Progress on the Navier-Stokes Problem

Over nine decades, mathematicians proved that weak solutions exist for all time (Leray), that any singularities must be extremely small (Caffarelli–Kohn–Nirenberg), which quantities must blow up at a first singularity (Beale–Kato–Majda and later criteria), and that energy and scaling alone cannot rule out blowup (Tao). In September 2026, OpenAI announced a proof of finite-time blowup for the 3D equations with a smooth external force, Clay’s breakdown alternatives, and Clay said the problem has “apparently been settled”. Whether flows without external force always stay smooth remains open.

This page follows that arc: the milestones, then the 2026 chapter and how it connects to them. For the dated status on its own, see Is Navier-Stokes solved?

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

Over nine decades, mathematicians proved that weak solutions exist for all time (Leray), that any singularities must be extremely small (Caffarelli–Kohn–Nirenberg), which quantities must blow up at a first singularity (Beale–Kato–Majda and later criteria), and that energy and scaling alone cannot rule out blowup (Tao). In September 2026, OpenAI announced a proof of finite-time blowup for the 3D equations with a smooth external force, Clay’s breakdown alternatives, and Clay said the problem has “apparently been settled”. Whether flows without external force always stay smooth remains open.

Below: the milestones in their precise form, then the claimed forced breakdown (manuscript Theorem 1.1 and Corollary 10.6) read against them. Dated status: Is Navier-Stokes solved?

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

The results that reshaped the field:

• 1934, Leray: Proved that global-in-time weak solutions exist for any reasonable initial data. Something persists forever. But does it stay smooth? That's the question Leray couldn't answer; for flows without forcing it is still open.
• 1982, Caffarelli, Kohn, Nirenberg: The set of possible singularities is extremely small: in the parabolic geometry natural to these equations, it has zero one-dimensional size. Vanishingly small. If blowup happens, it's sparse beyond imagination.
• 1984, Beale, Kato, Majda: Huge result. A smooth solution can only break down if the vorticity blows up, which gave the entire field one precise target: control the relevant vorticity norm strongly enough, and a smooth solution cannot break down at that time.
• 2016, Tao: Constructed blowup for an averaged Navier-Stokes sharing the same energy and scaling properties as the real thing, which means a proof for the real equation has to use finer structure than energy estimates and scaling alone. A barrier. Not a solution.
• 2019, Buckmaster, Vicol: Without any force, distributional weak solutions with finite energy can be non-unique. These solutions lie outside the Leray–Hopf class (Annals 2019).
• 2022, Albritton, Brué, Colombo: Leray-Hopf weak solutions aren't unique when you allow an external force. The energy-inequality class is not as tame as hoped, which sharpened the question of which weak solutions are physically meaningful.

How these results bear on the 2026 announcement is the subject of the next section.

• 1934, Leray: Global existence of weak solutions in $L^2$ for divergence-free data $u_0 \in L^2(\mathbb{R}^3)$, satisfying the energy inequality (Acta Mathematica).
• 1982, Caffarelli–Kohn–Nirenberg (CKN): Partial regularity; $\mathcal{P}^1(\mathrm{sing}\, u)=0$, meaning the one-dimensional parabolic Hausdorff measure of the singular set vanishes. Still the strongest general result we have (Comm Pure Appl Math).
• 1984, Beale–Kato–Majda (BKM): The continuation criterion that reshaped the field: a smooth solution on $[0,T)$ extends past $T$ if and only if $\int_0^T \|\omega(\cdot,t)\|_{L^\infty}\,dt < \infty$, reducing regularity to vorticity control. Originally for Euler; adapted to Navier-Stokes (Comm Math Phys).
• 2016, Tao: Finite-time blowup for an averaged Navier-Stokes equation obeying the same energy and scaling properties as the true system, meaning any regularity proof must exploit finer structure (J Amer Math Soc).
• 2019, Buckmaster–Vicol: Non-uniqueness of distributional weak solutions with finite kinetic energy for the unforced equations, via convex integration; the solutions do not satisfy the energy inequality and so lie outside the Leray–Hopf class (Annals 2019).
• 2022, Albritton–Brué–Colombo: Non-uniqueness of Leray-Hopf solutions for forced 3D Navier-Stokes, constructed via unstable self-similar solutions (Ann of Math).

The next section reads the 2026 claim against these results.

On September 8, 2026, OpenAI announced a proof that, for every positive viscosity, a smooth solution starting from rest and driven by a smooth, compactly supported external force develops unbounded velocity at a finite time while its kinetic energy stays bounded. It states that this establishes Clay’s alternative C, and its periodic counterpart D. On September 11, Clay said the problem has “apparently been settled”, with evaluation deliberately unhurried. Details and sources.

How it relates to earlier work. These points concern consistency and context; none of them verifies the proof.

• Leray’s question. Leray (1934) introduced weak (“turbulent”) solutions while leaving open whether smooth solutions can lose regularity, and he suggested a self-similar blowup scenario. That particular scenario was later excluded under natural integrability conditions (Nečas–Růžička–Šverák, 1996; Tsai, 1998). The 2026 claim answers the breakdown question in a forced setting; for flows without forcing it remains open.
• Partial regularity. Caffarelli–Kohn–Nirenberg limits how large a singular set can be; it never excluded singularities. A blowup construction is therefore not in conflict with it.
• Continuation criteria. Beale–Kato–Majda-type and Prodi–Serrin–Ladyzhenskaya criteria say which quantities must become unbounded at a first singular time. Clay’s statement notes that for Navier–Stokes the velocity becomes unbounded near a finite blowup time, and the announced theorem is stated in that form: the maximum velocity is unbounded as the singular time approaches, while the energy stays bounded.
• Tao’s barrier. Tao’s averaged equation showed that a regularity proof must use finer structure of the true nonlinearity than energy and scaling. The announced construction works with the true equations, but with forcing, so it does not tell us whether the unforced dynamics has structure that prevents blowup.
• Non-uniqueness. Buckmaster–Vicol (unforced, outside the Leray–Hopf class) and Albritton–Brué–Colombo (Leray–Hopf, with forcing) concern uniqueness of weak solutions. The 2026 claim concerns loss of smoothness for a solution that starts smooth: a distinct question.
• Euler. OpenAI also announced smooth-data blowup for unforced 3D Euler. That inviscid result does not establish blowup for unforced viscous Navier–Stokes.

What has not changed: the unforced global-regularity question (Clay’s A and B), uniqueness of Leray–Hopf solutions without forcing, and the partial-regularity and continuation-criteria programmes remain open research directions. See Subproblems and Approaches.

The claim. For every $\nu>0$ the manuscript constructs $f\in C_c^\infty(\mathbb{R}^3\times(0,\infty);\mathbb{R}^3)$ and smooth $(u,p)$ on $\mathbb{R}^3\times[0,1)$ with $u(\cdot,0)=0$, $\sup_{0\le t<1}\|u(t)\|_{L^2}<\infty$ and $\limsup_{t\uparrow1}\|u(t)\|_{L^\infty}=\infty$ (Theorem 1.1, alternative C); Corollary 10.6 gives the periodic case (alternative D). Clay (September 11): “apparently been settled”. Status and sources.

• Leray. Leray’s self-similar ansatz $u(x,t)=\lambda(t)U(\lambda(t)x)$ with $\lambda(t)=(2a(T-t))^{-1/2}$ was excluded for $U\in L^3(\mathbb{R}^3)$ by Nečas–Růžička–Šverák (1996) and under local energy bounds by Tsai (1998). These results constrain one blowup profile; they do not exclude singularities in general.
• CKN. $\mathcal{P}^1(S)=0$ restricts the size of the singular set of suitable weak solutions; it is compatible with isolated singular points, so it does not conflict with a blowup construction.
• Criteria. Beale–Kato–Majda ($\int_0^{T^*}\|\omega\|_{L^\infty}\,dt=\infty$), Prodi–Serrin–Ladyzhenskaya ($u\notin L^p_tL^q_x$, $2/p+3/q\le1$, $q>3$) and Escauriaza–Seregin–Šverák describe necessary behaviour at a first singular time. Fefferman’s statement records that for $\nu>0$ the velocity becomes unbounded near a finite blowup time; Theorem 1.1 is stated as $\limsup_{t\uparrow1}\|u(t)\|_{L^\infty}=\infty$ with bounded $L^2$ norm.
• Tao. The averaged model shows that energy identity and scaling do not by themselves prevent blowup. The forced construction uses the true nonlinearity; whether the unforced true dynamics admits a regularity mechanism beyond energy and scaling is unaffected.
• Non-uniqueness. Buckmaster–Vicol (2019): unforced, distributional, outside Leray–Hopf. Albritton–Brué–Colombo (2022): Leray–Hopf, forced. The 2026 claim is a regularity (breakdown) statement for a classical solution, not a uniqueness statement.
• Euler. OpenAI also announced smooth-data blowup for unforced 3D Euler. That inviscid result does not establish blowup for unforced viscous Navier–Stokes.

Unchanged open problems include A/B ($f\equiv0$), Leray–Hopf uniqueness for $f\equiv0$, and sharp continuation criteria. See Subproblems and Approaches.

This page is a map, not the territory. For the details:

For detailed treatment of specific research directions:

• Subproblems: resolved and partially resolved cases, including 2D global regularity, axisymmetric without swirl, critical-space results ($L^3$, $\dot{H}^{1/2}$, $BMO^{-1}$), and conditional regularity criteria beyond BKM.
• Approaches: the main proof strategies under active investigation, including energy and enstrophy methods, profile decomposition, mild solution theory, stochastic Navier-Stokes, convex integration programmes, and computational-assisted bounds.

Sources reviewed 2026-09-29 (JST). This page summarizes published literature and public statements; it is not an independent proof audit. The repository labels its review “self-assessed”; that neither establishes independent verification nor shows that nobody else has checked the work. The Lean release and manuscript provide material to inspect; they do not replace checking that the formal statements match the intended theorems.

2026 sources: Clay (2026-09-11) · OpenAI (2026-09-08) · Manuscript (PDF) · Lean / GitHub.

Sources reviewed 2026-09-29 (JST). This page summarizes published literature and public statements; it is not an independent proof audit. The repository labels its review “self-assessed”; that neither establishes independent verification nor shows that nobody else has checked the work. The Lean release and manuscript provide material to inspect; they do not replace checking that the formal statements match the intended theorems.

2026 sources: Clay (2026-09-11) · OpenAI (2026-09-08) · Manuscript (PDF) · Lean / GitHub.