Calculus of Variations and Evolutive Systems on the Wasserstein Space

这个项目的重点是分析与动力和机械系统有关的一系列变分问题。特别是，它寻求开发研究变分演算和Monge-Kantorovich理论的基本工具。在发现变分问题领域的新结果的过程中，它将有希望发现与其他科学和数学领域的新联系。在该项目中，考虑了由有限多个粒子组成并具有潜在泊松结构的哈密顿系统。当粒子的数量变得无限时，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。许多工作都致力于确定这种有限无限维系统的泊松结构。波恩和因菲尔德，以及独立的泡利，开始发展一种量子场论，其中换向子算子类似于切尔诺夫、马斯登、温斯坦和许多其他人研究的泊松括号。在与合作者的共同工作中，首席研究员检查了没有电场或磁场的物理系统。在这个简化模型中，他们得到了许多严谨的结果，可以用来处理一类包含奇异测度的偏微分方程。由首席研究员和其他人开发的一些概念在表述问题时很有用，例如与约束Navier-Stokes方程相关的相干结构的形成，该方程最近被Caglioti， Pulvirenti和Rousset所考虑。该项目将研究这些方程及其对不可压缩流体的二维欧拉方程的影响。这个项目背后的想法不难解释。考虑一个物理系统，该系统由有限维环面（想象一个甜甜圈的表面）上演化的有限多个粒子组成，并假设施加于该系统的力来自周期势。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，周期轨道和不变环面可以被明确地描述。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，人们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等价于Hamilton-Jacobi方程理论中出现的“单元问题”的解的存在。后一种解的图告诉我们什么是“好的”初始条件。KAM理论确定了被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（比如两次可微）。“弱”KAM理论考虑了一个更大的旋转向量类，并且对于它们中的每一个都建立了不太光滑的单元问题解的存在性（即，它们只是“Lipschitz”函数）。首席研究员计划继续他的研究，在极限系统中，粒子的数量变得无限。他预计他的研究将为我们对偏微分方程稳定性问题的理解带来新的曙光。该项目将特别关注学生的培养和大学数学的推广。

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.