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. ***

小行星9623220 统计力学中的一个基本策略 非齐次系统的一个重要特征是构造一个渐近精确的 少变量非线性偏微分方程问题， 原始的线性问题，其压倒性的许多变量。在 建议的第一部分涉及两个尚未解决的问题， 这种，即微正则系综：点涡;经典引力硬粒子。在这两个系统中， 由于熵的不确定性遇到了技术问题。 我们提出了一个新的策略，通过以下方式来规避这些问题： 在高维参数空间中重新安装Rounds，原始问题通过某种投影从该空间获得。 这一领域的进展应能大大促进我们的理解 无凸性的非线性偏微分方程及其与统计力学的关系。在第二部分的建议项目中，我们直接处理系统的非线性偏微分方程，牢固地建立在统计力学的关系。我们的主要兴趣在于 解的对称性。我们最近建造了夏普 二维椭圆型偏微分方程组的等周估计， 将它们与Rellich-Pohozaev类型的先验恒等式进行比较，从而 首先得到解是径向对称的条件 而不需要唯一性或最小化属性。我们想 继续这项研究，并优化条件，二维 椭圆系统，如Ginzburg-Landau和班尼特方程， 将该技术应用于更高维度，并将其应用于Escherichas-Fermi模型，最后将该技术扩展到抛物型输运方程，如Landau-Boltzmann方程，特别是应该产生一个全局 存在是时间的结果。 %长寿命涡流是一种普遍存在的特征 在湍流中，在地球的天气系统中， 木星大气中的大红斑就是两个突出的例子。 在超流体、超导体中，涡旋也会作为结构缺陷出现。 除了具有科学意义外，它还具有紧迫的技术和 气象的重要性，以准确地了解条件下， 这样的漩涡形成。拟议项目目标的一个整体部分 为这一奋进做出重大贡献。数学 框架由某些系统的非线性偏微分 这些方程深深植根于静力学的科学学科中。第一部分应发展的技术 将允许我们提取相关的差异 在更现实的条件下， 比目前治疗的要多在第二部分中，我们将进一步发展我们最近的一种技术，该技术产生关于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，这降低了某些方程的复杂性 显著除了旋涡，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，以下类别：带电粒子束，这是 对技术开发的各个领域都有着浓厚的兴趣; 恒星结构，这是基本的天体物理学的兴趣。最后， 我们看到了将我们的技术扩展到一个动态的 等离子体的输运方程组，具有重要意义，特别是和平热核能源的研究。 ***