Navier–Stokes Existence and Smoothness

Let \(u_0\) be smooth, divergence-free initial data of finite energy on \(\mathbb{R}^3\) (or the 3-torus), and consider the incompressible Navier–Stokes equations \(\partial_t u + (u \cdot \nabla) u = -\nabla p + \nu \Delta u\), \(\nabla \cdot u = 0\). Either prove that a smooth solution exists for all time for every such \(u_0\) — global existence and smoothness — or produce smooth finite-energy data whose solution develops a singularity in finite time. Both eternal regularity and finite-time blow-up count as resolutions of the Clay problem.

The equations date to Navier (1822) and Stokes (1845), but the regularity question is 20th century. Leray (1934) constructed global weak solutions in 3D and Hopf (1951) extended them; Ladyzhenskaya proved full smoothness in 2D. Scheffer (1976, 1980) began partial regularity, culminating in Caffarelli–Kohn–Nirenberg (1982): the singular set of a suitable weak solution has one-dimensional Hausdorff measure zero. Beale–Kato–Majda (1984) gave the sharp blow-up criterion via vorticity. Escauriaza–Seregin–Šverák (2003) closed the endpoint Ladyzhenskaya–Prodi–Serrin regularity class. Tao (2016) showed an averaged Navier–Stokes system can blow up — a barrier result: any proof of global regularity must use structure the averaging destroys. Convex integration then broke the uniqueness side: Buckmaster–Vicol (2019) built non-unique weak solutions, and Albritton–Brué–Colombo (2022) non-unique forced Leray solutions — so if smooth global solutions exist, it is not because weak solutions are unique. In September 2026 OpenAI announced a claimed resolution: a finite-time singularity from smooth data under smooth finite-energy forcing — statements C and D of the Clay formulation — produced by a ~10,000-agent system in 88 hours with Lean formalization, alongside a claimed unforced Euler blow-up. OpenAI states it does not intend to claim the prize. The announcements triggered a priority and data-provenance dispute — Alpöge–Buckmaster allege the result follows their unpublished approach and question its data provenance; OpenAI denies seeing their work — and independent mathematical verification remains pending. Both claims are recorded here as open pending review.