An Exact Solution of the 3-D Navier-Stokes Equation

TL;DR

This paper presents an exact, turbulence-free solution to the 3-D Navier-Stokes Equation using an improved operator formalism and specific initial conditions, satisfying Clay Institute criteria.

Contribution

It introduces a novel operator formalism to derive an exact, turbulence-free solution to the 3-D Navier-Stokes Equation under specific initial conditions.

Findings

Exact solution satisfies Clay Mathematics Institute criteria.

No turbulence observed in the solution.

Provides insights into the mathematical structure of Navier-Stokes solutions.

Abstract

We continue our work reported earlier (A. Muriel and M. Dresden, Physica D 101, 299, 1997) to calculate the time evolution of the one-particle distribution function. An improved operator formalism, heretofore unexplored, is used for uniform initial data. We then choose a Gaussian pair potential between particles. With these two conditions, the velocity fields, energy and pressure are calculated exactly. All stipulations of the Clay Mathematics Institute for proposed solutions of the 3-D Navier-Stokes Equation are satisfied by our time evolution equation solution. We then substitute the results for the velocity fields into the 3-d Navier-Stokes Equation and calculate the pressure. The results from our time evolution equation and the prescribed pressure from the Navier-Stokes Equation constitute an exact solution to the Navier-Stokes Equation. No turbulence is obtained from the solution. A philosophical discussion of the results, and their meaning for the problem of turbulence concludes this study.