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 for 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. Finally, the paper will provide a brief discussion of the results.

The purpose of this study is to: 1.) find solutions to the NSEs with or without an external force; and 2.) see if the exact solutions satisfied some of the criteria established by the Clay Mathematics Institute. The generating function technique (GFT) is used in finding exact solutions to the NSEs.

A short synopsis on the NSEs and its relevant vector fields.
The NSEs are a group of nonlinear PDEs involving vector fields [1,3]. The three equations are defined as follows: respectively. Focus of the above equations will be limited to the following: (2.6) , and (2.7) .

Application of GFT.
GFT is a method for solving [non]linear PDEs via the usage of a general solution ug that is comprised of [truncated] Laurent series sets of combinatorial number or trigonometric-based generating functions [4]. First, it requires an individual to determine the maximal and minimal power degree, ps, through which the Laurent series is cut short. Then, one must solve a linear auxiliary/characteristic ordinary differential equation to obtain a function  that is used in the transformed general solution Ug for the principle nonlinear PDE that is in question. The transformed general solution Ug can take the following form: where the expression Sk(0) is the square root of the k-th Fibonacci number at/about zero, or

Finding, then describing the solutions to the NSEs (via GFT).
Mathematica® was used to derive solutions to the NSEs with and without an external force. Mathematica® spreadsheets for the two sets of problems were provided in the supplementary data of this study. 4 First, an individual must solve a particular order homogeneous auxiliary/characteristic ordinary differential equation like the following:

Derivation of solutions for the NSEs with and without an external force.
whose solution  is given as: Next, one determines the maximal and minimal power for truncation ps, which is equal to the value of 1. Then using equation (3.1), (s)he establishes the transformed general solutions for the internal pressure P and the vector fields and , respectively: Note: the index j was limited to -1, all bij and dij coefficients were set to the value of zero to limit the computational cost for deriving the putative exact solutions. Also, parenthesis was used for indexing some coefficient. Finally, the external force involved a 3-dimensional Gaussian diffusion process.
The transformed NSEs with external force is defined as follows:  (4.1.13) .
After plugging in the transformed general solutions into the above equations, one extracts a total of thirtyseven algebraic equations associated with . Then (s)he uses the algebraic equations to solve for constants whenever possible. One set of solved constants yields the following results for the velocity vector field and internal pressure p:  To ascertain the smoothness of the solutions, one must determine if the solutions are functions.
A function is differentiable for all degrees of differentiation. Since this type of functions and all its derivatives lack corners, it is considered smooth. Also, this type of function is considered continuous due to it and all of its derivatives do not possess any abrupt/discontinuous jumps/drops in value to infinity in both directions. As one can see, both sets of velocity vector fields and the internal pressures p are linear combinations of functions (i.e. e +ax ) concerning time and 3D-space; therefore, they are likely functions themselves. To determine whether the set of solutions exists as defined by the Clay Mathematics Institute an individual must assess whether kinetic energy for a scenario is globally bound. This kinetic energy is less than or equal to some constant E and is the right side of the following equation: Where the magnitude of the velocity vector field is defined as: The kinetic energy for the NSEs without external force is zero while the kinetic energy for the NSEs with external force is either nonintegrable or can be represented by the following expression if the constant c1 is set to zero: (Note: if the constants   and  are imaginary numbers, the right side of the above expression is likely to grow exponentially become greater than the constant E at some point in time.) This data suggests the kinetic energy for the NSEs without external force is globally bound. In contrast, the kinetic energy for the NSEs with external force is NOT globally bound.

Conclusion
The Utilization of the Generating Function Technique in the Discovery of Solutions for the Three-Dimensional Navier-Stokes Equation System 8 If an external force is lacking in the NSEs, the velocity vector field and the internal pressure p are both smooth and continuous or functions. Also, the kinetic energy of the velocity profile for this system is zero, which is less than some constant E; thus, the kinetic energy is globally bound for the system. In other words, both the Millennium prize criteria are satisfied. On the other hand, if the NSEs do possess an external force, the velocity vector field and the internal pressure p which are still both smooth and continuous, or functions. However, the kinetic energy of the velocity profile for this system can eventually obtain values higher than some constant E; therefore, the kinetic energy for this system is not globally bound. This individual data would suggest that only one of the criteria established by the Millennium prize can be met for NSEs with an external force . In other words, via complementary logic, there are no velocity vector fields , under an external force , that can satisfy both criteria set by the Clay Mathematics Institute for such a system.