Navier-Stokes Existence and Smoothness: The Official Clay Problem Statement, Explained

The Clay Mathematics Institute’s Navier–Stokes problem, written by Charles Fefferman for the Millennium Prize Problems announced in 2000, asks whether the 3D incompressible Navier–Stokes equations have smooth solutions for all time, or whether smooth solutions can break down. This page explains exactly what the official statement requires: the equations, which starting flows and forces are allowed, what counts as a “physically reasonable” solution, and the four alternatives, any one of which resolves it.

In September 2026, OpenAI announced a proof of breakdown alternatives C and D, and Clay said the problem has “apparently been settled”; its evaluation is deliberately unhurried and no prize has been awarded. The unforced existence alternatives A and B are not decided by that result.

What was announced and how it is being evaluated: is Navier-Stokes solved? (sources reviewed 2026-09-29)

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

Fefferman’s official statement concerns the 3D incompressible Navier–Stokes system with viscosity $\nu>0$ on $\mathbb{R}^3$ or on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$. It defines admissible initial data and forces, a class of physically reasonable solutions (smooth for all $t\ge0$, with bounded energy on $\mathbb{R}^3$), and asks for a proof of one of four statements A–D.

In September 2026, OpenAI announced a proof of breakdown alternatives C and D, and Clay said the problem has “apparently been settled”; its evaluation is deliberately unhurried and no prize has been awarded. The unforced existence alternatives A and B are not decided by that result.

Status and sources: is Navier-Stokes solved? (sources reviewed 2026-09-29)

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

The Clay problem is not asking whether fluid simulations work, whether engineers can solve pipe-flow examples, or whether weak solutions exist. Those are separate questions. The prize question is about global smoothness or finite-time breakdown for the three-dimensional incompressible equations.

A resolution has to land in one of two buckets:

1. Global smoothness: show that, with no external force, every admissible smooth initial flow stays smooth for all future time.
2. Breakdown: give an admissible smooth setup, a starting flow and, if wanted, a smooth rapidly decaying force, for which a globally smooth solution cannot exist.

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 core system is the 3D incompressible Navier-Stokes equation

$$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f, \qquad \nabla \cdot u = 0.$$

Clay's formulation separates whole-space and periodic cases. In the whole-space existence direction, the initial velocity is smooth, divergence-free, and rapidly decaying; the target is a globally smooth solution with finite energy for all time. In the breakdown direction, the task is to construct admissible data, under one of the official alternatives, for which the required globally smooth solution fails to exist.

Existence (A, B) is posed with $f\equiv0$ and quantifies over all admissible data; breakdown (C, D) asks for one admissible pair $(u_0,f)$, with forcing allowed under the decay conditions below.

1. (A) Smoothness on all of space. With no external force, every smooth, divergence-free starting flow whose derivatives decay rapidly at infinity has a smooth solution for all time with bounded total energy.
2. (B) Smoothness in a periodic box. The same, with no external force, for smooth divergence-free flows that repeat periodically in each direction.
3. (C) Breakdown on all of space. Some admissible smooth starting flow and some smooth external force, whose size and every derivative decay rapidly in space and time, admit no smooth, bounded-energy solution for all time.
4. (D) Breakdown in a periodic box. Some smooth periodic starting flow and smooth periodic force, decaying rapidly in time, admit no smooth periodic solution for all time.

Proving any one of the four resolves the problem as posed. They are not arranged as statement and negation: C is weaker than “A is false”, because C also lets the solver choose a force. The official text describes the four options as giving “reasonable leeway to solvers while retaining the heart of the problem”.

Following Fefferman's formulation on $\mathbb{R}^3$ with $f \equiv 0$:

Hypotheses: Let $u_0 \in C^\infty(\mathbb{R}^3)$ be divergence-free. Assume for every $\alpha$ and $K$ there exist constants $C_{\alpha,K}$ such that

$$|\partial^\alpha u_0(x)| \leq \frac{C_{\alpha,K}}{(1 + |x|)^K} \quad \text{on } \mathbb{R}^3.$$

Conclusion (to prove): There exist $p \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ and $u \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ satisfying the Navier-Stokes equations, $u(x,0) = u_0(x)$, and the energy bound

$$\int_{\mathbb{R}^3} |u(x,t)|^2 \, dx < C \quad \text{for all } t \geq 0.$$

Breakdown (C): there exist a smooth, divergence-free $u_0$ satisfying the decay condition above and a smooth $f$ on $\mathbb{R}^3\times[0,\infty)$ satisfying condition (5),

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

for all $\alpha, m, K$, for which no $(p,u)$ with $p,u\in C^\infty(\mathbb{R}^3\times[0,\infty))$ satisfies the equations, the initial condition and the energy bound.

Periodic versions (B, D): $u_0$ and $f$ are unit-periodic in each coordinate; the force condition becomes $|\partial_x^\alpha\partial_t^m f(x,t)|\le C_{\alpha mK}(1+|t|)^{-K}$, condition (9); solutions must be smooth and periodic in $x$, and the appended erratum requires the pressure to be periodic as well. B takes $f\equiv0$ and asks for a global smooth periodic solution for every smooth periodic divergence-free $u_0$; D asks for an admissible periodic pair $(u_0,f)$ with no global smooth periodic solution. The periodic alternatives carry no separate energy bound.

The announced whole-space theorem starts from zero velocity, which meets the initial-data conditions, and uses a smooth force supported in a bounded region of space and time, which meets the force-decay condition. It concludes that no globally smooth solution with bounded energy exists for that data, which is what C requires; a corollary transfers the construction to the periodic box for D. Checking these data conditions is elementary; it does not verify the proof itself.

The correspondence concerns C and D only. A and B, global smoothness without forcing, are neither proved nor disproved by it, and Clay’s statement does not require them once C or D is established. Whether the proof is correct is a matter for expert checking and for Clay’s procedure; see the status explainer.

• Datum. $u_0=0$ is smooth, divergence-free, and satisfies the decay condition (4) and the periodicity condition (8) trivially.
• Force for C. $f\in C_c^\infty(\mathbb{R}^3\times(0,\infty);\mathbb{R}^3)$ has compact support in space–time, so each derivative is bounded and vanishes outside a compact set; (5) holds.
• Conclusion for C. Theorem 1.1 gives smooth $(u,p)$ on $[0,1)$ with $\sup_{0\le t<1}\|u(t)\|_{L^2}<\infty$ and $\limsup_{t\uparrow1}\|u(t)\|_{L^\infty}=\infty$, and states that no smooth solution on $\mathbb{R}^3\times[0,\infty)$ with the same datum and force has uniformly bounded kinetic energy, which is the class required by (6)–(7).
• D. Corollary 10.6: for every $\nu>0$, a smooth force on $\mathbb{T}^3\times[0,\infty)$, compactly supported in time (hence satisfying (9)), zero initial velocity, velocity blowup at $t=1$, and no global smooth periodic solution for the same datum and force; velocity and pressure are periodic, as the erratum requires.
• Limits. These are the manuscript’s statements, and the data-class checks above are elementary; they do not verify the construction, the Lean formalization, or its correspondence with the official text.

Manuscript: OpenAI — Navier–Stokes (PDF), Theorem 1.1 and Corollary 10.6.

Three things put Navier-Stokes on that shortlist:

• Practical importance. These equations run most of fluid dynamics: aircraft design, climate models, blood flow, ocean currents. Even without a complete proof, engineers use these equations successfully in many regimes; the open problem is about whether the 3D equations can always be justified mathematically.
• Mathematical depth. It draws on analysis, geometry, topology, and physics simultaneously.
• Lasting importance. Clay selected Navier–Stokes as one of its seven Millennium Prize Problems in 2000 because it combines fundamental mathematical difficulty with the equations’ central role in fluid dynamics. The September 2026 announcement changes the status, not those reasons for its importance. See also: explore why

The problem's difficulty is rooted in the supercritical nature of the 3D equations. The natural energy estimate

$$\frac{1}{2}\|u(t)\|_{L^2}^2 + \nu \int_0^t \|\nabla u(s)\|_{L^2}^2 \, ds \leq \frac{1}{2}\|u_0\|_{L^2}^2$$

places $u$ in $L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$, which is below the critical scaling. The Navier-Stokes equations are invariant under

$$u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t), \quad p(x,t) \mapsto \lambda^2 p(\lambda x, \lambda^2 t)$$

and the critical space is $L^3(\mathbb{R}^3)$ (or $\dot{H}^{1/2}$). The energy class $L^2$ is supercritical. It lies below the critical scaling threshold and does not by itself control the small-scale nonlinear cascade, leaving a gap that all existing techniques struggle to bridge.

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

The essential milestones:

• 1822: Navier derives the equations from molecular considerations.
• 1845: Stokes gives the modern derivation from continuum mechanics.
• 1934: Leray proves that "weak" solutions always exist. A massive result, but these solutions might not be smooth.
• 1982: Caffarelli, Kohn, and Nirenberg prove that singularities (more on partial regularity), if they exist, are extremely small: in the parabolic geometry natural to these equations, the singular set has zero one-dimensional parabolic Hausdorff measure.
• 1984: Beale, Kato, and Majda prove (originally for Euler, with Navier-Stokes analogues) that blowup can only happen if the vorticity becomes infinite.
• 2000: Clay names it a Millennium Problem.
• 2026: OpenAI announces a proof of smooth-forced breakdown claiming alternatives C and D; Clay says the problem has “apparently been settled”, with evaluation deliberately unhurried.

See also: critical-space approaches

Foundational results, selectively:

• Leray (1934): Global weak solutions $u \in L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$ exist, proved via compactness. He introduced the Leray projector and the concept of turbulent solutions. The starting gun for everything that followed.
• Hopf (1951): Extended Leray's construction to bounded domains.
• Ladyzhenskaya, Prodi, Serrin (1960s): Regularity criteria. If $u \in L^p_t L^q_x$ with $2/p + 3/q \leq 1$, $q > 3$, then the solution is smooth. Escauriaza, Seregin, and Šverák settled the endpoint $L^\infty_t L^3_x$ case in 2003.
• Caffarelli, Kohn, Nirenberg (1982): $\mathcal{P}^1(\Sigma) = 0$. The singular set has zero one-dimensional parabolic Hausdorff measure.
• Beale, Kato, Majda (1984): Originally proved for incompressible Euler: blowup happens iff $\int_0^{T^*} \|\omega(\cdot,t)\|_{L^\infty} \, dt = \infty$. Analogous criteria hold for Navier-Stokes.
• Koch, Tataru (2001): Global well-posedness for small data in $\mathrm{BMO}^{-1}$, the largest of the standard critical spaces in which such a result is known.
• Seregin (2012): At a blowup time $T^*$, the $L^3$ norm must diverge: $\|u(t)\|_{L^3} \to \infty$ as $t \to T^*$. Strictly stronger than ESS (2003), which only showed failure of uniform boundedness.
• 2026: Announced forced breakdown (Theorem 1.1 and Corollary 10.6 of the manuscript), claiming C and D; see the correspondence above.

This article is part of The Problem.

If you came here wondering whether someone already solved it, start with Is the Navier-Stokes Problem Solved?.

Then explore why it's so hard, or see how mathematicians have broken it into subproblems. For the structural reasons the 2D problem is tractable while 3D remains open, see Why 2D Is Easier Than 3D.

This article is part of The Problem.

Want the short answer on whether it's been solved? See Is the Navier-Stokes Problem Solved? That page also clarifies the gap between weak existence and global smooth regularity.

The mathematical obstacles are laid out in Why It's Hard. For a decomposition into tractable pieces (weak solutions, partial regularity, blowup classification), see Subproblems. And for why the 2D case is settled while 3D isn't, see Why 2D Is Easier Than 3D.