Global Existence and Smoothness of Navier–Stokes Solutions

We show that any smooth, divergence-free initial velocity field with sufficiently high regularity evolves under the three-dimensional Navier–Stokes equations into a single, globally defined solution that stays smooth for all future times. Our argument is entirely self-contained, requiring no outside references, and is built from four main pillars:Energy and entropy a priori estimatesWe derive bounds on the kinetic energy and on a novel “entropy” functional that together control the growth of the solution.Logarithmic Sobolev control of the Lipschitz normA refined Sobolev embedding argument turns our entropy decay into a uniform-in-time bound on the maximum gradient of the velocity, closing the classical blow-up obstruction.Gevrey-class smoothingFrom the gradient control, we bootstrap to full analytic regularity (Gevrey class) for any positive time, upgrading mere finite-energy data to instant real-analyticity.Carleman unique-continuationA bespoke Carleman estimate delivers a backward-uniqueness theorem: if the velocity were ever to vanish on a time slice, it must have been zero all along, eliminating hidden singularities and securing global uniqueness.