This paper establishes the global existence and C-infinity smoothness of solutions to the three-dimensional incompressible Navier-Stokes equations on T³ and R³ for all initial data u0 in Hˢ, s > 5/2, without any restriction on the size of the data. The author, F. A. Lopez, approaches this Millennium Prize Problem through the Israel-Stewart (IS) theory of causal viscous hydrodynamics, introducing a relaxation parameter tau > 0 that transforms the parabolic system into a symmetric hyperbolic-parabolic system. Key Contributions: Incompressible K-S Condition: The first verification of the Kawashima-Shizuta dissipativity condition for the incompressible Israel-Stewart equations. mu-tau Cancellation: A critical algebraic identity where a specific calibration of the coupling parameter causes the relaxation-time denominator to cancel, producing energy estimates independent of tau. L1-Free Fourier Splitting: An adaptation of the Schonbek technique that eliminates the traditional L1 hypothesis on initial data by applying a Plancherel estimate directly to the IS Lyapunov functional. Parabolic Splicing: A method using classical parabolic smoothing and Serrin-Prodi uniqueness to extend regularity from H⁵ to the full range s > 5/2 required by the Clay Millennium Problem. Convergence Rates: Proof that the IS-regularized solutions converge to the Navier-Stokes solution in H⁵ at a rate of O (tau¹/2) as tau approaches 0. The argument is entirely self-contained. This work is part of the Neoatomist Research Programme, a unified framework seeking to resolve Millennium Problems through the thermodynamics of open quantum systems.
Fdo Andres Lopez (Mon,) studied this question.