Global smoothness of periodic three-dimensional Navier–Stokes solutions

Fix a viscosity $\nu >0$. Let $u_0:{\mathbb{R}}^3\to {\mathbb{R}}^3$ be smooth, divergence-free, and periodic with period one in each spatial coordinate: $u_0(x+e_j)=u_0(x)$ for the standard coordinate vectors $e_j$, $j=1,2,3$. Periodic functions here can equivalently be viewed on the quotient ${\mathbb{R}}^3\mathrm{/}{\mathbb{Z}}^3$, where points differing by an integer vector are identified.

Does every such $u_0$ admit a velocity $u:{\mathbb{R}}^3\times [0,\mathrm{\infty })\to {\mathbb{R}}^3$ and pressure $p:{\mathbb{R}}^3\times [0,\mathrm{\infty })\to \mathbb{R}$, both smooth up to $t=0$ and periodic in each spatial coordinate, satisfying the unforced incompressible system for every $t\ge 0$?

${\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u=\nu \mathrm{\Delta }u-\mathrm{\nabla }p,\mathrm{div}u=0,u(x,0)=u_0(x)\mathrm{.}$

Smoothness up to the time boundary uses local smooth extension as in Smooth functions on relatively open half-space sets. At a fixed time, divergence and the gradient have the coordinate meanings of Divergence and curl of a $C^1$ vector field; the Laplacian is componentwise as in The Laplacian of a $C^2$ function and of a $C^2$ vector field. Explicitly, $((u\cdot \mathrm{\nabla })u{)}_i={\sum }_{j=1}^3{u}_j{\mathrm{\partial }}_{x_j}{u}_i$ and $(\mathrm{\Delta }u{)}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}^2{u}_i$. This is Fefferman's periodic global-smoothness formulation (B), including spatial periodicity of the pressure specified by the errata. It does not assert global regularity as a fact.

Status checked against the cited official records on 12 October 2026: Clay's problem page labels the problem Active. Its 11 September 2026 announcement describes an apparent settlement and an evaluation process that is deliberately unhurried. That announcement is a status report, not a proof of the formulation above; this library accepts no solution or proof premise from it. The present item records the unresolved/pending question and must be revisited if authoritative acceptance changes its status.

The equations themselves are a mathematical formulation. Established local, small-data, two-dimensional, weak-solution, or partial-regularity results require separate precise result items and proofs. A numerical computation or a theorem about a discretization does not settle this question. No item may use this open problem in deps, justified_by, or as a proof premise.