Date of Award

8-16-2024

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Changhui Tan

Abstract

Macroscopic traffic flow models describe the evolution of a function ρ(t, x), which represents the traffic density at time t and location x according to a differential equation (typically a conservation law). Numerous models have been introduced over the years which capture the phenomenon of shock formation in which the solution develops a discontinuity. This presents difficulties from the standpoint of mathematical analysis, necessitating the consideration of weak solutions. At the same time, this undesirable mathematical behavior corresponds to unsafe driving conditions on real roadways, in which the heaviness of traffic may vary abruptly and dramatically. This thesis introduces and explores the behavior of new first and second order onlocal macroscopic traffic flow models in order to address this concern.

While several nonlocal models have been proposed throughout the years, the wellposedness of these models has primarily been studied in the context of the weak formulation. The primary contributions herein are the proposal and introduction of two classes of first and second order traffic flow models and a detailed analysis of these models which shows that their structure can give rise to global well-posedness (that is, existence and uniqueness of smooth classical solutions for all time). The nonconvexity of the flux and the structure of the nonlocal contribution distinguish these traffic flow models as unique within the context of existing literature.

By establishing regularity criteria for each model and studying their behavior in the phase plane, we will show that each model admits a class of nontrivial initial data for which the phenomenon of shock formation is suppressed.

Rights

© 2024, Thomas Joseph Hamori

Download

Included in

Mathematics Commons

Share

COinS