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PDE的存在和规律性以及这些方程的推导。更确切地说，研究人员建议研究两组问题。第一组侧重于分析对超临界非线性波（NLW）和Schrodinger（NLS）方程的定期性。在过去的二十年中，在理解所谓的“批判”非线性PDE的全球解决方案的存在方面带来了许多进步，在这种意义上说，批判性是在PDE具有全球范围内数量的及时控制的意义上具有相同规律性的，该数量与一定的扩展不变规范具有相同的规律性。但是，获得超临界方程式的全球时间解决方案仍然是一个具有挑战性的问题。在这里，通过超临界方程式，我们的意思是保守数量的规律性低于缩放规范。一个著名的例子是3D Navier-Stokes方程，描述了粘性不可压缩流体的最基本特性。其他示例涉及各种非线性波方程，这些方程出现在一般相对论以及施罗宾格方程的背景下。调查员与她的合作者一起，提议研究三个项目，以获得超临界NLW和NLS的部分规律性结果，其灵感来自3D Navier-Stokes的相似结果。第二组问题的重点是与许多人体量子动力学的NLS衍生有关的项目。研究人员与Chen一起提议进一步开发他们最近在GROSS-PITAEVSKII（GP）层次结构的Cauchy问题上开始的工作，该层次结构是一种无限耦合的线性非均匀PDES系统，它出现在NLS的推导中。 GP层次结构描述了无限许多相互作用的玻色子气体的动力学，同时保留了分散PDE的某些特征。 基于这些分散特征，研究人员建议研究GP层次结构的解决方案，并说明，在某些情况下，可以使用分散PDE的方法进行概括来研究GP。解决问题的问题涉及重要的数学问题，例如存在描述各种波浪现象的PDES的存在和规律性的解决方案。例如，已经提出了NLS及其与Korteweg-de-vries和Wave方程的组合，作为许多基本波浪现象的模型。由于其身体意义，必须开发工具来了解这些非线性方程式的解决方案的行为，并计划通过调整她早期在流体运动方程（例如介绍粘性流体基本属性的Navier-Stokes方程）中朝着该方向朝着该方向工作的工具。另一方面，调查员计划继续她最近在许多身体玻色子系统的分散PDE上进行物理启发的问题。所提出的活动包含一种跨学科的方法，因为它有可能将分散PDE方法带到许多身体量子动力学水平，反之亦然。特别是，长期目标是尝试适应从分散PDE到许多身体系统的一些最新进展，在这些系统中，人们拥有与已知数学方法相关的物理相关问题。