The Utilization of the Generating Function Technique in the Discovery of Solutions for the Three-Dimensional Navier-Stokes Equation System

The derivation of solutions to the Navier-Stokes (system of) equations (NSEs), in three spatial dimensions, has been an enigma as time can tell. This study wishes to show how to eradicate this problem via the usage of a recently proposed method for solving partial differential equations called the Generating Function Technique, or GFT for short. The paper will first quickly define the NSEs with and without an external force, then provide a quick synopsis of the GFT. Next, the study will derive solutions to these two major problems and give an analysis of the data concerning a specific set of criteria established by the Clay Mathematics Institute to determine the smoothness and existence of solutions. Results via GFT will show one can easily prove the existence of solutions to the NSEs with or without the presence of an external force. However, only the solutions to the NSEs will be globally bound.