MATHPH In the literature: open

Global regularity or blowup of unforced 3D Navier-Stokes flows

In plain words

Prove that every smooth, finite-energy flow of a viscous incompressible fluid in 3D stays smooth forever, or find one that develops infinite velocity in finite time with no outside push. This is the unforced half of the Clay Millennium problem.

Precise statement

Incompressible Navier-Stokes du/dt + (u.grad)u = nu Lap u - grad p, div u = 0, on $R^3$ or the torus $T^3$ with $\nu > 0$, external force $f = 0$ and smooth divergence-free initial data (Schwartz class on $R^3$). Answer: a proof that the smooth solution exists for all $t > 0$ (Clay statements A and B), or smooth data whose solution has $\operatorname{limsup}_{t \to T} \mid\mid u(t)\mid\mid_{\operatorname{inf}} = \infty$ at a finite $T$.

What would settle it

A refereed, independently checked proof of global regularity or an explicit blowup construction with $f = 0$.

Status in the literature

Unverified note

OpenAI announced in September 2026 a blowup with a smooth time-dependent force (Clay statements C and D); Constantin, Ignatova and Vicol (arXiv 2609.20803, 2026) proved that for constructions of that type (anisotropic Type II angular mean, exactly axisymmetric collapsing core) the force can neither vanish identically near the singular point nor be real analytic in space, so such constructions cannot be made unforced.