EXISTENCE AND SMOOTHNESS OF NAVIER–STOKES SOLUTIONS: A COMPLETE MATHEMATICAL PROOF

Abstract

In the continuum of fluid, average properties, pressure, density, velocity, and temperature, are evaluated over a small volume with a large number of particles of fluid. These properties vary continuously in space and time. Mathematical fluid dynamic models of these properties give rise to the continuity equation, momentum equation, energy equation, Euler’s equation, Cauchy’s equation of fluid motion, and the Navier-Stokes equations. These known existing equations add meaning to understanding the mechanics of fluid in science and engineering, geophysics, climate science, and computational fluid dynamics (CFD). Despite their long history, the analytical structure of the equations remains partially understood; famously, the Clay Mathematics Institute lists the existence of smoothness of solutions in three dimensions as one of the Millennium Prize Problems. However, to solve the Navier-Stokes equations, we must dig down to the very minimum force by which an infinitesimal fluid particle (Quantum molecule) moves around its volume mass under gravity, in alignment with quantum theory. In this paper ,a solution to the Navier-Stokes equation on , is put forward. A novel analytical framework for solving the Navier-Stokes equations by introducing the concept of a minimum force - the smallest quantifiable force acting on a quantum fluid particle under gravity. The analysis quantize the fundamental forces (momentum, pressure, and shear) acting on an infinitesimal fluid element, leading to discrete quantum numbers that characterize each force ( . These quantum values offer new solutions for both linear and non-linear terms of the Navier-Stokes equations on a torus. A general quantum number emerges ( , determining fluid smoothness or turbulence: positive values correspond to smooth flow, while negative values represent chaotic outbursts and vorticity. The resulting solutions provide insight into local and convective accelerations, vortex formation, and turbulence behaviour, revealing a natural logarithmic structure underpinning vortex dynamics. This approach merges classical fluid dynamics with quantum theory and relativity, offering new pathways for addressing one of the millennium problems-the existence and smoothness of Navier-Stokes equations.