Hyperbolic Conservation Laws and 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.