Dissertation
Some existence results for nonlinear hyperbolic systems of balance laws arising in biology and traffic flow
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2026
DOI: 10.25820/etd.008472
Abstract
We investigate nonlinear hyperbolic systems of balance laws in one dimension that arise in biology and traffic flow models. More specifically, we rigorously prove existence results for three different systems motivated by attractive chemotaxis, traffic flow models with relaxation, and blood flow. This thesis is divided into three parts.
In the first part, we prove the global existence of a unique C1 solution for the Cauchy problem associated with a system of mixed-type originating from Keller-Segel models in attractive chemotaxis. We also provide sharp conditions for shock formation, illustrating the critical threshold phenomenon.
In the second part, we prove the existence of entropy weak solutions for the Cauchy problem of a non-strictly hyperbolic system arising in traffic flow models with relaxation terms. We use the vanishing viscosity method and compensated compactness arguments to show the existence of weak solutions and to establishthe existence and uniqueness of the zero-relaxation limit with large initial data.
In the last part, we show the existence of entropy weak solutions for the initial-boundary value problem of a non-strictly hyperbolic system arising in blood flow models. By using the theory of divergence measure fields, we formulate a suitable notion of weak solution for the problem with homogeneous Dirichlet conditions in the half-line. We prove the existence of entropy weak solutions when the initial data is rough, large, and contains vacuum.
Details
- Title: Subtitle
- Some existence results for nonlinear hyperbolic systems of balance laws arising in biology and traffic flow
- Creators
- José David Beltrán Lizarazo
- Contributors
- Tong Li (Advisor)Palle Jørgensen (Committee Member)Lihe Wang (Committee Member)Xiaoyi Zhang (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2026
- DOI
- 10.25820/etd.008472
- Publisher
- University of Iowa
- Number of pages
- ix, 71 pages
- Copyright
- Copyright 2026 José David Beltrán Lizarazo
- Language
- English
- Date submitted
- 04/08/2026
- Description illustrations
- Graphs, charts
- Description bibliographic
- Includes bibliographical references (pages 63-71).
- Public Abstract (ETD)
Systems of balance laws are systems of partial differential equations commonly used in the mathematical description of complex processes in diverse disciplines. One of the most representative features of these systems is the formation of singularities in finite time. In many situations classical solutions break down and become discontinuous, and the existence of solutions for these systems becomes an interesting and challenging problem.
In this thesis, we present new contributions towards the analysis of systems of balance laws arising in traffic flow and biology. More specifically, we prove the existence of solutions for systems motivated by models describing bacteria movement, traffic flow dynamics, and blood flow.
Our results illustrate the versatility of hyperbolic systems of balance laws to describe seemingly unrelated scenarios, and they also highlight some of the main obstacles that arise in the study of this field.
- Academic Unit
- Mathematics
- Record Identifier
- 9985176976202771
Metrics
1 File views/ downloads
1 Record Views