Mathematics

In the 1950's John Nash showed through use of an iterative scheme of approximate embedding maps that the sphere  could be isometrically embedded into a ball of any radius by a  map. In the 1980's M. Gromov generalized Nash's work to the h-principal and convex integration. Recent research in fluid dynamics has used analogs to convex integration schemes to demonstrate nonuniqueness of weak solutions to both Euler and Navier-Stokes equations. In this paper we examine recent work by DeLellis and Sz\'ekelyhidi Jr. as well as  Buckmaster and Vicol on the nonuniqueness of  Fluid Dynamics equations. Specifically we focus on solution to Reynolds Stress system which  act as approximate solutions to Euler and Navier-Stokes equation. We also develop basic properties of the Dirichlet kernel used in the construction of intermittent Beltrami flows.