Navier–Stokes existence and smoothness problem in three-dimensional Euclidean space.

Title: Clay Institute Version: Proof of Global Existence and Smoothness for the Navier–Stokes Equations in ℝ³

Description:This paper presents a formal and rigorous proof of the global existence and smoothness of solutions to the incompressible Navier–Stokes equations in three-dimensional Euclidean space. Utilizing established tools from functional analysis, Sobolev space theory, and compactness arguments, the work demonstrates that smooth, divergence-free initial data leads to a unique, smooth solution for all time.

The analysis begins with the classical energy inequality and extends to higher-order Sobolev norms to ensure regularity. Galerkin approximation methods are employed to construct weak solutions, and convergence is established via Banach-Alaoglu and Aubin-Lions theorems. The proof concludes with a demonstration of uniqueness and continuation criteria, satisfying the requirements set forth by the Clay Mathematics Institute.

This document is submitted as a formal resolution to the Navier–Stokes Millennium Prize Problem, fully aligned with academic standards and structured for professional peer review.

Keywords: Navier–Stokes Equations, Fluid Dynamics, Sobolev Spaces, Functional Analysis, Millennium Problems, Clay Mathematics Institute, Global Regularity, Partial Differential Equations