Mathematical Sciences: Nonlinear Partial Differential Equations and Statistical Physics

非齐次系统统计力学中的一个基本策略是构造一个具有少量变量的渐近精确非线性偏微分方程问题，该问题近似于具有绝大多数变量的原始线性问题。在第一部分中，我们处理了这类问题的两个未解决的问题，即点涡的微正则系综；对于经典的引力硬粒子。在这两个系统中，由于熵的非凹性，都遇到了技术问题。我们提出了一种新的策略，通过在高维参数空间中重新安装凹性，从而绕过这些问题，原始问题通过某种投影得到。这一领域的进展将极大地促进我们对非凸非线性偏微分方程及其与统计力学的关系的理解。在计划的第二部分，我们直接处理非线性偏微分方程系统的性质，这些性质牢固地建立在它们与统计力学的关系中。我们主要感兴趣的是解的对称性。我们最近构造了二维椭圆型PDE系统的尖锐等周估计，并将其与Rellich-Pohozaev型先验恒等式进行了比较，从而首次获得了解是径向对称的条件，而不需要唯一性或最小化性质。我们希望继续这项研究，优化二维椭圆系统（如Ginzburg-Landau方程和Bennett方程）的条件，将该技术扩展到更高的维度并应用于Thomas-Fermi模型，最后将该技术扩展到抛物输运方程（如Landau-Boltzmann方程），特别是应该在时间上产生全局存在性结果。长期存在的漩涡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，涡旋也作为结构缺陷出现。除了具有科学意义外，精确地了解这种涡旋形成的条件在技术和气象方面也具有紧迫的重要性。拟议项目的一个完整部分旨在为这一努力作出重大贡献。数学框架由若干非线性偏微分方程组组成，这些方程组深深植根于统计力学这一科学学科。将在拟议项目的第一部分中发展的技术将使我们能够在比迄今为止处理的更为现实的条件下从统计力学中提取有关的微分方程。在第二部分中，我们将进一步发展我们最近的一项技术，该技术对非线性方程解的对称性性质给出定性和定量的表述。在有利的情况下，这大大降低了某些方程的复杂性。除了涡旋，我们的目标是将我们的技术应用于控制比以前处理过的更现实的等离子体结构的问题，包括以下类别：带电粒子束，这在技术发展的各个分支中都是非常感兴趣的；恒星结构，这是基本的天体物理学兴趣。最后，我们看到了将我们的技术扩展到等离子体的一组动力学输运方程的可能性，这对和平的热核能研究具有重要意义。＊＊＊

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. ***