Qualitative analysis focused on some nonlinear systems

PI建议研究一些著名的非线性偏微分方程解的定性性质。这些方程式自然产生于几何学、流体力学、物理学、化学和生物学。其一是关于三维不可压N-S方程解的全局稳定性问题。Navier-Stokes方程组是不可压缩流动动力学的导引系统。这一全球稳定性问题与“千年”悬而未决的问题密切相关，即三维Navier-Stokes方程能否从合理的初始数据发展成有限时间奇点。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了解当然是第一步，也是最基本的一步。另一种是在几何分析、泛函分析和量子物理问题中出现的Hardy-Littlewood-Soblev类型。PI建议研究轴对称的流体流动，并推导出解的某些渐近估计。这是一种可能的方式来表明，对于具有紧支撑的轴对称光滑初始数据，相应的解是全局适定的(因此不会产生奇异性)。该项目还探索了研究轴对称Navier-Stokes方程以及完整的Navier-Stokes方程的局部和全局结构的其他方法，并寻求在相关研究领域中的应用。对于Hardy-Littlewood-Sobolev型系统，解的分类导致了数学和物理科学中著名的非线性偏微分方程组和积分方程组的新结构的发现。以Lane-Emden猜想为特例的Liouvile型定理被广泛用于建立解的各种形式的先验估计。这是微分方程组或系统的非线性分析的核心。这种本质的唯一性与某些物理束缚态及其量子化密切相关。逐点或以积分形式导出新的和固有的估计，并获得该项目中一些关键的积分发挥中心规则的渐近估计。在这两个问题上，PI强调寻找看似独立的现象之间的新的内在联系，并为问题提供一些新的视角。拟议的研究项目还积极吸引研究生和年轻数学家参与。并为他们在数学分析、物理建模和数值模拟方面提供了坚实的训练。正在进行的非正式研讨会“非线性偏微分方程的分析”每周举行一次，以培训研究生和初级教员，这也是该项目的一个重要部分。

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.