Graduate Studies
First Advisor
Petronela Radu
Degree Name
Doctor of Philosophy (Ph.D.)
Committee Members
George Avalos, Huijin Du, Mikil Foss, Florin Bobaru
Department
Mathematics
Date of this Version
7-2024
Document Type
Dissertation
Citation
A dissertation presented to the faculty of the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy
Major: Mathematics
Under the supervision of Professor Petronela Radu
Lincoln, Nebraska, July 2024
Abstract
Conservation laws are fundamental principles that play an important role in modeling various phenomena in physics, chemistry, and biology. However, their limitations, such as the development of shocks despite smooth initial conditions, are well known. The nonlocal model framework can be used to overcome these challenges. Nonlocal frameworks utilize integral operators that mimic differential operators but also incorporate long-range interactions within a finite horizon. This approach not only allows for non-smooth solutions, but also provides flexibility in modeling different phenomena. This study investigates the convergence of nonlocal divergence operators, defined with a general flux density function, to their classical counterparts. We show that under the assumption of differentiable flux density functions with Hölder continuous derivatives, the nonlocal divergence of a differentiable function converges to the classical divergence as the horizon parameter vanishes. Additionally, we analyze the convergence of solutions for nonlocal conservation laws to those of their classical equivalents.
Advisor: Petronela Radu
Recommended Citation
Vo, Anh Thuong, "Nonlocal Frameworks for Nonlinear Conservation Laws and Advection-Diffusion Processes" (2024). Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–. 187.
https://digitalcommons.unl.edu/dissunl/187
Comments
Copyright 2024, Anh Thuong Vo. Used by permission