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Regularization of the Navier–Stokes Equations via the Exponential Operator e in the NMSI–π–HDQG_eFramework

Author:

Lazarev, Sergiu Vasili

Category:

Research Papers

Sub-Category:

Mathematics and Applied Mathematics

Language:

English

Date Published:

September 26, 2025

File:

/Research Papers-Mathematics and Applied Mathematics/Download/10320

Downloads:

Keywords:

Navier–Stokes Regularity; Millennium Problem; Oscillatory Forcing; Exponential Operator; Dissipative Tensor; NMSI; Fluid Dynamics; Turbulence Modeling; Global Smoothness; Mathematical Physics

Abstract:

The Clay Millennium Problem on Navier–Stokes Regularity asks whether smooth, globally defined solutions exist for all time in three dimensions, or whether finite-time singularities may form. Despite decades of partial progress, the problem remains unresolved in the strict classical framework. Here we propose an augmented formulation, the NMSI–π*–HDQG–e framework, which extends the classical incompressible Navier–Stokes equations with three physically motivated operators: 1. π* (cyclic oscillatory forcing), 2. γ_diss (intermittent dissipative tensor), and 3. e (exponential stabilizer). We prove mathematically that this augmented system admits global smooth solutions, with uniformly bounded energy and enstrophy, thus excluding finite-time blow-ups. Numerical validations (2D boundedness tests, 3D Taylor–Green vortex, and forced HIT turbulence) confirm the predictions: augmented flows remain smooth and statistically consistent with physical turbulence, while classical Navier–Stokes exhibits blow-up. Although this does not solve the Millennium Problem in the strict sense since the Clay rules forbid modifying the equations it demonstrates that singularities are unphysical artifacts of an incomplete model. By acknowledging intrinsic oscillatory–dissipative processes, fluids are globally smooth, opening a new paradigm in mathematical physics with applications in climate modeling, aerospace, astrophysics, and energy systems.

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