Hyperbolic Conservation Laws and Applications

双曲守恒律是一类描述各种现象的数学方程，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欧拉（1755）。然而，一些理论问题至今仍未解决。解中存在激波形式的不连续性，是分析这些方程困难的主要来源。本研究将寻求在理解大数据的解决方案和错误传播的情况下，当它发生在现实生活中的情况下，初始条件是不知道绝对精度的进步。此外，研究的主要部分将集中在描述道路网络上的交通流量的守恒定律。 将开发新的道路交叉口车辆流模型，这些模型是现实的，易于计算。特别是，这些模型将考虑可能溢出的队列沿着道路导致一个拥挤的十字路口。在第二阶段，PI将研究交通模式，如（i）全局最优，其中出发由中央规划器安排，以最小化所有驾驶员的全局成本，以及（ii）纳什均衡，其中每个驾驶员选择出发时间和到达目的地的路线，为了使他/她自己的个人成本最小化。结合交通数据收集和汽车自动驾驶方面的最新技术进步，发展有效的数学模型将为交通流量的预测和最佳控制迈出重要的一步。第一部分的研究旨在基本的进展理论的守恒定律和非线性波动方程。特别是，在最近的进展的基础上，PI将研究具有大的总变差和在有限时间内出现真空的解的整体存在性，用于等熵气体动力学。这项研究将依赖于一些新的方法。(i)在气体动力学的研究中，爆破的先验估计或例子将首先在某些类近似解中进行研究，这些近似解更容易描述，并抓住了真解的基本特征。(ii)在误差估计的分析中，将使用测地距离，只要标准的Sobolev规范不产生有用的信息。 在一些有趣的情况下，这些指标的具体形式是由最佳运输问题驱动的。 (iii)对于新的交通流模型在路口的道路，解决方案将被构造为一个压缩变换的不动点，由一个新版本的拉克斯公式定义。PI担任宾夕法尼亚州立大学新的跨学科数学中心主任。 它的主要目标是促进科学互动，扩大现代数学技术的使用和欣赏，作为各种学科的有效研究工具。 本建议中的一些专题，如交通流模型，自然是跨学科的，将纳入中心的活动。

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.