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

研究者计划研究非线性色散偏微分方程解的存在性和正则性，以及这些方程的推导。更准确地说，这位研究人员建议研究两组问题。第一组集中于分析超临界非线性波(NLW)和薛定谔(NLS)方程解的正则性。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整体存在性方面取得了许多进展，其中临界性被理解为：一个偏微分方程解的全局控制量与某个标度不变范数具有相同的规律性。然而，在时间上获得超临界方程的全局解仍然是一个具有挑战性的问题。这里的超临界方程指的是守恒量比标度不变范数具有更低的正则性。一个著名的例子是3D Navier-Stokes方程，它描述了粘性不可压缩流体的最基本性质。其他例子包括在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。这位研究人员与她的合作者建议开展三个项目，以获得超临界NLW和NLS的部分正则性结果，灵感来自3D Navier-Stokes背景下的类似结果。第二组问题集中在与从多体量子动力学推导NLS有关的项目上。这位研究人员和Chen一起，建议进一步发展他们最近开始的关于Gross-Pitaevskii(GP)族的Cauchy问题的工作，GP是一个由耦合的线性非齐次偏微分方程组组成的无限系统，出现在NLS的推导中。GP层次描述了由无限多个相互作用的玻色子组成的气体的动力学，同时保留了色散PDE的一些特征。基于这些色散特征，研究者建议研究GP族的解，并说明在某些情况下，GP可以用色散偏微分方程解的推广来研究。所建议的问题涉及重要的数学问题，例如描述各种波动现象的偏微分方程解的存在性和正则性。例如，NLS及其与Korteweg-de-Vries和波动方程的组合已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，开发工具来了解这些非线性方程的解的行为是必不可少的，研究人员计划将她早期在流体运动方程(如描述粘性流体基本性质的Navier-Stokes方程)方面的一些工具应用到色散方程的背景下，从而朝着这个方向工作。另一方面，这位研究人员计划继续她最近关于物理启发问题的工作，这些问题与从许多身体玻色子系统派生色散偏微子有关。拟议的活动包含了一种跨学科的方法，因为它有可能将分散的PDE方法带到多体量子动力学的水平上，反之亦然。特别是，长期目标是试图将最近的一些进展从分散的PDE应用到许多身体系统中，在这些系统中，一个人有一些物理相关的问题，这些问题超出了已知的数学方法的范围。

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.