Graduation Semester and Year
Spring 2026
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics
First Advisor
Yue Liu
Second Advisor
Tuncay Aktosun
Third Advisor
Hristo Kojouharov
Fourth Advisor
Ren-Cang Li
Abstract
We consider a two-dimensional shallow water model equation in the context of full water waves. From the incompressible and irrotational governing equations in the three-dimensional water waves, we show an equation arises in the modeling of the propagation of shallow water waves over a flat bed. This two-dimensional model contains the weakly transverse behavior associated with the Kadomtsev-Petviashvili (KP) equation but captures a wave breaking property which exists in the b-family of shallow water wave equations. This new asymptotic model called the b-KP family of equations, is analogous to the two-dimensional Green-Naghdi equations that model the propagation of waves in a two-dimensional shallow water.
In this thesis, we derive the two-dimensional asymptotic model of the b-KP family of equations from the two-dimensional Green-Naghdi equations using the Camassa Holm regime. Then, we verify the Green-Naghdi equations are wellposed through an energy estimate argument. Furthermore, a rigorous justification showing the solutions of the Green-Naghdi equations tend to the associated solution of the b-KP family of equations under specific boundary conditions.
Keywords
shallow water model, b-family of equations, Green-Naghdi equations, Kadomtsev-Petviashvili equation
Disciplines
Analysis | Partial Differential Equations
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Martinez, Christina M., "On An Asymptotic Two-Dimensional Shallow Water Model" (2026). Mathematics Dissertations-Archive. 272.
https://mavmatrix.uta.edu/math_dissertations/272