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

Robert Lloyd Jackson

doi:10.4236/ojfd.2022.121005

Abstract

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.

Highlights

The motion of viscous fluids is depicted through Euler and Navier-Stokes equations (NSEs) [1]
Where vector field v is the velocity profile of the incompressible homogenous fluid, vector field f is an external force, function p is the internal pressure, coefficient ρ is the density of the fluid, and coefficient v is the kinematic viscosity of the fluid
If an external force is lacking in the Navier-Stokes (system of) equations (NSEs), the velocity vector field v and the internal pressure p are both smooth and continuous or C∞ functions

Summary

Introduction

The motion of viscous fluids is depicted through Euler and Navier-Stokes equations (NSEs) [1]. The educational organization wanted to know whether solutions existed for certain conditions (i.e., the kinetic energy of a fluid substance should never exceed a particular value) It wanted to know if these solutions maintained their smoothness and continuous nature at all points under an infinite number of differentiations. It sees if the exact solutions satisfied some of the criteria, such as smoothness and existence, established by the Clay Mathematics Institute in subsection 4.2.

A Short Synopsis on the NSEs and Its Relevant Vector Fields

Application of GFT

Derivation of Solutions for the NSEs with and without an External Force

The Assessment of Smoothness and the Existence of Both Sets of Solutions

Conclusion