Linear and nonlinear problems in dispersive Partial Differential Equations

色散偏微分方程中的线性和非线性问题

• 批准号: 1600942
• 负责人: Rowan Killip
• 金额: $ 24万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600942&HistoricalAwards=false
• 关键词: Linear; nonlinear; problems; dispersive; Partial

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.

波的频散，即不同频率的波以不同速度传播的现象，是波在介质中的常态。非线性效应正是那些在这些波之间产生相互作用的效应。该项目的主要目标是研究结合这两种现象的模型。然而，很明显，在这些问题上取得进一步进展需要对纯粹的色散效应有更深的了解；相应地，项目的很大一部分是朝着这个目标进行的。特别地，主要研究者将关注在有限范围内或在异质介质中发生的色散行为复杂的情况。虽然该项目侧重于简单的模型，但这些模型体现了在波动理论中出现的更广泛的基本障碍。所考虑的所有模型都可以产生为机械系统。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。项目的一部分是（第一次）展示这样的限制方程的粒子的无限系统，分布在一个无限的体积。该项目涉及色散偏微分方程中的几个主题，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封装了很多关于线性流的内容。在包括布尔甘和德墨忒尔在内的最近重大进展的基础上，将进一步研究在紧形流形上设置薛定谔方程的这种估计。尺度临界非线性方程的处理不仅需要了解线性流的估计，而且需要了解其紧致性的缺陷。在这个方向上，主要研究者寻求获得非常系数模型的质量临界剖面分解，建立在这个方向上的一些初步成功。现在转向真正的非线性问题，非挤压结果将在两个不同的无限体积设置中寻求。回想一下，非挤压是一般有限维哈密顿流的一个特殊性质，Gromov说没有球可以完全流入一个（辛）截面半径较小的圆柱体。过去的工作仅限于环面，这在有限维近似的发展中有重要的帮助。由于完全可积性性质的本质差异，Korteweg-de Vries方程的低正则性理论在环面上比在全线上得到了进一步的发展。该项目试图在线路的低正则性问题上取得一些进展，建立在最近开发的Sobolev/Besov规范全局控制方法的基础上，该方法同时适用于两种几何形状。