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

提案DMS-0354729 PI：W.Gangbo（格鲁吉亚理工学院）、L.Caffarelli（德克萨斯大学）、L.C.Evans（加州伯克利大学）、M.费尔德曼（U威斯康星州），R。多伦多）标题：运输理论在非线性动力学中的应用摘要本项目的重点是分析一系列以最优运输为主题的变分优化和动态演化问题--当演化局部保持标量时，最优运输就进入动态环境。中心问题可以用漫画的形式描述如下：给定铁矿在农村的分布，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，以使总运输成本最小化。 这里，每吨矿石从x点的矿山运输到y点的工厂的成本由函数c（x，y）确定，所以这个问题可以用线性规划来表示。 然而，当矿山和工厂连续分布在欧氏空间或弯曲的景观与障碍物-和成本有关的距离在这个景观，那么这个问题有一个丰富的结构和深刻的联系，几何和非线性偏微分方程，才刚刚开始探索。 这一问题的具体表现体现在当前各种现象的模型中。 沿着关于最优映射的结构和定性特征的基本问题，提出的研究解决了大气中锋面形成的模型，动力学理论中的耗散平衡，流体流动和颗粒材料以及几何和动力学不等式。经过半个世纪的数学忽视，过去十年见证了对最优运输的兴趣的复兴，并看着它发展成为一个肥沃的调查领域，以及一个充满活力的工具，探索内外数学的各种应用。这种转变的发生部分是因为长期存在的问题终于得到了解决，但也因为人们发现了意想不到的联系，将这些问题与物理学，几何学，计算机视觉，偏微分方程，地球科学和经济学中的问题联系起来。 时机已经成熟，在国际范围内合作努力，探索现有的连接和挖掘新的，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，他们除了科学合作外，还将在未来几年内共同努力，创造交通研究蓬勃发展所需的研究环境和人力。 为了实现这一目标，我们计划就这一主题的不同方面举办两个讲习班。 此外，我们计划通过使用赠款的资金来支持年轻研究人员，从而分担培养研究生和博士后研究员的责任。 这一独特的安排将使参与者能够接触到异常广泛的观点和专业知识。此外，我们相信，一个为期三年的培养窗口，让年轻研究人员学习这一主题并参与其中--如果现在继续下去--最终将进一步推动该领域十多年的进步。

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.