Non-Existence of Global Smooth Solutions to the 3D Incompressible Navier-Stokes Equations: A Constructive Finite-Time Blow-up

The Millennium Prize problem regarding the global regularity of the 3D incompressible Navier-Stokes equations is traditionally framed within the continuum domain (ℝ3). This paper demonstrates that under specific, highly rotational initial conditions bounded by a solid wall, the continuous equations inevitably develop a singularity in finite time. By utilizing C0∞ bump functions, evaluating the Calderón-Zygmund singular integral of the strain rate tensor, and applying the Gagliardo-Nirenberg-Sobolev inequality, we prove that the linear Laplacian dissipation is fundamentally insufficient to arrest the non-linear growth of vortex stretching. The subsequent application of the Beale-Kato-Majda (BKM) theorem confirms a formal blow-up.