New methods for the study of supercritical wave equations

在这个项目中，主要研究人员将研究色散临界偏微分方程和超临界偏微分方程组，进一步发展最近引入的几种方法。偏微分方程提供了许多重要的物理、生态和经济理论的严格的数学描述(例如，爱因斯坦的广义相对论，金融中期权定价的布莱克-斯科尔斯模型)。对它们进行分析和数值研究，以确定解的长期行为及其对初始数据的依赖。在许多情况下，有物理意义的概念，如能量、质量和动量，扮演着重要的角色，因为很好地控制这些量可以使人证明解在时间上是全局存在的，并研究它们的行为。然而，研究这些量不足以支配解的方程(称为能量超临界方程)是非常有意义的。这类高度非线性的方程包括许多著名和重要的物理现象的模拟(例如，爱因斯坦方程、纳维斯托克斯方程、杨-米尔斯方程和欧拉方程)。该项目的目标是尽可能地了解超临界类型波动方程的“大”解的行为，目前对这种解知之甚少。主要研究者对这一课题的潜在贡献基于两种方法：(1)将解分成传入和传出部分；(2)波动方程的比较原理。他的初步结果只适用于一些(典型的)模型案例，但似乎很有希望。将它们推广到非径向情况，所有的维度，以及其他类型的非线性和方程，将是很有趣的。该项目的一个目标是采用专门研究椭圆型和抛物型方程的技术(比较原理、下解和上解)，并将它们以有意义的方式应用于超临界波动方程。作者的方法不同于Kenig和Merle以及其他人(陶渊明、Krieger和Schlag、Li、Wang和Yu)的方法。它导致了无条件的结果，并在几种情况下导致了尖锐的分散标准。另一种研究方法是由含时势驱动的线性和非线性发展方程。势可以是确定性的或随机的(例如，布朗运动)。这在孤子稳定性的研究和一些物理问题上都有明显的应用。主要研究人员介绍的方法，如波算符的结构公式和使用抽象的维纳定理来证明色散估计，通过允许势的更一般的时间依赖性，提供了一种看待此类方程的改进方法，并允许得出更强有力的结论。

In this project the principal investigator will study dispersive critical and supercritical partial differential equations, further developing several recently introduced methods. Partial differential equations provide a rigorous mathematical description of many important physical, ecological, and economical theories (e.g., Einstein's theory of general relativity, the Black-Scholes model of option pricing in finance). They are studied both analytically and numerically in order to determine the long-term behavior of solutions and their dependence on initial data. In many cases, physically meaningful concepts, such as energy, mass, and momentum, play important roles, since having good control of these quantities may enable one to prove that solutions exist globally in time and to study their behavior. However, it is of great interest to study equations (called energy-supercritical) for which such quantities are insufficient to govern the solution. This class of highly nonlinear equations includes many famous and important ones that arise from modeling physical phenomena (e.g., Einstein's equations, the Navier-Stokes equation, the Yang-Mills equations, and Euler's equations).The goal of the project is to understand, as far as possible, the behavior of "large" solutions to wave equations of supercrticial type, about which little is currently known. The principal investigator's potential contribution to the subject is based on two methods: (1) splitting solutions into incoming and outgoing parts and (2) a comparison principle for the wave equation. His preliminary results apply only to some (canonical) model cases, but seem promising. It will be interesting to generalize them to the nonradial case, to all dimensions, and to other types of nonlinearities and equations. One objective of the project is to take techniques (comparison principles, sub- and supersolutions) specific to the study of elliptic and parabolic equations and apply them in a meaningful way to the supercritical wave equation. The author's approach is different from those of Kenig and Merle and of others (Tao, Krieger and Schlag, Li, Wang and Yu). It leads to unconditional results and to sharp scattering criteria in several cases. Another avenue of investigation is that of linear and nonlinear evolution equations driven by time-dependent potentials. The potential can be deterministic or random (e.g., Brownian motion). This has clear applications to the study of soliton stability and some physical problems. The methods that the principal investigator has introduced, such as a structure formula for wave operators and the use of an abstract Wiener theorem to prove dispersive estimates, provide an improved way of looking at such equations, by allowing a more general time dependence of the potential, and permit stronger conclusions to be reached.