Hyperbolic Conservation Laws and Applications

双曲守恒定律及其应用

• 批准号: 1411786
• 负责人: Alberto Bressan
• 金额: $ 31.51万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411786&HistoricalAwards=false
• 关键词: Hyperbolic; Conservation; Laws; Applications

Hyperbolic conservation laws are a class of mathematical equations describing a wide variety of phenomena, including gas dynamics, water waves, liquid crystals, and vehicle flow. Their study dates back to Euler (1755). However, several theoretical problems remain unresolved to this day. The presence of discontinuities in the solutions, in the form of shock waves, is a major source of difficulties in the analysis of these equations. The present research will seek advances in the understanding of solutions with large data, and error propagation in case when, as it happens in real-life situations, the initial conditions are not known with absolute precision. In addition, a major portion of the research will focus on conservation laws describing traffic flow on a network of roads. New models for vehicle flow at road intersections will be developed, which are realistic and easily computable. In particular, these models will account for the possible spill-back of queues along roads leading to a congested intersection. At a second stage, the PI will study traffic patterns arising as(i) global optima, where departures are scheduled by a central planner in order to minimize a global cost to all drivers, and (ii) Nash equilibria, where each driver selects a departure time and a route to destination, in order to minimize his/her own individual cost.In connection with recent technological advances in the collection of traffic data and the automatic driving of cars, developing efficient mathematical models will provide an important step toward the prediction and the optimal control of traffic flow. The first part of the research aims at fundamental advances in the theory of conservation laws and nonlinear wave equations. In particular, building upon recent progress, the PI will study the global existence of solutions with large total variation and the appearance of vacuum in finite time, for isentropic gas dynamics. The research will rely on a number of new approaches.(i) In the study of gas dynamics, a priori estimates or examples of blow-up will first be studied within certain classes of approximate solutions, which are easier to describe and that capture the essential features of true solutions. (ii) In the analysis of error estimates, geodesic distances will be used, whenever the standard Sobolev norms do not yield useful information. In some interesting cases, the specific form of these metrics is motivated by optimal transportation problems. (iii) For the new models of traffic flow at a junction of roads, a solution will be constructed as the fixed point of a contractive transformation, defined by a novel version of the Lax formula.The PI serves as Director of a new Center for Interdisciplinary Mathematics at Penn State. Its primary goal is to foster scientific interactions, widening the use and appreciation of modern mathematical techniques as effective research tools in a variety of disciplines. Some of the topics of the present proposal, such as traffic flow models, are naturally interdisciplinary and will be included in the activities of the Center.

双曲线保护定律是描述各种现象的一类数学方程，包括气体动力学，水波，液晶和车辆流动。他们的研究可以追溯到Euler（1755）。但是，直到今天，几个理论问题仍未解决。在冲击波的形式中，解决方案中的不连续性是对这些方程式分析的主要困难来源。本研究将寻求在对解决方案的理解方面的进步，并且在现实生活中发生的情况下，最初的条件是绝对精确的。此外，研究的很大一部分将集中于描述道路网络上交通流量的保护法。 将开发出道路交叉路口的车辆流动的新型号，这些模型是现实且易于计算的。特别是，这些模型将解释导致交叉路口的道路上可能溢出的排队。在第二阶段，PI将研究以（i）全球优化的方式出现的交通模式，其中中央计划者计划的出发人员为了最大程度地减少所有驾驶员的全球成本，以及（ii）NASH均衡，每个驾驶员都选择一个出发时间和到达目的地的途径，以使自己/她/她自己的个人成本的范围最小化，并在驱动器上的数字范围内，并且在驱动器上的数学范围以及众所周知的数据范围，并且在收集方面的数据范围，并且在收集方面的效果，并且在收集方面的收集范围众所周知，该型号的范围众所周知，该技术的范围众所周知，该杂志的收集范围众所周知，该数据的范围众所将为预测和对交通流量的最佳控制提供重要的一步。研究的第一部分旨在旨在在保护法理论和非线性波方程理论方面的基本进步。特别是，在最近的进步基础上，PI将研究具有较大总变化的溶液的全球存在，并且在有限时间内，对于等质气体动力学而言，真空的出现。该研究将依靠多种新方法。（i）在对气体动力学的研究中，先验估计或爆炸的示例将首先在某些类似的近似解决方案中进行研究，这些解决方案易于描述，并捕获真正解决方案的基本特征。 （ii）在分析误差估计值时，只要标准SOBOLEV规范未产生有用的信息，就会使用地球距离。 在某些有趣的情况下，这些指标的具体形式是由最佳运输问题激发的。 （iii）对于道路交界处的新交通流量模型，解决方案将被构造为缩水式转换的固定点，这是由新颖版本的LAX公式定义的。 它的主要目标是促进科学互动，扩大和欣赏现代数学技术作为各种学科的有效研究工具。 本提案的某些主题（例如交通流模型）自然是跨学科的，将包括在中心的活动中。