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 Kiessling 非齐次系统统计力学的基本策略是构造一个具有很少变量的渐近精确非线性 PDE 问题，该问题近似于具有绝大多数变量的原始线性问题。在该提案的第一部分中，我们处理此类未解决的两个问题，即微正则系综： 点涡旋； 对于经典的引力硬粒子。在这两个系统中，由于熵的非凹性，人们都遇到了技术问题。 我们提出了一种新策略，通过在高维参数空间中重新设置凹性来规避这些问题，原始问题通过某种投影从该空间中获得。 这一领域的进展将显着增进我们对无凸性非线性偏微分方程及其与统计力学关系的理解。在拟议项目的第二部分中，我们直接处理非线性偏微分方程系统的性质，这些性质在其与统计力学的关系中牢固确立。我们主要感兴趣的是解的对称性。我们最近构建了二维椭圆偏微分方程系统的尖锐等周估计，并将其与 Rellich-Pohozaev 类型的先验恒等式进行比较，从而首先获得解径向对称的条件，而不需要唯一性或最小化属性。我们希望继续这项研究并优化二维椭圆系统（例如 Ginzburg-Landau 和 Bennett 方程）的条件，将该技术扩展到更高维度并将其应用于 Thomas-Fermi 模型，最终将该技术扩展到抛物线输运方程，例如朗道-玻尔兹曼方程，尤其应该及时产生全局存在结果。 %%% 长寿命涡旋是湍流中普遍存在的特征，地球天气系统中的飓风和木星大气层中的大红斑就是两个突出的例子。 涡旋也作为超流体、超导体中的结构缺陷而出现。 除了具有科学意义外，准确了解此类涡流形成的条件也具有紧迫的技术和气象重要性。拟议项目的一个完整部分旨在为这一努力做出重大贡献。该数学框架由某些深深植根于统计力学科学学科的非线性偏微分方程组组成。在拟议项目的第一部分中将开发的技术将使我们能够在比迄今为止所处理的更为现实的条件下从统计力学中提取相关的微分方程。在第二部分中，我们将进一步开发我们最新的技术，该技术可以产生有关非线性方程解的对称性的定性和定量陈述。 在有利的情况下，这会显着降低某些方程的复杂性。除了涡流之外，我们的目标是将我们的技术应用于控制比以前处理的更真实的等离子体结构的问题，其中包括以下类别：带电粒子束，这在技术开发的各个分支中都受到了极大的关注； 具有基本天体物理学意义的恒星结构。最后，我们看到了将我们的技术扩展到等离子体动态输运方程组的可能性，这对于和平的热核能研究具有重要意义。 ***