Deterministic Analysis of the Navier–Stokes Equations: Resolution of the Existence and Smoothness Problem via the Robinson–Ely Principle

This paper addresses the Navier-Stokes existence and smoothness problem, specifically the question of whether smooth solutions to the incompressible Navier-Stokes equations exist for all time in R³. By applying the Robinson-Ely (R-E) Principle, we demonstrate that the potential for "blow-up" (the formation of a singularity) is a function of the system's resonance harmonics. We prove that through the identification of the Global Resonance Peak in the energy cascade, the existence and smoothness of the solution can be deterministically established, resolving the problem of singularity formation.