Calculus of Variations and Evolutive Systems on the Wasserstein Space

Wasserstein 空间上的变分和演化系统微积分

• 批准号: 0901070
• 负责人: Andrzej Swiech
• 金额: --
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901070&HistoricalAwards=false
• 关键词: Calculus; Variations; Evolutive; Systems; Wasserstein

This project focuses on the analysis of a collection of variational problems in connection with dynamical and mechanical systems. In particular, it seeks to develop basic tools for studying the calculus of variations and the Monge-Kantorovich theory. In the process of discovering new results in the variational problem realm, it will hopefully unearth new connections with other areas of science and mathematics. In the project, Hamiltonian systems that consist of finitely many particles and possess underlying Poisson structures are considered. When the number of particles becomes infinite, these finite dimensional systems may converge in an appropriate sense to infinite dimensional systems that are encoded as partial differential equations. Much work has been devoted to identifying the Poisson structures for such limiting infinite dimensional systems. Born and Infeld, and independently Pauli, started to develop a quantum field theory in which the commutator operator is analogous to the Poisson bracket studied by Chernoff, Marsden, Weinstein, and many others. In joint work with collaborators, the principal investigator has examined physical systems with no electric or magnetic fields. In this simplified model, they obtained many rigorous results that can be used to handle a class of partial differential equations involving singular measures. Some of the concepts developed by the principal investigator and others are useful in formulating problems such as the formation of coherent structures in connection with the constrained Navier-Stokes equations that has been considered recently by Caglioti, Pulvirenti, and Rousset. The project will investigate these equations and their implications for the two-dimensional Euler equations of incompressible fluids.The ideas that underlie this project are not difficult to explain. Consider a physical system that consists of finitely many particles evolving on a finite-dimensional torus (think of the surface of a doughnut) and assume that the forces applied to the system are derived from a periodic potential. One of the central issues in dynamical system is the search for periodic orbits and so-called invariant measures. In the simple case where there is no force, the periodic orbits and invariant tori can be described explicitly. The celebrated KAM (Kolmogorov-Arnold-Moser) theory ensures that, if the potential is small, then for certain initial conditions one can describe the solutions of the system explicitly in conveniently chosen new coordinates. It is well-known that the existence of these suitable new coordinates is equivalent to the existence of solutions of a "cell problem" that arises in the theory of Hamilton-Jacobi equations. The graph of the latter solution tells one what the "good" initial conditions are. The KAM theory identifies parameters, called rotation vectors, for which smooth solutions (twice-differentiable, say) of the cell problem exist. The "weak" KAM theory considers a larger class of rotation vectors and for each one of them establishes the existence of solutions of the cell problem that are not quite smooth (i.e., that are only "Lipschitz" functions). The principal investigator plans to continue his investigation on the limiting systems where the number of particles becomes infinite. He anticipates his study will shed new light on our understanding of stability issues for partial differential equations. The project will pay special attention to the training of students and the promotion of mathematics in colleges.

这个项目的重点是分析一系列与动力和机械系统有关的变分问题。特别是，它旨在开发研究变分法和蒙格-康托洛维奇理论的基本工具。在发现变分问题领域的新结果的过程中，它将有望挖掘出与其他科学和数学领域的新联系。在该项目中，考虑了由多个粒子组成的具有潜在泊松结构的哈密顿系统。 当粒子的数量变成无穷大时，这些有限维系统可以在适当的意义上收敛到编码为偏微分方程的无限维系统。许多工作一直致力于确定这种限制无限维系统的泊松结构。 玻恩和因菲尔德，并独立泡利，开始制定一个量子场论，其中的换向器运营商是类似的泊松括号研究的巴夫，马斯登，温斯坦，和许多其他人。在与合作者的联合工作中，首席研究员研究了没有电场或磁场的物理系统。在这个简化模型中，他们得到了许多严格的结果，可以用来处理一类涉及奇异测度的偏微分方程。一些概念开发的主要研究者和其他人是有用的制定问题，如形成的相干结构与约束的Navier-Stokes方程，最近已被认为是由Caglioti，Pulvirenti，和Cagliset。本计画将探讨这些方程式及其对二维不可压缩流体之欧勒方程式之意涵，其基本思想不难解释。 考虑一个由有限多个粒子组成的物理系统，该系统在有限维环面上演化（想象一下甜甜圈的表面），并假设施加到系统上的力源自周期势。动力系统的中心问题之一是寻找周期轨道和所谓的不变测度。在没有力的简单情况下，可以明确地描述周期轨道和不变环面。 著名的KAM（Kolmogorov-Arnold-Moser）理论保证，如果势很小，那么对于某些初始条件，可以在方便选择的新坐标中明确地描述系统的解。众所周知，这些合适的新坐标的存在性等价于在Hamilton-Jacobi方程理论中出现的“胞腔问题”的解的存在性。 后一个解的图形告诉我们什么是“好的”初始条件。 KAM理论确定了称为旋转向量的参数，对于这些参数，存在细胞问题的光滑解（例如，二次可微）。“弱”KAM理论考虑了更大类的旋转向量，并且对于它们中的每一个，建立了不太光滑的单元问题的解的存在性（即，这些函数都是“Lipschitz”函数。首席研究员计划继续研究粒子数变为无穷大的极限系统。他预计他的研究将为我们理解偏微分方程的稳定性问题提供新的思路。该项目将特别注意学生的培训和在大学推广数学。