Research

My research focuses on fluid dynamics, particularly the study of self-similar solutions to the 2D Boussinesq equations with Voigt regularization. Working with Mihaela Ignatova and Peter Constantin, I explore how these equations—analogous to the 3D axi-symmetric Euler equations—can be analyzed through Voigt regularization to understand self-similar blow-up phenomena. Our findings aim to enhance our comprehension of blow-up scenarios across various well-known fluid dynamics equations.

At the University of Arkansas, my research with Zachary Bradshaw delved into the 3D Navier-Stokes partial differential equations. Together, we have published two papers and are preparing two more, focusing on the approximation and decay properties of Navier-Stokes solutions. Our work encompasses weak Besov solutions and discretely self-similar local energy solutions. We have investigated regularity in the far-field, decay properties of solutions in with data in subcritical Lebesgue spaces, and approximation characteristics with Besov and Lorentz space data. Notably, our research has established a bound on the rate at which two distinct solutions to the 3D Navier-Stokes equations can diverge locally.

Publications 

1. Bradshaw, Z. and Phelps, P. "Spatial decay of discretely self-similar solutions to the Navier-Stokes equations", Pure and Applied Analysis, Volume 5 Number 2, 2023. https://msp.org/paa/2023/5-2/p05.xhtml

2. Bradshaw, Z. and Phelps, P. "Estimation of non-uniqueness and short-time asymptotic expansions for Navier-Stokes flows", Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 2023. https://ems.press/journals/aihpc/articles/11101927 

Thesis

3. Phelps, P. "Asymptotic Properties and Separation Rates for Navier-Stokes Flows", April 2023.