Numerical Resolution of the Navier-Stokes Equations in 3D Using the Finite Volume Method: Application to the Millennium Problem

Abstract

We present a robust numerical framework for solving the three-dimensional incompressible Navier–Stokes equations using the finite volume method. Our Python-based implementation employs explicit time integration, pressure correction via a Poisson solver, and advanced 3D visualization tools—including vortex identification and particle tracking. The simulations capture the formation, evolution, and dissipation of vortex structures, with a monotonic decay of kinetic energy consistent with the physics of viscous incompressible flows. While this work does not constitute a formal proof, our results provide new insights into the regularity and energy properties of solutions, directly addressing the Clay Mathematics Institute’s Millennium Problem. All code and visualization tools are openly available to ensure full reproducibility and to foster further research on the existence and smoothness of Navier–Stokes solutions in three dimensions.While the numerical methods employed are well established, this work distinguishes itself by providing a fully open-source, Python-based 3D framework—complete with advanced visualization, detailed documentation, and explicit orientation towards the Millennium Problem.