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Canonical Regularization of the Navier-Stokes Equations: Clay Institute Approach

Author:

Lazarev, Sergiu Vasili

Category:

Research Papers

Sub-Category:

Mathematics and Applied Mathematics

Language:

English

Date Published:

October 14, 2025

File:

/Research Papers-Mathematics and Applied Mathematics/Download/10338

Downloads:

360

Keywords:

Navier-Stokes Equations; Canonical Regularization; Clay Millennium Problem; Existence and Uniqueness; Smoothness; Topological Operators; Vorticity Control; Ternary Oscillations; Plasma Modeling; Astrophysical Fluids

Abstract:

We present a canonical reformulation of the three-dimensional incompressible Navier-Stokes equations (NSE) that directly addresses the Clay Millennium Problem of existence, uniqueness, and smoothness. Our framework replaces phenomenological closure operators (π*, γ_diss, e*) with canonical topological operators (∇, ∇·, ∇×), thereby ensuring invariance under Galilean transformations and compatibility across bounded and unbounded domains. We construct explicit a priori estimates in critical norms, provide uniqueness proofs via control of the vortex-stretching term, and demonstrate smoothness through higher-order derivative bounds with explicit constants C(s, ν). The canonical approach eliminates ad hoc dissipation models and yields a mathematically rigorous closure of NSE. Applications to plasma modeling, astrophysical turbulence, and hypersonic atmospheric flows are discussed, with numerical illustrations confirming boundedness of critical norms. This work establishes a structured path toward resolving the Navier–Stokes regularity problem within the Clay Institute framework.

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