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

本提案所支持的研究涉及非线性色散方程和波动方程的研究。它们作为各种物理系统的基本模型出现，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诸如解的存在性和唯一性等重要问题，以及在定性和定量水平上相应的演化行为，是一个具有基础科学重要性的问题。作为一个特殊的例子，本研究中要研究的几个问题涉及研究非线性薛定谔方程和非线性波动方程从“一般”随机选择的初始数据演化的解的长时间性质。除了具有重要的数学意义外，对这些问题的研究还解决了一个更广泛的科学问题，即数学公式中出现的可能的奇点是否会在物理相关的环境中发生。此外，所研究的主题与偏微分方程，概率和谐波分析中的广泛问题密切相关，所开发的思想和技术将为未来相关问题的研究提供重要的数学工具。本研究项目的具体范围是研究非线性色散方程和波动方程的局部和全局适定性问题，重点研究非线性薛定谔方程、非线性波方程和Korteweg-de Vries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，并且每个方向都有助于更广泛的主题，即理解局部和全局解决方案的精确动态行为。前两个方向关注解的全局时间存在性，特别是首先关注适应能量超临界环境的全局适定性结果的发展，其次关注围绕端点和极限情况下的概率全局适定性结果的重要问题（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）。并通过排除以小概率出现的集合得到结果)。第三个兴趣方向转向解的局部时间稳定性性质的研究，特别关注低正则性的初始数据。第四个研究方向是研究具有高阶非局部项的色散模型，其中一个关键因素将是适应具有类似非局部特征的非线性椭圆型和抛物型偏微分方程的最新研究进展。在这些设置中，PI和合作者将结合谐波分析，概率和频谱理论的技术来分析演化中涉及的动态特征。

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.