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 () as an axiomatic constraint on the fluid domain's microstructure. We demonstrate that kinematic viscosity  is not a passive scalar constant, but a spectral regularization operator that imposes a rigid ultraviolet cutoff, satisfying the inequality  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  remains uniformly bounded, thereby guaranteeing compliance with the Beale-Kato-Majda non-blow-up criterion: . This result transforms the Navier-Stokes equations into a -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.