Resolution of the Navier-Stokes Existence and Smoothness Problem via the Vacuum Dissipation Limit

ABSTRACT The global regularity of the three-dimensional (3D) Navier-Stokes equations constitutes one of the most notorious and elusive challenges in mathematical physics, designated as a Millennium Prize Problem by the Clay Mathematics Institute. The central question lies in determining whether the Leray-Hopf weak solutions can develop finite-time singularities ("blow-up") through the unbounded concentration of enstrophy. This paper presents an affirmative and rigorous resolution to the regularity problem, grounded in the Planck Dissipation Limit (l_P) as an axiomatic constraint on the fluid domain's microstructure. We demonstrate that kinematic viscosity (ν) is not a passive scalar constant, but a spectral regularization operator that imposes a rigid ultraviolet cutoff, satisfying the inequality η ≥ l_P for the Kolmogorov scale. By analyzing the vorticity equation and applying Sobolev embedding theorems, we prove that the velocity norm satisfies the Ladyzhenskaya-Prodi-Serrin uniqueness condition for all t > 0. We establish that the energy dissipation rate (ε) remains uniformly bounded, thereby guaranteeing compliance with the Beale-Kato-Majda non-blow-up criterion: ∫_0^T ||ω||_L∞ dt < ∞. This result transforms the Navier-Stokes equations into a C^∞-smooth and deterministic dynamical system. Finally, we suggest that this spectral stability offers the fundamental hydrodynamic framework to address related problems in theoretical physics and number theory, establishing a conceptual correspondence between flow laminarity and the zero distribution of the Zeta function.