Deterministic and probabilistic dynamics of nonlinear dispersive PDEs

非线性色散偏微分方程组是模拟波动现象的时间演化方程，如非线性波动方程和非线性薛定谔方程。它们普遍存在于物理和工程的各个分支中，如非线性光学、等离子体物理、水波和电信系统。它们的有效性得到了数值和实验证据的广泛认可和支持。一方面，非线性色散偏微分方程组的数学理论研究对应用科学具有重要意义，因为它为这些模型的验证和适用性提供了坚实的基础。另一方面，这一理论研究被证明对数学本身非常有价值。事实上，在过去的三十年里，非线性色散偏微分方程组提出了非常困难和有趣的挑战，推动了数学分析中许多新思想和新技术的发展。非线性色散偏微分方程解的丰富性的一个来源是每一类方程都有其自身的困难，因此需要专门的工具.本文的目的是从确定性和概率的角度用数学分析的方法来探索非线性色散偏微分方程组的动力学.在确定性背景下，这一建议集中于(I)构造一类非线性薛定谔方程的特解和(Ii)证明具有非常量涡度的等离子体物理方程的解的长期存在性。主要研究人员(PI)计划将偏微分方程组技术与调和分析和谱理论的工具相结合。在传统的(确定性)非线性发展方程研究中，目标是为所有初始数据构造给定偏微分方程组的解。然而，在应用中，人们往往满足于了解典型解决方案的行为，而忽略了罕见的病理行为。这一观点可以通过使用概率论而变得严谨，并在过去十年中取得了令人兴奋的发展。特别是，它使我们能够超越确定性分析的限制。这一建议的一个方面是从概率的角度研究非线性色散偏微分方程组的动力学。更具体地说，PI将通过结合概率论和非常活跃的奇异随机偏微分方程领域的思想和工具，专注于构建具有粗糙和随机初始数据的定义良好的动力学。

Nonlinear dispersive partial differential equations (PDEs) such as the nonlinear wave equations and the nonlinear Schrodinger equations, are time evolution equations modelling wave phenomena. They appear ubiquitously in various branches of physics and engineering such as nonlinear optics, plasma physics, water waves and telecommunication systems. Their validity is widely recognised and supported by numerical and experimental evidences.On the one hand, the mathematical theoretical research of nonlinear dispersive PDEs is important for applied sciences since it has provided solid foundations for the verification and applicability of these models. On the other hand, this theoretical research has proved to be very valuable for mathematics itself. Indeed, over the last thirty years, nonlinear dispersive PDEs have presented very difficult and interesting challenges, motivating the development of many new ideas and techniques in mathematical analysis. One of the sources of richness of nonlinear dispersive PDEs is that each subclass of equations poses its own difficulties, thus requiring the elaboration of specific tools.The aim of this proposal is to explore the dynamics of nonlinear dispersive PDEs using mathematical analysis from both deterministic and probabilistic points of view. In the deterministic setting, this proposal focuses on (i) constructing special solutions to a class of nonlinear Schrodinger equations and (ii) proving the long-time existence of solutions to an equation from plasma physics with non-constant vorticity. The principal investigator (PI) plans to combine PDE techniques with tools from harmonic analysis and spectral theory.In the traditional (deterministic) study of nonlinear evolution equations, one aims to construct solutions to a given PDE for all initial data. In applications, however, one is often content with understanding the behaviour of typical solutions, neglecting rare pathological behaviours. This point of view can be made rigorous by employing probability theory and has led to exciting developments over the last decade. In particular, it has allowed us to go beyond the limits of deterministic analysis. One aspect of this proposal is to investigate dynamics of nonlinear dispersive PDEs from a probabilistic point of view. More specifically, the PI will focus on constructing well-defined dynamics with rough and random initial data by incorporating ideas and tools from probability theory and the very active field of singular stochastic PDEs.