Calculus of Variations and Hamilton-Jacobi Equations in the Wasserstein Space

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本课题旨在研究与流体力学和量子力学中某些基本偏微分方程系统相对应的Wasserstein概率测度空间上的一类动力系统。Wasserstein空间方法的一个很好的特性是它处理奇异数据和奇异解的能力。此外，还可以用相同的形式来处理离散模型和连续模型。该计划的核心在于在Wasserstein空间中发展适当的变分演算和Hamilton-Jacobi方程理论。在这个项目中，主要研究者将研究以下主题：在概率测度的Wasserstein空间的绝对连续路径集上定义的某些物理上有意义的拉格朗日量的作用泛函的最小化；Wasserstein空间中Hamilton-Jacobi方程黏性解的存在唯一性；以及这些Hamilton-Jacobi方程的齐次化。这些问题的一些最具挑战性的方面是在概率测度空间中缺乏局部紧性以及拉格朗日量和哈密顿量中存在强奇点。变分法和哈密顿-雅可比方程在包括物理学、经济学和工程学在内的许多科学领域发挥着核心作用。特别是，它们是经典力学研究的主要对象。期望本课题所研究的变分问题和Hamilton-Jacobi方程解的性质将为理解流体力学和量子力学中出现的一些无限维动力系统的动力学提供重要的一步。首席研究员的沃瑟斯坦形式主义方法也有望给出一个统一的框架，在这个框架中，粒子的经典和量子行为都可以被描述。该活动将与其他一些理论和应用数学学科，如最优质量运输、守恒定律的双曲系统、概率论和无限维最优控制理论等进行富有成效的互动。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project aims to study a class of dynamical systems on the Wasserstein space of probability measures corresponding to some fundamental systems of partial differential equations in fluid and quantum mechanics. A nice feature of the Wasserstein space approach is its ability to handle singular data and singular solutions. In addition, there is also a possibility of handling discrete and continuous models with the same formalism. The core of this program lies in the development of an appropriate theory for calculus of variations and Hamilton-Jacobi equations in the Wasserstein space. In this project, the principal investigator will investigate the following topics: minimizing the action functional for certain physically meaningful Lagrangians defined on the set of absolutely continuous paths in the Wasserstein space of probability measures; existence and uniqueness of viscosity solutions for Hamilton-Jacobi equations in the Wasserstein space; and homogenization of these Hamilton-Jacobi equations. Some of the most challenging aspects to these problems are the lack of local compactness in the space of probability measures and the presence of strong singularities in the Lagrangians and Hamiltonians.Calculus of variations and Hamilton-Jacobi equations play a central role in many areas of sciences including physics, economics, and engineering. In particular, they are the main object of study in classical mechanics. It is expected that the properties of solutions of the variational problems and the Hamilton-Jacobi equations to be studied in this project will provide a significant step in understanding the dynamics of some infinite-dimensional dynamical systems arising in fluid and quantum mechanics. The principal investigator's Wasserstein formalism approach is also expected to give a unifying framework in which both classical and quantum behavior of particles can be described. The proposed activity will provide a fruitful interaction with some other disciplines of theoretical and applied mathematics such as optimal mass transportation, hyperbolic systems of conservation laws, probability theory, and infinite-dimensional optimal control theory.