How a smooth force can break a fluid

Start a fluid in a smooth state, with finite energy, and let it evolve under the Navier–Stokes equations. Does it stay smooth forever, or can the velocity become infinite somewhere at a finite time?

In two dimensions the answer has been known for decades: solutions stay smooth. In three dimensions it is open, and it is one of the seven problems the Clay Mathematics Institute set in 2000. Leray showed in 1934 that weak solutions exist for all time, but not that they stay smooth or are unique. Since then the field has collected partial results, criteria that any singularity must satisfy, and numerical candidates, but no proof either way for the equations as they stand.

In September 2026 several preprints appeared that construct singularities for fluid equations by a common route: add a smooth external force, designed together with the flow. The table separates them, because they are different theorems about different equations.

Three distinctions do most of the work in reading these results, and the rest of the page is organised around them.

• Forced or unforced. The Clay text allows a smooth force in its two breakdown statements and forbids one in its two smoothness statements (§03). A forced construction addresses the first pair, not the second.
• Navier–Stokes or Euler. Euler is the same equation without viscosity. It is not a Clay problem, and viscosity is exactly what a Navier–Stokes construction has to beat (§06).
• Proof, formal proof, refereed proof. All of the September results are preprints. Several come with Lean formalizations, which check a formal statement mechanically. None had been through peer review as of this writing.

Navier–Stokes is Newton's second law written for a parcel of fluid. For an incompressible flow with velocity field u(x, t) and pressure p:

• ∂_tu: how the velocity at a fixed point changes in time.
• (u·∇)u, advection. Velocity carrying velocity. It is quadratic in u, which is where the difficulty comes from: it lets a fast region make itself faster.
• ν∆u, viscosity. Linear and smoothing. It removes energy from small scales at a rate that grows like the square of the wavenumber.
• −∇p, pressure. Not an independent unknown: it takes whatever value keeps ∇·u = 0, instantly and everywhere.
• f, the external force. Gravity, a stirring rod, anything applied from outside. Setting it to zero gives the unforced problem. Allowing it changes the question, as §05 explains.

Drop viscosity (ν = 0) and you get the Euler equations for an ideal fluid with no friction. Euler is older, is not a Clay problem, and in three dimensions has nothing at all opposing the nonlinearity.

The competition in one dimension

Burgers' equation keeps only the advection and viscous terms:

Fast parts of the wave overtake slow parts. With ν = 0 the profile steepens until the slope is vertical and ∂_xu becomes infinite: a finite-time singularity from smooth data. With viscosity on, diffusion meets the steepening, the wave settles into a travelling front and the slope stops growing.

Charles Fefferman's official problem description for the Clay Institute does not pose one question. To give solvers some leeway, it asks for a proof of any one of four statements: two that solutions stay smooth, and two that they can break down. One on all of space, one on the periodic box, for each.

The smoothness statements, (A) and (B), require the force to be identically zero. The breakdown statements, (C) and (D), ask for smooth initial data and a smooth force, both decaying as the text specifies, for which no smooth solution with bounded energy exists for all time. A smooth force is permitted there.

So a forced blowup construction is, on the text, a proof of (C) or (D). The OpenAI Navier–Stokes paper states that its Theorem 1.1 establishes (C) and its Corollary 10.6 establishes (D). The question most people have in mind, whether an unforced fluid can break down on its own, is (A) and (B), and a forced construction does not address it.

The same description adds, right after the four statements, that the corresponding questions for the Euler equations are also open and important, but that Euler is not on the Clay prize list. An unforced Euler blowup, however significant, does not bear on the prize.

The prize itself

Clay's rules require a solution to be published in a refereed journal of worldwide repute, at least two years to pass after publication, and the solution to be generally accepted, with the Institute deciding whether the official problem has been answered. OpenAI's announcement says it does not intend to claim the prize for this result.

A singularity has to announce itself. The cleanest criterion is due to Beale, Kato and Majda (1984), stated for the Euler equations: a smooth solution on [0, T) can be continued smoothly past T if and only if

So it is not enough for the maximum vorticity to grow without bound. It has to grow fast enough that its time integral diverges. If ‖ω‖_∞ grows like (T − t)^−α, the integral is finite for α < 1 and infinite for α ≥ 1. The borderline is exactly the rate that self-similar collapse produces: a velocity scale U over a length scale L gives a gradient U/L, and for a structure that shrinks in a time comparable to L/U, that gradient is of order 1/(T − t).

Navier–Stokes has criteria of the same kind. Escauriaza, Seregin and Šverák (2003) showed that a solution which stays bounded in L³ cannot blow up, so at a singularity the L³ norm must become unbounded. Caffarelli, Kohn and Nirenberg (1982) showed that the set of singular points of a suitable weak solution is small: it has zero one-dimensional parabolic Hausdorff measure, so it cannot contain a curve in space-time. Both are reasons any blowup has to concentrate into tiny regions, which is what the constructions below do.

The OpenAI Euler paper states its theorem with the Beale–Kato–Majda integral directly: its solution has ∫‖curl u‖_∞ dt = ∞ at the breakdown time. The Alpöge–Buckmaster Euler paper states the same divergence for its forced solution.

Given any smooth velocity field u and pressure p at all, you can define the force to be whatever is left over when you plug them into the equation:

With that choice, Navier–Stokes holds by construction. So with a force allowed, the problem is not to find a solution that blows up: any divergence-free field that blows up "solves" the equation with some force. The problem is to find a blowing-up flow whose residual ℛ stays smooth, with all of its space and time derivatives bounded, all the way through the singular time. The OpenAI paper defines its force exactly this way (its equation (3.1) and §10), and almost all of its length goes into making the residual smooth.

That requirement is strong. Near the singularity the velocity changes on shorter and shorter scales, and every derivative of ℛ involves derivatives of u. A naive collapsing flow produces a residual that blows up along with it. The constructions differ in how they cancel that residual, and §06 and §07 describe the two main approaches.

The OpenAI Navier–Stokes paper builds its singularity around a single concentrating vortex. Before following it, it helps to see why a vortex does not normally collapse.

Stretching against viscosity: the Burgers vortex

The Burgers vortex is an exact, steady solution of Navier–Stokes. A straining flow draws fluid inward at radial speed u_r = −a r/2 and pushes it out along the axis at u_z = a z. That stretching concentrates the swirl, while viscosity spreads it out. The two balance at a core radius r_c = √(4ν/a), and the swirl is

The Burgers vortex is the obstacle in miniature. Stretching alone concentrates vorticity, and viscosity alone would win at small enough scales. A blowup construction for Navier–Stokes needs a vortex that keeps shrinking, so the stretching has to grow as the core shrinks, and viscosity has to be outpaced in at least one direction.

A core that shrinks at two rates

In the OpenAI construction, fluid spirals inward toward the axis near a mid-plane and flows outward along the axis above and below it. Write τ = 1 − t for the time left before the singularity at t = 1. To leading order the core is self-similar, the same shape at each instant in shrinking units, and it shrinks at two different rates (the paper's §2.1):

Both lengths go to zero, but their ratio ℓ_r/ℓ_z ≍ τ^h goes to zero too, so the core becomes a needle rather than a point. The paper records the consequences: the radial Reynolds number stays of order one, so viscosity remains fully active across the core, while the swirl Reynolds number Re_θ ≍ τ^−h grows without bound. Diffusion along the axis, at rate ν/ℓ_z², falls behind diffusion across it, at rate ν/ℓ_r², by the factor ℓ_r²/ℓ_z² ≍ τ^2h.

Dividing the speed by the radius gives the size of the velocity gradient, τ^−1/2−h/τ^1/2 = τ^−1−h. That is just past the borderline α = 1 of Figure 3, so its time integral diverges.

Cancelling the residual

The bare self-similar vortex does not have a smooth residual: equation (4) applied to it diverges as τ → 0 in a ring between the core and the smooth flow outside it. The paper's fix is to add spatially oscillating pulses in that ring. Averaged over their oscillation, the pulses' quadratic self-interaction produces a stress, and the amplitudes are tuned so that this stress cancels the singular part of the residual. The correction leaves its own smaller error, which is cancelled the same way, and so on. The paper's Figure 6 describes each cycle as improving the decay exponent of the residual by 1/10, and the limit is a force that is smooth through the singular time.

The technique of cancelling an error with the averaged stress of high-frequency oscillations comes from convex integration, and the paper cites the work of Daneri and Székelyhidi on Euler and of Buckmaster and Vicol on Navier–Stokes in that line.

Finally, one construction covers every viscosity. If u, p, f work for ν = 1, then u_ν(x, t) = √ν u(x/√ν, t), p_ν = ν p(x/√ν, t) and f_ν = √ν f(x/√ν, t) work for viscosity ν, with the same singular time and the support shrunk by √ν.

The other route, used in most of the September papers, comes from a programme of Diego Córdoba and Luis Martínez-Zoroa. Over 2023 to 2025 they and coauthors proved finite-time blowup with forcing for 3D Euler, for the incompressible porous medium equation, for the 2D Boussinesq system, and for Navier–Stokes with weakened (hypodissipative) viscosity. The Alpöge–Buckmaster papers present themselves as completing steps of this programme, and the OpenAI papers cite it.

Terence Tao's blog post of 7 September 2026 describes the strategy abstractly. Write the equation as N(u) = f. Suppose you have a forced solution at low frequency, N(u_lo) = f_lo. Look for

with u_hi at a much higher frequency and relatively large, and f_hi tiny. Keeping f_hi small means u_hi must nearly solve the equation linearised around u_lo. So the background is designed to make that linearised equation unstable: a correction that starts exponentially small grows large just as it is needed, timed so the nonlinearity does not spoil the analysis. Repeat at ever higher frequencies and pass to the limit. The force stays smooth because each increment is tiny; the solution blows up because each stage amplifies gradients further.

Each stage rides on the shear of all the stages before it. In the toy below, the initial amplitudes fall off fast enough in frequency that the data are smooth, while the amplified gradients diverge at t = 1.

The qualifiers in these statements are where the content is. They are restated from the papers, with the paper's own numbering.

The fluid starts at rest. The force is compactly supported in space and time (the paper realises it inside K × [0, 2]), so it switches on, acts and switches off. The singular time is fixed at t = 1 for every viscosity, by the rescaling in §06.

For comparison, the paper sets out the earlier unforced Euler results it extends. Elgindi (2021) proved blowup on all of ℝ³ for velocities in C^1,α, below smooth. Córdoba, Martínez-Zoroa and Zheng (2025) used data smooth everywhere except one point. Chen and Hou proved blowup from smooth data, but in a cylinder with a solid wall, combining analysis with rigorous computer-verified estimates. The new claim is smooth, compactly supported data with no boundary.

Formal proofs

Both groups released Lean 4 formalizations: openai/NavierStokesAndEuler covers the two OpenAI results, and tristanbuckmaster/fluid_lean covers forced Boussinesq and forced Euler. Each says its proofs use only Lean's standard axioms and are set up for Comparator, a checker that confirms a submitted proof proves a given challenge statement.

A formal proof certifies the formal statement. Whether that statement says the same thing as the theorem in the paper is a separate check, made by reading both. As of this writing that comparison, and refereeing of the papers, were still to be done.

The September 2026 papers sit at the end of a long line. Filter it by equation, or hide the numerical work to see only theorems. Model equations are simpler systems with the same kind of nonlinearity (Boussinesq, the porous medium equation, an averaged Navier–Stokes).

Directly, very little. Computational fluid dynamics has integrated these equations numerically since the 1960s, and its limits are set by grid resolution, turbulence models, boundary conditions and measurement, not by whether smooth solutions exist for all time. A solver discretises and runs, and its errors come from modelling choices far larger than anything a regularity theorem would touch.

A forced construction is further still from application, because of where its force acts.

The longer-term interest is in the methods. Weak solutions, partial regularity and the Beale–Kato–Majda criterion all came out of work on this problem and are used far beyond it. If the multiscale ladder turns out to be a general way of building singularities in nonlinear evolution equations, that would matter more widely than any one theorem.

The obvious next question is whether the force can be removed. One scaling heuristic, posted by Stan Palasek in reply to Tao on 8 September 2026, suggests a ladder of instabilities cannot do it for Navier–Stokes. At stage k, at frequency N_k, the new component can grow by at most a factor of ∫‖∇u_<k‖_∞ dt from the stages below it. Bernstein's inequality and the energy bound make that factor O(N_k^3/2), while viscosity drains the stage at a rate of order N_k². Without a force to top it up, the drain wins.

The argument is a heuristic about one kind of construction, not a theorem about all of them. It also says nothing against Euler, which has no viscous drain, and it does not apply in high dimensions, as Palasek notes.