Zach practicing presenting his main theorem

Author: Zachary Akridge | Major: Math and Physics | Semester: Spring 2024

My name is Zachary Akridge. I am a Junior in the Fulbright College of Arts and Sciences majoring in math and physics. My mentor is Dr. Zachary Bradshaw, an associate professor of mathematics. My semester of research was Spring 2024. Next year I plan to continue this work with Dr. Bradshaw. After I graduate I would like to attend graduate school with the goal of getting a PhD in mathematics.

The dynamics of the ocean and the atmosphere can, in certain settings, be understood through the use of surface temperature as a conserved quantity. The evolution of the surface temperature over time and space is modeled using the surface quasi-geostrophic equations (SQG). This model has applications in forecasting of the atmosphere and oceans at large length scales. They are of mathematical interest as toy models for the incompressible Euler equations and the Navier-Stokes equations, the latter of which is the subject of one of the famously unsolved Millenium problems. The purpose of my research is to study these equations through the lens of frequency sparseness, which quantifies which length scales dominate in the activity of the solution.

My work thus far has been to show that under the assumption of sufficiently sparse surface temperature functions cause the norm of the function to shrink. This simple proposition has many implications such as the extension of solutions to larger time intervals. The main theorem, which is pictured, shows that solutions of SQG which cannot be extended to larger time intervals necessarily have large activity on the lower frequencies. Further work on this project is still needed however as there are many restrictions and there is still lots of exciting work to do. Currently I am working on extending these methods to work on the uniqueness problem for the SQG equation. For the Navier-Stokes equations it is known that if a certain relationship holds between the low frequencies and the high frequencies, then the equations have a unique solution. It is expected that this holds for the SQG equation as well and is what I am currently investigating.

I am extremely grateful to Dr. Bradshaw for continuing to guide me through this project. I first got interested in non linear PDEs as a freshman while talking to a professor about the Navier-Stokes equations, and he recommended that I ask Dr. Bradshaw about his research and if he had any projects I could work on. He did and after one year of studying the prerequisite material, I’ve been working on extending his work the the Navier-Stokes equations to the SQG equation. We’ve met once a week for the past year and these meetings have been extremely helpful for when I get stuck on a specific problem. My work this semester culminated in a talk I gave at the MAA sectional meeting which was hosted here at the University of Arkansas for which my talk was selected as one of the best three presentations out twenty four that were given.

I am also very grateful to receive this grant as it allows me to focus more on research and not have to work a job while doing so. I am also grateful for the recognition that comes with it as after graduating I would like to attend graduate school for mathematics and this looks great on my resume!