴趣的第三个方向转向研究当地的时间稳定性的解决方案，特别是集中在初始数据的低正则性。 研究的第四个方向调查色散模型与高阶非局部项，其中一个关键成分将是适应最近的发展，在研究非线性椭圆和抛物偏微分方程具有类似的非局部功能。 在每一种情况下，PI和合作者将结合谐波分析，概率和谱理论的技术来分析演变中涉及的动力学特征。

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.