FRG: Collaborative Research: Applications of Transportation Theory to Nonlinear Dynamics

提案DMS-0354729PI：W.Gangbo(佐治亚理工学院)，L.Caffarelli(得克萨斯大学)，L.C.Evans(加州大学伯克利分校)，M.Feldman(威斯康星大学)，R.McCann(多伦多大学)标题：运输理论在非线性动力学中的应用本项目专注于分析一系列变分优化和动态演化问题，其中心是最优运输--每当进化保持局部标量时进入动态设置。核心问题可以用讽刺的方式来描述：鉴于铁矿在农村的分布，以及需要铁矿石的工厂的分布，决定哪些矿山应该向每个工厂供应铁矿石，以便将总运输成本降至最低。这里，从x的矿场到y的工厂的每吨矿石的运输成本由一个函数c(x，y)来表示-所以这个问题可以用一个线性规划来表示。然而，当矿山和工厂连续分布在欧几里德空间或带有障碍物的弯曲景观中-成本与该景观上的距离有关时，问题就具有丰富的结构和与几何和非线性PDE的深刻联系，而这些才刚刚开始探索。这个问题的化身嵌入到当前的模型中，产生了令人惊讶的各种现象。除了与最优映射的结构和质量特征有关的基本问题外，拟议的研究还涉及大气中锋面形成的模型、动力学理论中的耗散平衡、流体流动、颗粒物质以及几何和动力学不平等。在半个世纪的数学忽视之后，过去十年见证了对最优运输的兴趣的复苏，并见证了它发展成为一个肥沃的研究领域，以及一个探索数学内外各种应用的充满活力的工具。这一转变的发生部分是因为长期存在的问题终于可以得到解决，但也因为发现了意想不到的联系，将这些问题与物理、几何、计算机视觉、偏微分方程式、地球科学和经济学的问题联系起来。现在是时候在国际范围内开展合作，探索现有的联系并挖掘新的联系，同时发展最佳地图的基本理论，并向学生和同事介绍该领域的挑战和前景-从而形成一个具有这些目标的重点研究小组。我们计划的核心是安排小组成员之间和周围的持续互动，他们除了进行科学合作外，还将在未来几年共同努力，为交通研究的蓬勃发展创造必要的研究环境和人力。为了实现这一目标，我们计划举办两次研讨会，讨论这一主题的不同方面。此外，我们计划分担培养研究生和博士后研究员的责任，使用补助金中的资金支持年轻研究人员。这一独特的安排将使参与者有机会接触到异常广泛的视角和专业知识。此外，我们相信，年轻研究人员学习这一主题并参与其中的三年培养窗口-如果现在继续-最终将在该领域进一步推进十多年。

Proposal DMS-0354729PIs: W.Gangbo (Georgia Tech), L.Caffarelli (U Texas), L.C.Evans(U California Berkeley), M. Feldman (U Wisconsin), R.McCann (U. Toronto)Title: Applications of Transportation Theory to Nonlinear DynamicsABSTRACTThis project focuses on the analysis of a collection of variationaloptimization and dynamical evolution problems centered around the themeof optimal transportation --- which enters the dynamical settingwhenever the evolution conserves a scalar locally. The central problem can be caricatured as follows:Given a distribution of iron mines throughout the countryside, and a distribution of factories which require iron ore, decide which mines should supply ore to each factory in order to minimize the totaltransportation costs. Here the cost per ton of ore transported from themine at x to factory at y is specified by a function c(x,y) --- so the problem can be formulated as a linear program. However, when the mines andfactories are distributed continuously throughout Euclidean space ora curved landscape with obstacles --- and the cost is related to thedistance on this landscape, then the problem has a rich structure anddeep connections to geometry and non-linear PDE which have only begun to be explored. Incarnations of this problem embed in current models forsurprisingly diverse phenomena. Along with basic questions concerning the structure and qualitative features of optimal mappings, the proposedresearch addresses models for front formation in the atmosphere, dissipative equilibration in kinetic theory, fluid flow, and granular materials and geometric and dynamical inequalities.After half a century of mathematical neglect, the past decade witnessed a revival of interest in optimal transportation, and watched as it blossomed into a fertile field of investigation as well as a vibrant tool for exploring diverse applications within and beyond mathematics. The transformation occurred partly because long-standing issues could finally be resolved, but also because unexpected connections were discovered which linked these questions to problems in physics, geometry, computer vision, partial differential equations, earth science and economics. The time is ripe for a collaborative effort on aninternational scale to explore existing connections and unearth new ones, while simultaneously developing the basic theory of optimal maps and introducing students and colleagues to the challenges and promise of the field --- thus for the formation of a focused research group with these goals. The core of our plan is to arrange sustained interactions between and around members of the group, who in addition to collaborating scientifically, will work together over the next several years to create the research environment and manpower necessary for transportation research to flourish. To achieve this goal, we plan to organize two workshops on different aspects of the subject. Furthermore, we plan to share the responsibilities of training graduate students and postdoctoral fellows, by using funds from the grant to support young researchers. This unique arrangement will give participantsaccess to an unusually broad assortment of perspectives and expertise.Moreover, we believe a three-year nurturing window for young researchers to learn the subject and become involved --- if continued now --- will ultimately further advance progress in the field by more than a decade.