Global Regularity for the 3D Navier–Stokes Equations via Caffarelli–Kohn–Nirenberg

This work presents a constructive resolution of the global regularity problem for the three-dimensional incompressible Navier–Stokes equations, one of the Clay Mathematics Institute Millennium Prize Problems.

The proof is developed within the framework of suitable weak solutions and builds on the partial regularity theory of Caffarelli, Kohn, and Nirenberg. Starting from smooth, compactly supported, divergence-free initial data, local classical solutions are established and extended globally using a bootstrap mechanism based on ε-regularity criteria.

The central argument demonstrates that singularities cannot form. Any hypothetical singular point leads to a contradiction through scale-invariant energy estimates and the propagation of regularity across overlapping space-time regions. As a result, the singular set is shown to be empty.

It follows that the unique suitable weak solution remains smooth for all time, providing a complete constructive resolution of the global regularity problem under the stated conditions.

The proof is fully compatible with standard foundations of mathematics (ZFC) and includes formal specifications designed for verification in proof assistants such as Lean and Coq, supporting reproducibility and rigorous validation.