Disproof of the 3D Navier-Stokes equations' global existence and smoothness of solutions - Applying a new analytical approach with and without external forces

The Navier-Stokes equations are used to describe the relationship between velocity, pressure and density of moving fluid. In the beginning of the 19th century, these partial differential equations for modeling fluid dynamics were discovered independently from each other by French and Irish physicists Claude-Louis Navier and George Gabriel Stokes. The main goal of this research paper is to discover whether globally defined and smooth solutions to the 3-dimensional Navier-Stokes partial differential equations could always exist or that the solutions to these PDEs could break down under certain circumstances. By using a unique and entirely analytical approach, it shows that under certain conditions, the solutions of the 3-dimensional Navier-Stokes equations will always blow up which provides a counterexample that disproves the global existence and smoothness of solutions for the 3D Navier-Stokes equations.