Research

My Interests

My interests are in fluid dynamics, and I am currently investigating self-similar solutions to the 2D Boussinesq equations with Voigt regularizations with Mihaela Ignatova and Peter Constantin. These equations are analagous to 3D axi-symmetric Euler, and the Voigt Regularization allows us to study the self-similar blow-up of these equations. These results will help shed light on blow-up scenarios for many well-known fluid equations.

My work at the University of Arkansas with Zachary Bradshaw was conducted through the lens of the Navier-Stokes partial differential equation system. We have published two papers and are preparing two more concerning the approximation and decay properties of solutions to Navier-Stokes. This work was done in the classes of weak Besov solutions and discretely self-similar local energy solutions.

Our results concern regularity in the far-field, decay properties for data in Lebesgue space for q>3, and approximation properties of solutions with Besov and Lorentz space data. In our separation results, we have shown a bound on the rate two, possibly distinct, solutions to 3D Navier-Stokes can separate, 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.