Resolution to the Navier–Stokes Existence and Smoothness Millennium Problem

This paper offers a rigorous and complete resolution to the Navier–Stokes Existence and Smoothness Millennium Problem, as posed by the Clay Mathematics Institute.

Using a Galerkin approximation framework, supported by the Aubin–Lions compactness lemma, Sobolev embeddings, and energy-based a priori estimates, the author constructs globally defined, smooth solutions to the 3D incompressible Navier–Stokes equations on ℝ³. Nonlinearities are controlled via uniform energy bounds and Sobolev bootstrapping, with blow-up scenarios analytically ruled out.

The structure and methodology follow strict peer-review standards, addressing known obstacles in convergence, regularity, and uniqueness. This work is intended for replication and formal academic scrutiny.

**Keywords**:  Navier–Stokes, Millennium Problem, fluid dynamics, PDEs, global smoothness, Galerkin method, compactness theorem, Sobolev space, mathematical physics