Dissertation
Study on systems of nonlinear conservation laws arising in chemotaxis and traffic flow
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2023
DOI: 10.25820/etd.007098
Abstract
We study nonlinear conservation laws in partial differential equations (PDEs). In particular, we investigate systems of conservation laws in biology and traffic flow. We solve the Riemann problem for a system modeling chemotaxis and prove the existence of global BV solutions to the Cauchy problem for a system of balance laws arising in traffic flow.
For our first problem, we study the Riemann problem for a system arising in chemotaxis. The system is of mixed-type and transitions from a hyperbolic to an elliptic region. We solve the Riemann problem in the physically relevant region up to the non-strictly boundary that occurs between the hyperbolic and elliptic regions. While solving this problem, we encounter classical shock and rarefaction waves in the hyperbolic region as well as contact discontinuities in the linearly degenerate region.
For the second problem, we establish global well-posedness and asymptotic behavior of BV solutions to a system of balance laws modeling traffic flow with nonconcave fundamental diagram. This problem is of specific interest since nonconcave fundamental diagrams arise naturally in traffic flow. We prove the results for the system with concave fundamental diagram by finding a convex entropy-entropy flux pair and verifying the Kawashima condition, the sub-characteristic condition, and the partial dissipative inequality in the framework of Dafermos. We then extend the results to nonconcave fundamental diagram by perturbation analysis.
Details
- Title: Subtitle
- Study on systems of nonlinear conservation laws arising in chemotaxis and traffic flow
- Creators
- Nitesh Mathur
- Contributors
- Tong Li (Advisor)Lihe Wang (Committee Member)Xiaoyi Zhang (Committee Member)Xueyu Zhu (Committee Member)Palle Jorgensen (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2023
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.007098
- Number of pages
- xi, 59 pages
- Copyright
- Copyright 2023 Nitesh Mathur
- Language
- English
- Date submitted
- 04/19/2023
- Date approved
- 04/20/2023
- Description illustrations
- graphs
- Description bibliographic
- Includes bibliographical references (pages 55-59).
- Public Abstract (ETD)
- We study problems in the field of nonlinear conservation laws, an important branch of partial differential equations (PDEs) in mathematics. PDEs are widely studied since they are useful in modeling and describing physical phenomenon found in nature. In this thesis, we analyze solutions to specific PDEs problems with applications in chemotaxis (biology) and traffic flow. The first problem we study is the mathematical theory of chemotaxis, which is made up of two phrases – ‘chemo’ meaning ‘chemicals’ and ‘taxis’ meaning the ‘movement of’. Chemotaxis is the movement of organisms due to chemical response. Examples of chemotaxis include movement of bacteria toward areas of higher food molecules and cell migration in angiogenesis. The Keller-Segel model most famously described chemotaxis into a mathematical model. We study a modified version of the Keller-Segel model and solve a problem that describes how organisms behave as the chemical concentration changes. The second problem we study is on a traffic flow model. Traffic flow is based on the interactions between vehicles, drivers, and roadways as well as other factors like nonlinear dynamics and human behavior. In our model, the acceleration factor consists of relaxation to the static equilibrium speed-density relation and an anticipation factor which expresses the effect of drivers reacting to conditions downstream. Under these circumstances, we solve an open problem and study how traffic flow behaves as time evolves.
- Academic Unit
- Mathematics
- Record Identifier
- 9984425200102771
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