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。更确切地说，研究者建议研究两组问题。第一组着重于分析超临界非线性波（NLW）和薛定谔（NLS）方程解的规律性。在过去的二十年里，在理解所谓的“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，其中临界性被理解为偏微分方程具有在时间上全局受控的量，该量具有与某个标度不变范数相同的规律性。然而，获得超临界方程的全局时间解仍然是一个具有挑战性的问题。这里的超临界方程是指守恒量低于标度不变范数。一个著名的例子是三维Navier-Stokes方程，它描述了粘性不可压缩流体的最基本性质。其他的例子包括在广义相对论中出现的各种非线性波动方程以及薛定谔方程。与她的合作者，研究人员提出了三个项目的工作，以获得超临界NLW和NLS的部分规律性结果的启发，在3D Navier-Stokes的背景下可用的类似结果。第二组问题集中在从多体量子动力学推导NLS相关的项目上。研究人员，与陈，建议进一步发展的工作，他们最近开始的柯西问题的Gross-Pitaevskii（GP）的层次，这是一个无限的系统耦合线性非齐次偏微分方程，出现在推导的NLS。GP层次描述了无限多相互作用玻色子的气体动力学，同时保留了色散PDE的一些特征。 基于这些分散的功能，研究者提出调查解决方案的GP层次结构，并说明，在某些情况下，GP可以使用分散PDEs.Suggested问题的方法进行研究，涉及重要的数学问题，如存在性和规律性的解决方案描述各种波动现象的PDE。例如，NLS及其与Korteweg-de-弗里斯和波动方程的组合已经被提出作为许多基本波动现象的模型。由于它们的物理意义，开发工具来理解这些非线性方程的解的行为是至关重要的，研究人员计划通过调整她早期关于流体运动方程（如描述粘性流体基本性质的Navier-Stokes方程）的一些工具来朝着这个方向工作。另一方面，研究人员计划继续她最近的工作，从许多身体玻色子系统的色散偏微分方程的推导有关的物理启发的问题。拟议的活动包含一个跨学科的方法，在这个意义上说，它有可能使色散偏微分方程方法的水平，许多机构的量子动力学，反之亦然。特别是，长期的目标是试图适应一些最近的进展，从分散的偏微分方程的许多身体系统，其中一个物理相关的问题，是超出了已知的数学方法。