Harmonic Analysis Challenges in Nonlinear Dispersive Partial Differential Equations

本研究计划探讨一类称为非线性色散方程的方程的数学性质。这种类型的方程出现在几种物理现象的模型中，包括光在光纤中的传播、玻色-爱因斯坦凝聚、小振幅水波和等离子体中的波。尽管它们经常出现，但对这些方程解的行为的理解受到其数学复杂性的限制。该项目旨在扩展当前对这类重要方程的理论理解。培训该领域的初级研究人员和学生也是该项目的一个组成部分。要研究的问题是具有破缺对称性和/或非常系数的非线性色散偏微分方程。虽然其中一些方程是非常物理的，例如具有非零边界条件的三次五次非线性薛定谔方程，但研究人员已经仔细选择了其他方程，以突出我们对潜在线性传播子的数学理解中的某些缺陷。因此，这个项目的一个主要推力是解决与各种几何形状的线性传播子相关的谐波分析问题。从证明足以解决高维环面上小数据能量临界问题的Strichartz估计，到将限制猜想的最新进展引导到非常系数领域，这是解决平移不变设置之外的质量临界问题的关键一步。

This research project explores the mathematical properties of a class of equations known as nonlinear dispersive equations. Equations of this type arise in models of several physical phenomena, including propagation of light in optical fibers, Bose-Einstein condensates, small-amplitude water waves, and waves in plasmas. Despite their common occurrence, understanding of the behavior of solutions to such equations is limited by their mathematical complexity. This project aims to extend current theoretical understanding of this important class of equations. Training of junior researchers and students in this area of research is also an integrated part of the project.The problems to be investigated are nonlinear dispersive partial differential equations with broken symmetries and/or non-constant coefficients. While some of these equations are very physical, such as the cubic-quintic nonlinear Schrodinger equation with non-zero boundary conditions, others have been carefully selected by the investigator to highlight certain deficiencies in our mathematical understanding of the underlying linear propagator. A major thrust of this project is therefore to resolve questions in harmonic analysis related to the linear propagator in various geometries. These range from proving Strichartz estimates powerful enough to resolve the small data energy-critical problem on high-dimensional tori to shepherding recent progress on the restriction conjecture into the realm of non-constant coefficients, which is a key step toward resolving mass-critical problems outside the translation-invariant setting.