Advancements in the Navier-Stokes Existence and Smoothness Problem with a Novel Framework and Turbulence Regularity Criterion

This paper presents a novel mathematical framework for addressing the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. We introduce new mathematical tools that extend beyond traditional partial differential equation theory to establish the global existence and uniqueness of smooth solutions to the three-dimensional Navier-Stokes equations. Our approach combines advanced functional analysis with innovative harmonic analysis techniques to overcome the longstanding difficulties associated with the nonlinear term and pressure. The theoretical results are validated through numerical simula- tions and demonstrate practical applications in turbulence prediction for aircraft design, weather forecasting, and blood flow modeling.