Qualitative analysis focused on some nonlinear systems

PI提出研究一些著名的非线性偏微分方程解的定性性质。这些方程自然产生于几何学、流体动力学、物理学、化学和生物学。一个是三维不可压Navier-Stokes方程在合理基准下解的整体稳定性。Navier-Stokes方程组是不可压缩流动动力学的指导系统。这一全球稳定性问题与三维纳维尔-斯托克斯方程能否从合理的初始数据发展出有限时间奇异性这一“千年”未决问题密切相关。许多物理现象都与流体流动密切相关。对不可压缩流体流动的理解无疑是第一步，也是最基本的一步。另一种是Hardy-Littlewood-Sobolev型，它出现在几何分析、泛函分析和量子物理问题中。PI提出研究轴对称的流体流动，并导出解的渐近估计。这是一种可能的方式来表明，对于一个轴对称的光滑的初始数据与紧支持的相应的解决方案是全球适定（因此发展没有奇异性）。本计画亦探讨其他研究轴对称Navier-Stokes方程及完整Navier-Stokes方程之局部与整体结构的方法，并寻求在相关研究领域的应用。对于Hardy-Littlewood-Sobolev型系统，解决方案的分类导致发现了数学和物理科学中著名的非线性偏微分和积分方程的新结构。Liouville型定理，以Lane-Emden猜想为特例，被广泛用于建立解的各种形式的先验估计。这是微分和积分方程或系统的非线性分析的核心。本质唯一性与某些物理束缚态及其量子化密切相关。导出了新的逐点或积分形式的内禀估计，并得到了一些关键积分的渐近估计，在本项目中起到了中心作用。在这两个问题上，PI强调寻找看似独立的现象的新的和内在的联系，并提供一些新的角度来解决问题。拟议的研究项目也积极参与研究生和青年数学家。并为他们提供了坚实的训练，在数学分析，物理建模和数值模拟。正在进行的非正式研讨会“非线性偏微分方程分析”也是该项目的一个重要组成部分，该研讨会每周举行一次，以培训研究生和初级教员。

The PI proposes to study qualitative properties of solutions to some well-known nonlinear partial differential equations. These equations arise naturally from geometry, fluid dynamics, physics, chemistry, and biology. One is about the global stability of solutions with reasonable datum to the three dimensional incompressible Navier-Stokes equations. The Navier-Stokes system of equations is the guiding system for the dynamics of the incompressible flows. This global stability problem is closely related to the `millennium' open question of whether the three dimensional Navier-Stokes equations can develop a finite time singularity from reasonable initial data. Many physical phenomena are heavily involved with fluid flows. The understanding of the incompressible fluid flows is certainly the first and the fundamental step. The other is the Hardy-Littlewood-Sobolev type which arises in geometric analysis, functional analysis, and in quantum physical problems.The PI proposes to work on fluid flows which are axisymmetric and to derive certain asymptotic estimate of the solutions. This is a possible way to show that for an axisymmetric smooth initial data with compact support the corresponding solution is globally well-posed (and thus develops no singularity). This project also explores other ways of studying the local and global structures of axisymmetric Navier-Stokes equations as well as the full Navier-Stokes equations and seek applications in related research fields. For the Hardy-Littlewood-Sobolev type systems, classification of solutions leads to discoveries of new structures of well-known nonlinear partial differential and integral equations arising from the mathematical and physical sciences. Liouville type theorems, with the Lane-Emden conjecture as a special case, are widely used for establishing various forms of a priori estimates for solutions. This is the core of nonlinear analysis of differential and integral equations or systems. The essential uniqueness are closely related to certain physical bound states and their quantization. Deriving new and intrinsic estimates point-wise or in integral forms, and getting asymptotic estimates of some key integral play central rules in this project. On both problems, the PI emphasizes on finding new and intrinsic connections of seemingly independent phenomena and on providing some new angles to the problems. The proposed research projects also actively involve graduate students and young mathematicians. And it provides a solid training for them in mathematical analysis, physical modeling and numerical simulation. The ongoing informal seminar `analysis of nonlinear partial differential equations' that meets weekly to train the graduate students and junior faculty members is also an important part of the project.