Is Navier-Stokes Solved? September 2026 Update

OpenAI has announced a proof that it says resolves the problem as Clay posed it, and Clay says the problem has “apparently been settled”. On September 8, 2026, OpenAI announced that a smooth solution of the 3D incompressible Navier–Stokes equations, starting from rest and driven by a smooth external force, develops unbounded velocity in finite time, and presented this as breakdown alternatives C and D of Clay’s official problem, which asks for any one of four possible answers. Clay’s statement came on September 11; evaluation and assignment of credit are deliberately unhurried, and no prize has been awarded.

In plain terms: the question was whether the equations that describe flowing water and air can ever “break”, producing infinite velocity from a perfectly smooth start. The announced proof says yes, when a smooth push from outside is applied in just the right way. It does not show the same without that push, and it does not predict that real water reaches infinite speed.

Source status reviewed 2026-09-29 (JST). This page reports what the primary sources state; it is not our own verification of the proof.

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

OpenAI has announced a proof that it says resolves the problem as Clay posed it, and Clay says the problem has “apparently been settled”. On September 8, 2026, OpenAI announced that a smooth solution of the 3D incompressible Navier–Stokes equations, starting from rest and driven by a smooth external force, develops unbounded velocity in finite time, and presented this as breakdown alternatives C and D of Clay’s official problem, which asks for any one of four possible answers. Clay’s statement came on September 11; evaluation and assignment of credit are deliberately unhurried, and no prize has been awarded.

The released manuscript claims (Theorem 1.1) that for every viscosity $\nu>0$ there are a force $f\in C_c^\infty(\mathbb{R}^3\times(0,\infty);\mathbb{R}^3)$, a compact set $K\subset\mathbb{R}^3$ and smooth $(u,p)$ on $\mathbb{R}^3\times[0,1)$ solving the equations with $u_0=0$, with the velocity $u$ and the pressure $p$ supported in $K$ at every time $0\le t<1$, with $\sup_{0\le t<1}\|u(t)\|_{L^2}<\infty$ but $\limsup_{t\uparrow1}\|u(t)\|_{L^\infty}=\infty$. It states as a consequence that no smooth solution on $\mathbb{R}^3\times[0,\infty)$ with the same force and initial datum has uniformly bounded kinetic energy: alternative C. Corollary 10.6 gives the periodic counterpart on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$, with a smooth force compactly supported in time and no global smooth periodic solution: alternative D. The force remains smooth through and beyond $t=1$. These are the authors’ theorem statements, transcribed rather than verified here: OpenAI — Navier–Stokes (PDF), 1.1 / 10.6.

Source status reviewed 2026-09-29 (JST). This page reports what the primary sources state; it is not our own verification of the proof.

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

• September 8, 2026 — OpenAI. Announced a proof of finite-time blowup for the 3D incompressible equations with a smooth external force, stating that it establishes alternative C (and also D) of Clay’s official formulation. It released a manuscript and a Lean formalization, and separately announced blowup for the unforced 3D Euler equations. OpenAI says it does not intend to claim the Millennium Prize.
• September 10, 2026 — repository. Date of the latest commit to the public Lean repository at our check. Its metadata still labels the review status “self-assessed”.
• September 11, 2026 — Clay Mathematics Institute. Said it “shares in the excitement of the global mathematical community as we contemplate the announcement that the Navier-Stokes problem has apparently been settled”, and that the process for evaluating the work and assigning credit “is deliberately unhurried”.
• September 29, 2026 — our source check. Clay’s announcement and news pages showed no later decision. This is a check of those pages only, not a claim about every institutional statement or every independent review.

• September 8, 2026 — OpenAI. Announced a proof of finite-time blowup for the 3D incompressible equations with a smooth external force, stating that it establishes alternative C (and also D) of Clay’s official formulation. It released a manuscript and a Lean formalization, and separately announced blowup for the unforced 3D Euler equations. OpenAI says it does not intend to claim the Millennium Prize.
• September 10, 2026 — repository. Date of the latest commit to the public Lean repository at our check. Its metadata still labels the review status “self-assessed”.
• September 11, 2026 — Clay Mathematics Institute. Said it “shares in the excitement of the global mathematical community as we contemplate the announcement that the Navier-Stokes problem has apparently been settled”, and that the process for evaluating the work and assigning credit “is deliberately unhurried”.
• September 29, 2026 — our source check. Clay’s announcement and news pages showed no later decision. This is a check of those pages only, not a claim about every institutional statement or every independent review.

In the Navier–Stokes equations, the force term $f$ stands for anything pushing the fluid from outside: gravity, a stirring paddle, a pump. With $f=0$ the fluid simply evolves from its starting state.

Clay’s official statement, written by Charles Fefferman, offers four ways to resolve the problem and says one is enough. Two are existence statements: every smooth, well-behaved starting flow stays smooth forever. They are posed with no external force (A on all of space, B in a periodic box). The other two are breakdown statements, and there the solver may choose both a smooth starting flow and a smooth external force, provided the force and all its derivatives decay rapidly: in space and time on all of space (C), and in time in a periodic box (D). The statement describes the four options only as giving “reasonable leeway to solvers while retaining the heart of the problem”.

No. The announced construction relies on a carefully chosen smooth external force. It does not decide whether every smooth flow without external forcing stays smooth forever (alternatives A and B), and Clay’s statement does not require that: it asks for one alternative, not all four.

The force in the announced construction is not a hidden singularity. It is smooth, confined to a bounded region of space and a bounded interval of time, and remains smooth at the moment the velocity becomes unbounded. Total kinetic energy stays bounded as well. Bounded energy does not prevent the maximum speed from becoming unbounded: the motion can concentrate in smaller and smaller regions.

Write the system as $\partial_t u + (u\cdot\nabla)u = -\nabla p + \nu\Delta u + f$, $\nabla\cdot u=0$. In A and B, $f\equiv0$ and the claim is universal: every admissible $u_0$ has a global smooth solution (with bounded energy on $\mathbb{R}^3$). In C and D, the claim is existential: some admissible pair $(u_0,f)$ has no global smooth solution in the required class. On $\mathbb{R}^3$ the force must satisfy condition (5),

$$|\partial_x^\alpha\partial_t^m f(x,t)|\le \frac{C_{\alpha m K}}{(1+|x|+t)^{K}},$$

for all multi-indices $\alpha$ and all $m,K$; on $\mathbb{T}^3$ it must satisfy condition (9), with decay in $t$ only. A smooth force with compact support in space–time satisfies (5).

So C is not the negation of A. The negation of A would be an admissible $u_0$ with $f\equiv0$ and no global smooth bounded-energy solution; C accepts a counterexample in which a smooth, rapidly decaying force is chosen as well. The official text gives its reason for the four-way formulation only as “reasonable leeway to solvers while retaining the heart of the problem”. A proof of C or D resolves the problem as posed while leaving the truth of A and B undecided.

No. The announced construction relies on a carefully chosen smooth external force. It does not decide whether every smooth flow without external forcing stays smooth forever (alternatives A and B), and Clay’s statement does not require that: it asks for one alternative, not all four.

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.

For the precise conditions (decay, the energy bound and periodicity) see the official Clay statement, explained, which also maps the announced theorem onto C and D line by line.

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.

Three different things are easy to run together:

• The mathematical claim. A long manuscript and a Lean formalization. The repository reports its main results as fully formalized, with no unfinished steps (“sorry”), and labels its review status “self-assessed”. That label is the authors’ own metadata: it neither establishes independent verification nor shows that nobody else has checked the work. A formal proof must also be matched against the intended theorem statement, which is a separate check.
• Clay’s response. The September 11 statement acknowledges an apparent settlement. It is not a certification or a prize award, and it names no recipient.
• The prize procedure. Clay’s rules list conditions including publication in a qualifying outlet, at least two years since publication, and general acceptance in the mathematics community. OpenAI has said it does not intend to claim the $1 million prize. Neither point implies a date or an outcome.

This site reports public sources. We have not audited the proof, built the Lean project or checked the correspondence between the formal and informal statements.

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

• Manuscript. Theorem 1.1 (whole space) and Corollary 10.6 (periodic), as quoted above. The periodic corollary does not print a separate energy inequality; the official D does not require one.
• Formalization. The repository metadata (commit f9e8bc5, September 10, 2026) aligns Theorem 1.1 and Corollary 10.6 with two Lean declarations, reports zero sorry and only the standard axioms propext, Classical.choice and Quot.sound, and takes its reference statements of C and D from the Formal Conjectures project. Its review status is “self-assessed”. These are author-supplied facts; a build, an axiom audit and a check that the formal statements match Fefferman’s text are independent tasks that this site has not performed.
• Institution and prize. Clay (September 11): “apparently been settled”; evaluation and credit “deliberately unhurried”. Clay’s rules list qualifying publication, a two-year interval and general acceptance among the conditions. OpenAI states it does not intend to claim the $1 million prize.

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

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. Without forcing, whether they stay smooth and unique is still open.
• 2D is settled. In two dimensions smooth solutions exist for all time; this has long been known, and Clay’s statement credits Ladyzhenskaya’s work. The difficulty is 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 for 3D flows without external forcing nobody has proved that smoothness always survives, and no unforced counterexample is known.

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 in this class for $f\equiv0$? Open.
• Two dimensions: for the 2D incompressible problem, smooth divergence-free data yield a unique global smooth solution; long known, with Clay’s statement citing Ladyzhenskaya.
• 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 divergence-free $u_0 \in H^s(\mathbb{R}^3)$ with $s > 3/2$, or in the scaling-critical space $\dot{H}^{1/2}(\mathbb{R}^3)$ (Fujita–Kato, 1964), there is a unique local strong (mild) solution on some interval $[0, T^*)$, smooth for positive times. Koch–Tataru (2001) proved global well-posedness for sufficiently small data in the critical space $BMO^{-1}$. Does $T^* = \infty$ for all admissible data?

Here's the gap: we know weak solutions exist globally, and we know strong solutions exist locally with full uniqueness. For $f\equiv0$, whether the strong solution can always be extended to all time remains open; the 2026 claim concerns forced data.

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

• Unforced global regularity. Does every smooth, finite-energy flow with no external force stay smooth forever (Clay’s A and B)? The announced result does not decide this.
• Uniqueness without forcing. With a specially designed force, Albritton, Brué and Colombo (2022) showed that Leray–Hopf weak solutions can be non-unique. Without forcing, uniqueness in that class is open.
• Checking and acceptance. Expert reading of the manuscript, checking the formal statements against the informal ones, and Clay’s evaluation under its rules are separate steps. The sources checked here report no completed prize evaluation.
• The mechanism. What the construction reveals about how viscous flows can concentrate, and whether anything similar can happen without forcing, are research questions.

Partial regularity, continuation criteria and weak-solution theory remain the working tools for these questions: see subproblems and the history of progress, including 2026.

• A/B. For $f\equiv0$ and every admissible $u_0$ (decaying on $\mathbb{R}^3$, or periodic), is there a global smooth solution? Neither a proof nor an unforced counterexample is known.
• Leray–Hopf uniqueness for $f\equiv0$. Open. Non-uniqueness is known with a suitable force (Albritton–Brué–Colombo, 2022) and, without forcing, for distributional weak solutions outside the Leray–Hopf class (Buckmaster–Vicol, 2019).
• Verification. Independent reading, formal-to-informal statement correspondence and Clay’s procedure are distinct from the announcement.
• Structure of the singularity. Rates, geometry and stability of the announced forced singularity, and their relation to Type I/Type II scenarios and to continuation criteria for the unforced problem.

See Subproblems and Progress.

Forcing means an external input to the velocity equation, not a force that becomes infinite. The Navier–Stokes equations are a continuum model: they treat a fluid as smooth matter and are not an accurate description at molecular scales. A singularity in the equations is a statement about the mathematics of that model. It does not predict that real water or air reaches infinite speed, and the announcement is not a closed-form formula for arbitrary flows or a new engineering method.

The question is still meaningful for physics and engineering: the equations are used every day, and knowing whether they can break down tells us what the model itself can and cannot guarantee.

The result concerns classical solutions of an idealized PDE: an incompressible Newtonian continuum with constant density and viscosity. Unbounded $\|u\|_{L^\infty}$ at $t=1$ is a property of that model; the model’s assumptions (continuum hypothesis, constitutive law, incompressibility) are not designed to hold at arbitrarily small scales. Physical relevance of any such mechanism is a separate question that the theorem does not address.

Nor is it a general solution formula: exact closed-form solutions remain special cases, and practical computation remains numerical. See Solving the Navier-Stokes Equations.

• 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’s monograph, which Clay’s statement cites for the two-dimensional theory; global regularity in 2D has long been known.
• 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 that energy and scaling alone cannot rule out singularities.
• 2026: OpenAI announces a proof of smooth-forced blowup claiming Clay’s C and D; Clay says the problem has “apparently been settled”, with evaluation deliberately unhurried.

For the full story, see Progress on the Navier-Stokes Problem.

• 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’s monograph; 2D global regularity long known.
• 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.
• 2026: Announced smooth-forced blowup (manuscript Theorem 1.1, Corollary 10.6) claiming C and D; Clay: “apparently been settled”.

Context and connections: Progress.

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.