Mathematical Sciences: Nonlinear Partial Differential Equations and Statistical Physics

数学科学：非线性偏微分方程和统计物理

• 批准号: 9623220
• 负责人: Michael Kiessling
• 金额: $ 6万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623220&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Partial; Differential

9623220 Kiessling A basic strategy in statistical mechanics of nonhomogeneous systems is to construct an asymptotically exact nonlinear PDE problem with few variables which approximates the original linear problem with its overwhelmingly many variables. In the first part of the proposal we deal with two unsolved problems of this kind, namely the microcanonical ensemble for: point vortices; for classical gravitating hard particles. In both systems one has encountered technical problems due to nonconcavity of the entropy. We propose a new strategy that circumvents these problems by reinstalling concavity in a higher-dimensional parameter space from which the original problem obtains by some kind of projection. Progress in this area should significantly advance our understanding of nonlinear PDE without convexity and their relation to statistical mechanics. In the second part of the proposed project we deal directly with properties of systems of nonlinear PDEs that are firmly established in their relationship to statistical mechanics. Our main interest is in the symmetry properties of solutions. We recently constructed sharp isoperimetric estimates for two-dimensional elliptic PDE systems and compared them to a priori identities of Rellich-Pohozaev type, thus firstly obtaining conditions under which solutions are radially symmetric without requiring uniqueness or minimizing properties. We want to continue this research and optimize the conditions for two-dimensional elliptic systems such as Ginzburg-Landau and Bennett equations, extend the technique to higher dimensions and apply it to Thomas-Fermi models, finally extend the technique to parabolic transport equations such as Landau-Boltzmann equations, which in particular should yield a global existence result in time. %%% Long-lived vortices are an ubiquitous feature in turbulent flows, with hurricanes in the Earth's weather system and the great red spot in Jupiter's atmosphere being two promin ent examples. Vortices also occur as structural defects in superfluids, superconductors. Beside being of scientific interest, it is of pressing technical and meteorological importance to understand precisely the conditions under which such vortices do form. An integer part of the proposed project aims at making a significant contribution to this endeavor. The mathematical framework consists of certain systems of nonlinear partial differential equations which are deeply rooted in the scientific discipline of statis- tical mechanics. The techniques which shall be developed in the first part of the proposed project will allow us to extract the relevant differential equations from statistical mechanics under far more realistic conditions than treated so far. In the second part, we shall further develop a recent technique of us that yields qualitative and quantitative statements about the symmetry properties of the solutions to the nonlinear equations. In favorable cases this reduces the complexity of certain equations significantly. Beside vortices, we aim at applying our techniques to the problem of controlling more realistic plasma structures than previously treated, of the following categories: charged particle beams, which are of preeminent interest in various branches of technology development; stellar structures, which are of basic astrophysical interest. Finally, we see the possibility of an extension of our techniques to a dynamical set of transport equations for plasmas which have significance, in particular, for peaceful thermonuclear energy research. ***

9623220非均匀系统的统计力学中的基本策略是构建一个渐近确切的非线性PDE问题，几乎没有变量，该变量近似于原始的线性问题，其绝大多数变量。在提案的第一部分中，我们处理了这类两个未解决的问题，即：点涡流的微型合奏； 用于经典的引力硬颗粒。在这两个系统中，由于熵的非洞穴，人们都遇到了技术问题。 我们提出了一种新的策略，通过重新安装凹度在高维参数空间中，从中通过某种投影从中获得的凹陷来解决这些问题。 该领域的进展应大大提高我们对非线性PDE的理解，而无需凸度及其与统计力学的关系。在拟议项目的第二部分中，我们直接处理在与统计力学关系中牢固确定的非线性PDE系统的属性。我们的主要兴趣是解决方案的对称特性。我们最近为二维椭圆形PDE系统构建了尖锐的等值估计，并将其与Rellich-Pohozaev类型的先验身份进行了比较，因此首先获得了溶液在不需要唯一性或最小化特性的情况下获得溶液对称的条件。我们希望继续进行这项研究，并优化二维椭圆系统的条件，例如Ginzburg-Landau和Bennett方程，将技术扩展到更高的维度并将其应用于Thomas-Fermi模型，最后将技术扩展到诸如Landau-Boltzmann方程，尤其是在全球存在中产生的抛物线运输方程，尤其是Landau-Boltzmann方程。 %% %%长寿命是动荡流中无处不在的特征，地球天气系统中的飓风和木星大气中的大红色点是两个突出的例子。 涡旋也作为超流体，超导体的结构缺陷出现。 除了具有科学兴趣之外，要精确理解这种涡流形成的条件是紧迫的技术和气象重要性。拟议项目的整数部分旨在为这项工作做出重大贡献。该数学框架由某些非线性偏微分方程的系统组成，这些方程式根源在统计学的科学学科中。在拟议项目的第一部分中应开发的技术将使我们能够从统计力学下从统计机制中提取相关的微分方程，而不是迄今为止处理的。在第二部分中，我们将进一步开发一种美国最新技术，该技术就非线性方程解决方案的对称性产生定性和定量陈述。 在有利的情况下，这大大降低了某些方程式的复杂性。除了涡旋外，我们旨在将技术应用于控制更现实的等离子体结构的问题，而不是以前的类别：带电的粒子梁，这对技术开发的各个分支都具有极大的兴趣； 恒星结构，具有基本的天体物理兴趣。最后，我们看到将技术扩展到具有重要性的动态传输方程的可能性，特别是对和平的热核能源研究具有重要意义。 ***