From many body quantum dynamics to nonlinear dispersive PDEs, and back

从许多体量子动力学到非线性色散偏微分方程，然后返回

• 批准号: 1101192
• 负责人: Natasa Pavlovic
• 金额: $ 19.85万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101192&HistoricalAwards=false
• 关键词: many; body; quantum; dynamics; nonlinear

The investigator plans to study existence and regularity of nonlinear dispersive PDEs as well as derivation of these equations. More precisely, the investigator proposes to study two groups of problems. The first group focuses on analyzing regularity of solutions to the super-critical nonlinear wave (NLW) and Schrodinger (NLS) equations. The last two decades brought numerous advances in understanding global existence of solutions to the so called "critical" nonlinear PDEs, where criticality is understood in the sense that a PDE possesses a quantity globally controlled in time which has the same regularity as a certain scaling invariant norm. However obtaining global in time solutions to super-critical equations remains a challenging problem. Here by a super-critical equation we mean that the conserved quantities are at lower regularities than the scaling invariant norm. A famous example is the 3D Navier-Stokes equations that describe the most fundamental properties of viscous incompressible fluids. Other examples involve various nonlinear wave equations that appear in the context of general relativity as well as Schrodinger equations. With her collaborators, the investigator proposes to work on three projects towards obtaining partial regularity results for super-critical NLW and NLS inspired by similar results available in the context of the 3D Navier-Stokes. The second group of problems focuses on projects related to derivation of the NLS from many body quantum dynamics. The investigator, together with Chen, proposes to develop further the work that they recently started on the Cauchy problem for the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDEs, that appear in the derivation of the NLS. The GP hierarchy describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE. Based on these dispersive features, the investigator proposes to investigate solutions to the GP hierarchy and illustrate that, in some instances, the GP can be studied using generalizations of methods of dispersive PDEs.Suggested problems involve important mathematical questions such as existence and regularity of solutions to PDEs that describe various wave phenomena. For instance, the NLS and their combinations with the Korteweg-de-Vries and wave equations have been proposed as models for many basic wave phenomena. Due to their physical significance, it is essential to develop tools to understand behavior of solutions to these nonlinear equations and the investigator plans to work in that direction via adapting some tools from her earlier work on equations of fluid motion (such as Navier-Stokes equations that describe fundamental properties of viscous fluids) to the context of dispersive equations. On the other hand, the investigator plans to continue her recent work on physically inspired questions related to derivation of dispersive PDEs from many body Boson systems. The proposed activity contains an interdisciplinary approach in the sense that it has potential to bring dispersive PDE methods to the level of many body quantum dynamics and vise versa. In particular, the long term goal is to try to adapt some of the recent advances from dispersive PDEs to the many body systems, where one has physically relevant questions that are beyond the reach of known mathematical methods.

研究人员计划研究非线性色散偏微分方程的存在性和规律性以及这些方程的推导。更准确地说，研究者建议研究两组问题。第一组重点分析超临界非线性波（NLW）和薛定谔（NLS）方程解的规律性。过去二十年在理解所谓“临界”非线性偏微分方程解的全局存在性方面取得了许多进展，其中临界性的理解是偏微分方程拥有一个全局控制的时间量，该量与某个标度不变量具有相同的规律性规范。然而，获得超临界方程的全局及时解仍然是一个具有挑战性的问题。这里，通过超临界方程，我们的意思是守恒量的规律性低于标度不变范数。一个著名的例子是 3D 纳维-斯托克斯方程，它描述了粘性不可压缩流体最基本的特性。其他例子涉及广义相对论背景下出现的各种非线性波动方程以及薛定谔方程。研究人员建议与她的合作者一起开展三个项目，以获得超临界 NLW 和 NLS 的部分正则性结果，其灵感来自于 3D Navier-Stokes 背景下的类似结果。第二组问题集中于与从许多体量子动力学推导 NLS 相关的项目。研究人员与陈一起建议进一步发展他们最近开始的关于 Gross-Pitaevskii (GP) 层次结构的柯西问题的工作，该问题是一个耦合线性非齐次偏微分方程的无限系统，出现在国家LS。 GP 层次结构描述了无限多个相互作用的玻色子的气体动力学，同时保留了色散偏微分方程的一些特征。 基于这些色散特征，研究者建议研究 GP 层次结构的解，并说明在某些情况下，可以使用色散 PDE 方法的推广来研究 GP。提出的问题涉及重要的数学问题，例如解的存在性和规律性描述各种波动现象的偏微分方程。例如，NLS 及其与 Korteweg-de-Vries 和波动方程的组合已被提议作为许多基本波动现象的模型。由于它们的物理意义，有必要开发工具来理解这些非线性方程解的行为，研究人员计划通过采用她早期流体运动方程（例如纳维-斯托克斯方程）工作中的一些工具来朝这个方向工作。描述粘性流体的基本属性）到色散方程的背景。另一方面，研究人员计划继续她最近对与从许多体玻色子系统推导色散偏微分方程有关的物理启发问题的研究。拟议的活动包含一种跨学科方法，因为它有可能将色散偏微分方程方法带到许多体量子动力学的水平，反之亦然。特别是，长期目标是尝试将色散偏微分方程的一些最新进展应用于许多身体系统，在这些系统中，人们遇到的物理相关问题超出了已知数学方法的范围。