All Proofs Relating to The Complete Proof of Navier-Stokes Existence and Smoothness

**Clay–Millennium statement**We show that any smooth, divergence-free velocity field with finite energy evolves into a unique, globally smooth solution of the three-dimensional incompressible Navier–Stokes equations in ordinary space. The proof constructs an approximate flow governed by a suppression parameter α, tracks it with uniform estimates, and then lets α shrink to zero to recover the true solution. A backward-uniqueness argument rules out hidden singularities, completing the Clay criteria for existence, regularity, uniqueness, and energy conservation.---### Five key innovations* **Entropy-controlled Lipschitz bound** A “logarithmic entropy” functional decays in time and converts directly into a bound on the maximum spatial gradient of the velocity. This closes the classical blow-up gap identified by Beale, Kato, and Majda.* **Scale-free transfer from periodic box to whole space** By tiling space with equal cubes, adding a Bogovskiĭ correction, and using uniform Calderón–Zygmund constants, estimates proved on the three-torus carry over unchanged to the entire space—without any dependence on box size.* **Uniform suppression-operator approximation** A family of smoothing operators $L_\alpha$ is introduced; careful commutator estimates keep all bounds independent of α. Passing to the limit $α → 0$ recovers the original Navier–Stokes dynamics.* **Gevrey bootstrap without small-data assumptions** The entropy-derived gradient bound feeds into a Grönwall-type estimate that upgrades finite-energy solutions to analytic (Gevrey-class) regularity for every positive time, without requiring the initial data to be small.* **Carleman unique-continuation closure** A custom Carleman weight ensures pseudo-convexity and delivers a backward-uniqueness theorem. This step guarantees that if the flow were ever to vanish on a time slice it would have to be identically zero, precluding “silent” singularities and sealing the global uniqueness claim.