Hyperbolic Conservation Laws and Applications

双曲守恒定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，到目前为止，仍有几个理论问题尚未解决。在这些方程的分析中，以激波形式存在的解的不连续性是一个主要的困难来源。本研究将寻求在理解大量数据的解决方案方面取得进展，并在实际情况下，当初始条件不是绝对精确已知的情况下，寻求错误传播。此外，研究的主要部分将集中在描述公路网上的交通流量的保护法上。将开发新的道路交叉口车流模型，该模型具有逼真和易于计算的特点。特别是，这些模型将考虑到通往拥堵十字路口的道路上可能出现的排队溢出。在第二阶段，PI将研究如下交通模式：(I)全局最优，其中发车由中央计划者调度，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，以最小化他/她自己的个人成本。结合最近在交通数据收集和汽车自动驾驶方面的技术进步，开发高效的数学模型将为预测和最优控制交通流量提供重要的一步。研究的第一部分旨在研究守恒定律和非线性波动方程理论的基本进展。特别是，在最新进展的基础上，PI将研究等熵气体动力学的整体解的存在性和有限时间内真空的出现。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，首先将在某些类近似解中研究爆破的先验估计或实例，这些类近似解更容易描述，并捕捉到真实解的本质特征。(2)在误差估计的分析中，只要标准索博列夫准则不能提供有用的信息，就会使用测地线距离。在一些有趣的情况下，这些指标的具体形式是由最优运输问题驱动的。(Iii)对于道路交界处的新交通流模型，将构造一个解作为收缩变换的不动点，由新版本的Lax公式定义。PI担任宾夕法尼亚州立大学一个新的跨学科数学中心的主任。它的主要目标是促进科学互动，扩大现代数学技术在各种学科中作为有效研究工具的使用和欣赏。本提案的一些专题，如交通流量模型，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.