Is Navier-Stokes Solved? September 2026 Update

Source status reviewed 2026-09-12 (JST). On September 8, OpenAI announced finite-time blowup for smooth-forced 3D incompressible Navier–Stokes, claiming Clay alternatives C and D, with a manuscript and Lean formalization. On September 11, Clay said the problem has “apparently been settled”; evaluation and assignment of credit are deliberately unhurried. This is not a prize award or our verification of the proof.

Primary sources: Clay (2026-09-11) · OpenAI (2026-09-08) · Lean / GitHub · Official Clay statement.

The unforced 3D Navier–Stokes global-regularity question remains open: the announced forced construction does not settle it. In the technical discussion below, unresolved regularity and uniqueness questions concern the unforced system unless another setting is explicitly stated.

See also: the official Clay Navier-Stokes problem statement explained

Forcing means an external input to the velocity equation, not a force that becomes infinite. The announced construction starts from rest and uses a smooth force. Its bounded total kinetic energy does not rule out velocity concentrating on smaller and smaller scales. This concerns an ideal mathematical fluid, not a prediction that real water reaches infinite speed. Clay’s $1 million prize procedure is separate; the September 11 statement does not award it.

Source status reviewed 2026-09-12 (JST). On September 8, OpenAI announced finite-time blowup for smooth-forced 3D incompressible Navier–Stokes, claiming Clay alternatives C and D, with a manuscript and Lean formalization. On September 11, Clay said the problem has “apparently been settled”; evaluation and assignment of credit are deliberately unhurried. This is not a prize award or our verification of the proof.

Primary sources: Clay (2026-09-11) · OpenAI (2026-09-08) · Lean / GitHub · Official Clay statement.

The unforced 3D Navier–Stokes global-regularity question remains open: the announced forced construction does not settle it. In the technical discussion below, unresolved regularity and uniqueness questions concern the unforced system unless another setting is explicitly stated.

The released manuscript claims the following for every viscosity $\nu>0$: smooth, compactly supported forcing $f\in C_c^\infty(\mathbb{R}^3\times(0,\infty);\mathbb{R}^3)$ and initial velocity $u_0=0$, with a smooth solution for $0\leq t<1$, uniformly bounded $L^2$ velocity, but $\limsup_{t\uparrow1}\|u(t)\|_{L^\infty}=\infty$. The force remains smooth through the singular time. Theorem 1.1 claims whole-space breakdown; Corollary 10.6 gives periodic breakdown. These are reported theorem statements, not our proof verification. OpenAI — Navier–Stokes (PDF), 1.1 / 10.6.

Precisely: given smooth, divergence-free initial data $u_0 \in C^\infty(\mathbb{R}^3)$ with suitable decay (or on $\mathbb{T}^3$), it's unknown whether a unique smooth solution $(u, p)$ exists for all $t \geq 0$ with $u \in C^\infty(\mathbb{R}^3 \times [0, \infty))$. No counterexample has been constructed.

It's not completely dark. Mathematicians have chipped away at this for over a century, and they've built up a surprisingly detailed picture of what's known and what isn't:

• Weak solutions exist (Leray, 1934). If you weaken the notion of solution, global solutions exist. But whether they stay smooth and unique is still open.
• 2D is solved (Ladyzhenskaya, 1969). Two dimensions? Done. Smooth solutions exist for all time, and the difficulty is entirely, stubbornly specific to 3D.
• Singularities are rare (Caffarelli-Kohn-Nirenberg, 1982). Even if singularities exist in 3D, they're confined to a set with zero one-dimensional parabolic Hausdorff measure, an extremely small set in the geometry natural to the equation.
• Short-time solutions exist. Smooth? Yes, at least briefly. The question: can they always be continued forever?

So the gap is narrow but deep. We know solutions start smooth and we know weak solutions persist globally, yet nobody can prove that smoothness survives for all time in three dimensions.

Key established results:

• Leray (1934): Global existence of weak (distributional) solutions satisfying the energy inequality $\|u(t)\|_{L^2}^2 + 2\nu \int_0^t \|\nabla u(s)\|_{L^2}^2 \, ds \leq \|u_0\|_{L^2}^2$. Uniqueness? Open.
• Ladyzhenskaya (1969): For the 2D incompressible problem, smooth divergence-free data yield a unique global smooth solution.
• Caffarelli-Kohn-Nirenberg (1982): Partial regularity for suitable weak solutions: the singular set has zero one-dimensional parabolic Hausdorff measure, $\mathcal{P}^1(S) = 0$.
• Local well-posedness: For $u_0 \in H^s(\mathbb{R}^3)$ with $s > 3/2$, a unique smooth solution exists on $[0, T^*)$; this also holds in the scaling-critical space $\dot{H}^{1/2}(\mathbb{R}^3)$ (Fujita-Kato, 1964), extending to strictly larger critical spaces including $BMO^{-1}$ (Koch-Tataru, 2001). Does $T^* = \infty$?

Here's the gap: we know weak solutions exist globally, and we know strong solutions exist locally with full uniqueness. Whether the strong solution can always be extended to all time is the question that remains wide open.

The energy estimate displayed here is for zero forcing. With forcing, the right-hand side also includes the work term $2\int_0^t\!\int f(x,s)\cdot u(x,s)\,dx\,ds$.

This is a review of public source statements, not an independent proof audit. The repository labels its review “self-assessed”; that does not establish that nobody else has checked it. A Lean release and a manuscript provide material to inspect, not a substitute for checking theorem correspondence. Clay’s evaluation and prize procedure are separate from the announced mathematical result.

OpenAI also announced smooth-data blowup for unforced 3D Euler. That inviscid result does not establish blowup for unforced viscous Navier–Stokes.

This is a review of public source statements, not an independent proof audit. The repository labels its review “self-assessed”; that does not establish that nobody else has checked it. A Lean release and a manuscript provide material to inspect, not a substitute for checking theorem correspondence. Clay’s evaluation and prize procedure are separate from the announced mathematical result.

OpenAI also announced smooth-data blowup for unforced 3D Euler. That inviscid result does not establish blowup for unforced viscous Navier–Stokes.

Clay asks for a proof of one of four alternatives, not all four: (A) global smooth existence without forcing on R³; (B) global smooth existence without forcing on the periodic torus T³; (C) breakdown on R³ with admissible smooth forcing allowed; (D) periodic breakdown with admissible smooth forcing allowed. A/B quantify over all admissible initial data; C/D require an admissible counterexample. Forced breakdown does not disprove A/B.

The following whole-space clauses summarize A and C, not the full official hypotheses. In A, $f=0$. In C, all spatial and time derivatives of the force must satisfy Clay’s decay conditions; the excluded global solution is in the smooth bounded-energy class of the official statement. The periodic alternatives are B and D.

Clay asks for a proof of one of four alternatives, not all four: (A) global smooth existence without forcing on R³; (B) global smooth existence without forcing on the periodic torus T³; (C) breakdown on R³ with admissible smooth forcing allowed; (D) periodic breakdown with admissible smooth forcing allowed. A/B quantify over all admissible initial data; C/D require an admissible counterexample. Forced breakdown does not disprove A/B.

1. (A) Existence and smoothness: For every $u_0 \in C^\infty(\mathbb{R}^3)$ with $\nabla \cdot u_0 = 0$ and $|\partial_x^\alpha u_0(x)| \leq C_{\alpha K} (1 + |x|)^{-K}$ for all $\alpha, K$, prove existence of $(u, p) \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ satisfying the equations, with $\int_{\mathbb{R}^3} |u(x,t)|^2 \, dx < C$ for all $t \geq 0$.
2. (C) Breakdown: Exhibit $u_0 \in C^\infty(\mathbb{R}^3)$ (divergence-free, with the prescribed decay) and $f \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ satisfying the force-derivative decay condition (5), such that no $(u,p) \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ with the specified initial data and uniformly bounded kinetic energy satisfies the system.

• 1822: Navier derives the equations from molecular considerations.
• 1845: Stokes gives them their modern form.
• 1934: Leray proves weak solutions exist globally. Huge.
• 1969: Ladyzhenskaya solves 2D.
• 1982: Caffarelli, Kohn, and Nirenberg prove partial regularity, establishing that any singularities must be extraordinarily rare, with zero one-dimensional parabolic Hausdorff measure.
• 1984: Beale, Kato, and Majda prove for the 3D Euler equations that breakdown of a smooth solution forces divergence of the vorticity time integral. Related continuation criteria also apply to Navier-Stokes.
• 2000: Clay names it a Millennium Problem. One million dollars.
• 2014: Tao constructs blowup for an averaged version of the equations (preprint; published 2016), showing there's no purely structural obstruction to singularity formation.
• September 2026: OpenAI announces smooth-forced 3D blowup (Clay C/D); Clay acknowledges apparent settlement. Review and prize award are separate.

• 1822: Navier. Molecular stress.
• 1845: Stokes. Continuum.
• 1934: Leray. The foundational result: global weak solutions in $L^2$, the Leray projector, and the energy inequality that would shape an entire century of mathematical fluid analysis and define every approach that followed.
• 1951: Hopf extends to bounded domains.
• 1962: Serrin establishes conditional regularity: smooth if $u \in L^p_t L^q_x$ with $2/p + 3/q \leq 1$ and $q > 3$; the endpoint $L^\infty_t L^3_x$ was resolved by Escauriaza-Seregin-Šverák in 2003.
• 1969: Ladyzhenskaya. 2D done.
• 1982: CKN. $\mathcal{P}^1(S) = 0$.
• 1984: Beale-Kato-Majda (for 3D Euler). If $T^* < \infty$, then $\int_0^{T^*} \|\omega(s)\|_{L^\infty} \, ds = \infty$. Analogous continuation criteria hold for Navier-Stokes.
• 2000: Clay.
• 2014: Tao. Averaged Navier-Stokes blowup (JAMS 2016), showing that any proof of regularity for the true equations must exploit finer features of the Navier-Stokes nonlinearity than those preserved by the averaged model.
• September 2026: OpenAI announces smooth-forced 3D blowup (Clay C/D); Clay acknowledges apparent settlement. Review and prize award are separate.

Part of The Problem.

Go deeper: why is the problem so hard?, what subproblems are mathematicians working on, and what approaches have they tried?

The formal Clay statement lives on the Millennium Problem page, and if you want to understand which version of the equations this problem actually targets, see Incompressible vs. Compressible Navier-Stokes.

Wondering how the equations get solved in practice while the problem stays open? See Solving the Navier-Stokes Equations.

Part of The Problem.

Details: Clay formulation. Obstacles: Why It's Hard.

For the full picture of partial results, open subquestions, and every strategy that's been attempted over the past century of work on this problem, see Subproblems and Approaches. Which formulation the Clay problem studies: Incompressible vs. Compressible.