Linear and nonlinear problems in dispersive Partial Differential Equations

波的频散，即不同频率的波以不同的速度传播的现象，是波在介质中传播的常态。非线性效应正是在这种波之间产生相互作用的那些效应。这个项目的主要目标是研究结合了这两种现象的模型。然而，很明显，在这些问题上取得进一步进展需要更深入地了解纯粹的分散效应；相应地，该项目的很大一部分是针对这一目标的。特别是，首席研究人员将专注于色散行为复杂的情况，这些情况要么发生在有限范围的区域中，要么发生在非均匀介质中。尽管该项目专注于简单的模型，但这些模型体现了波动理论中出现的更广泛的基本障碍。所有正在考虑的模型都可以作为机械系统出现。最近发现，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，远远超出了19世纪所发现的限制；然而，人们对这些限制的全部性质知之甚少。这个项目的一部分是(第一次)展示对无限大的粒子系统建模的方程的这种限制。该项目涉及几个关于色散偏微分方程组的主题，包括线性和非线性。Strichartz估计以一种很好地适应非线性理论的方式概括了许多关于线性流的信息。在紧致流形上薛定谔方程的背景下，这种估计的进一步研究将建立在最近的显著进展的基础上，包括Bourain和Demeter的进展。在处理尺度临界非线性方程时，不仅需要了解对线性流的估计，还需要了解其中紧致性的缺陷。在这个方向上，主要研究人员试图在这个方向的一些初步成功的基础上，获得具有非恒定系数的模型的质量临界剖面分解。现在转到真正的非线性问题，非压缩结果将在两个不同的无限大体积设置中寻找。回想一下，非压缩是格罗莫夫发现的一般有限维哈密顿流的一种特殊性质，他说没有任何球可以完全流入(辛)截面半径较小的圆柱体。过去的工作一直局限于环面，这对有限维近似的发展有很大帮助。由于完全可积性的性质有很大的不同，Korteweg-de Vries方程的低正则理论在环面上比在整条直线上进一步发展。该项目寻求在这条线路的低正则性问题上取得一些进展，建立在最近开发的一种全局控制Sobolev/Besov范数的方法上，该方法同时在两个几何图形上工作。

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.