Linear and nonlinear problems in dispersive Partial Differential Equations

波的色散，即不同频率的波以不同速度传播的现象，是介质中波的标准。 非线性效应正是那些在这些波之间产生相互作用的效应。 本项目的主要目标是研究将这两种现象联合收割机结合起来的模型。 然而，很明显，这些问题的进一步进展需要更深入地了解纯色散效应;相应地，该项目的很大一部分是针对这一目标的。 特别是，首席研究员将专注于色散行为是复杂的情况下，无论是发生在有限范围内的区域或在一个非均匀介质。 虽然该项目的重点是简单的模型，这些体现了基本的障碍出现在更广泛的波动理论。 所有被考虑的模型都可以作为机械系统出现。 最近人们发现，至少对于有限的粒子系统，这对可能的行为施加了相当大的限制，远远超过了世纪发现的限制;然而，人们对这些限制的全部性质知之甚少。 该项目的一部分是展示（第一次）这样的限制方程建模无限系统的粒子，分布在一个无限的体积。该项目涉及几个主题的色散偏微分方程，包括线性和非线性。 哈茨估计以一种非常适合于非线性理论的方式封装了许多关于线性流的信息。 进一步研究这种估计的设置薛定谔方程紧流形将进行，最近的戏剧性进展，包括布尔甘和德米特的建设。 尺度临界非线性方程的处理不仅需要理解线性流的估计，而且还需要理解其中紧性的缺陷。 朝着这个方向，主要研究人员寻求获得质量临界配置文件分解模型与非常数系数，在这个方向上的一些初步成功的基础上。 现在转向真正的非线性问题，将在两种不同的无限体积环境中寻求非压缩结果。 回想一下，非压缩性是一般有限维哈密顿流的一个特殊性质，格罗莫夫曾说过，没有球能完全流入（辛）截面半径较小的圆柱体。 过去的工作一直局限于环面，这大大有助于发展有限维近似。 由于完全可积性的本质有很大的不同，Korteweg-de弗里斯方程的低正则性理论在环面上比在整条直线上得到了进一步的发展。 该项目旨在对该线的低正则性问题取得一些进展，建立在最近开发的Sobolev/Besov范数全局控制方法的基础上，该方法同时适用于两种几何形状。

Wave dispersion, the phenomenon of waves with differing frequencies traveling with different speeds, is the norm for waves in media. Nonlinear effects are precisely those that generate interactions between such waves. The principal objective of this project is to study models that combine both phenomena. However, it has become evident that further progress on these problems requires a deeper understanding of the purely dispersive effects; correspondingly, a significant fraction of the project is directed toward this goal. In particular, the principal investigator will focus on situations where the dispersive behavior is complicated either by taking place in a region of finite extent or in a heterogeneous medium. Although the project focuses on simple models, these embody fundamental hurdles appearing much more broadly in the theory of wave motion. All the models under consideration can arise as mechanical systems. It has recently been discovered that, at least for finite systems of particles, this places considerable restrictions on the possible behavior, far beyond those discovered in the nineteenth century; however, the full nature of these restrictions is poorly understood. Part of the project is to exhibit (for the first time) such restrictions for equations modeling infinite systems of particles, spread over an infinite volume.The project concerns several topics in dispersive partial differential equations, both linear and nonlinear. Strichartz estimates encapsulate much about the linear flow in a manner well-adapted to the nonlinear theory. Further study of such estimates in the setting of Schrodinger equations on compact manifolds will be undertaken, building on recent dramatic advances, including those of Bourgain and Demeter. The treatment of scaling-critical nonlinear equations requires one to understand not just estimates for the linear flow, but also the defects of compactness therein. Toward this direction, the principal investigator seeks to obtain mass-critical profile decompositions for models with nonconstant coefficients, building on some initial successes in this direction. Turning now to truly nonlinear questions, nonsqueezing results will be sought in two distinct infinite-volume settings. Recall that nonsqueezing is a peculiar property of general finite-dimensional Hamiltonian flows uncovered by Gromov saying that no ball can flow wholly into a cylinder whose (symplectic) cross-section has lesser radius. Past work has been restricted to tori, which aids significantly in the development of finite dimensional approximations. Due to substantial differences in the nature of complete integrability, the low-regularity theory of the Korteweg-de Vries equation is further advanced on the torus than on the whole line. The project seeks to make some inroads on the low-regularity problem for the line, building on a recently developed method for global control of Sobolev/Besov norms that works simultaneously in both geometries.