Current Project

Nonunique solutions to Navier-Stokes: My current (masters) project is focused on nonuniqueness of weak solutions to the Navier-Stokes equation under the supervision of Dr. Enrique Thomann. The objective is the project is to gain a understanding of modern analytic techniques in weak form solutions to Navier-Stokes through such devices as highly oscilitory Beltrami flows.

Past Projects

Stokes's Theorem and Differential Forms: The purpose of this project was to explore differential forms as a consquence of stokes thoerem. Specifically Stokes says, $$ \int_{\partial \Omega} w = \int_{\Omega} dw $$ where \(w\) is an \(n-\)form and \(d\) is the exterior derivative. Dividing through by vol\(( \Omega)\) and taking the limit as the volume decreases we would expect from Lebesgue that $$\lim_{vol( \Omega ) \rightarrow 0} \frac{1}{vol ( \Omega )} \int_{\partial \Omega} w = d w, $$ leading to a new characterization of the differential form. All of the work of this project was tackled from a path integral point of view with minor diversions to homology. This project was supervised by Dr. Steven Gubkin as my Senior Thesis at Cleveland State University.