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ns — the blowup, taken apart

finite time blowup for navier–stokes · openai · 2026 · alternatives (c) and (d)

pages: core · pulse · stress · exterior · bands · solver · explorer

theorem 1.1, in words

For every viscosity ν > 0 there is a smooth force, compactly supported in space and time, under which the three-dimensional incompressible Navier–Stokes equations, started from rest, have a smooth solution on [0, 1) whose velocity and pressure stay inside one fixed compact set, whose kinetic energy stays bounded — and whose maximum speed goes to infinity as t ↑ 1. Consequently no smooth solution with the same force and datum can exist for all time with bounded energy.

∂tu + (u·∇)u − ν∆u + ∇p = f,   ∇·u = 0,   u(·,0) = 0   ⟹   supt<1 ‖u(t)‖L² < ∞,   limsupt↑1 ‖u(t)‖L∞ = ∞

The paper is 166 pages, posted as a PDF by OpenAI with no named authors. This pack does not verify it. What it does is take the construction apart — a collapsing vortex, a ring of shear-fed pulses whose averaged momentum flux supplies the force the vortex is missing, and an exterior that is an exact heat flow — and implement each piece in the browser, with the paper's own formulas and exponents, so you can turn the knobs the proof fixes.

What it claims, and what it does not

Fefferman's statement of the Millennium problem lists four alternatives and asks for a proof of any one of them. The two the paper claims are the forced breakdown ones. The unforced problem — smooth initial data, no force — is untouched.

alternativestatementstatus
(A)Whole space ℝ³, no force: every smooth, decaying initial velocity gives a smooth global solution with bounded energy.open
(B)Periodic box 𝕋³, no force: same conclusion.open
(C)Whole space, with a smooth force: some smooth datum and force admit no smooth global solution with bounded energy.claimed — Theorem 1.1, with zero initial datum
(D)Periodic box, with a smooth force: same breakdown.claimed — Corollary 10.6, by periodising the compactly supported fields

The force is not a trick played on the equations: for any divergence-free u and pressure p, defining f as the residual ∂tu + (u·∇)u − ν∆u + ∇p makes Navier–Stokes hold by construction. The whole difficulty, and the whole paper, is choosing a flow that blows up while that residual — with every one of its derivatives — stays smooth through the singular time (§2).

The anatomy

A radial–axial section through the flow at time remaining τ = 1 − t. Three regions, each doing one job. Scrub τ to watch the core collapse onto the origin; switch to similarity units and the picture stops moving — that is what "self-similar" means.

time remaining · τ = 1 − t
units
exponent h
The proof takes 0 < h < 1/100. At that size the core's slenderness ℓz/ℓr ≍ τ−h is invisible to the eye, so the exaggerated setting is the one Figure 1 of the paper also uses.
core: spiral in, flow out axially, spin up annulus: two families of pulses (rings) exterior: pure swirl, exact heat flow

Region radii Xa, Xb, Xc and the swirl shading are illustrative — the paper proves such radii exist and fixes them by estimates, it does not print them. The meridional arrows are exact for the paper's reference axis profile U = 4η (see core → meridional flow).

The construction, in six moves

  1. Build a collapsing axisymmetric vortex. In similarity coordinates X = r²/2q, η = z/qD, with τ = q(1−η²), the velocity is uθ = q−AE(X,η), uz = q−AU(X,η), r ur = V₀(X,η), where A = ½ + h and D = ½ − h. Radius shrinks like τ½, height like τ½−h, swirl and axial speed grow like τ−½−h. Fluid spirals inward toward the axis and leaves axially on both sides of a dividing layer near z = 0; angular momentum carried inward spins the core up. Kinetic energy of the core ≍ τ½−3h → 0.§2.1, §3.1, Theorem 4.6 · core
  2. Push the leftover force into a thin annulus and write it as a stress divergence. The core satisfies its leading momentum balance exactly, and the exterior is chosen to satisfy Navier–Stokes exactly; the imbalance lives only where they join, Xa < X < Xb. There the residual is −(∂r + 2/r)Trθ, −(∂r + 1/r)Trz for a stress T that vanishes at both edges — plus a "flat" remainder all of whose derivatives vanish faster than every power of q. Corrections in powers q2nh fix the background to every order.§3.2, Proposition 5.5 · bands
  3. Let the fluid supply that stress itself. Add spatially oscillatory pulses — complete rings, localized in radius, height and time. Their velocities have zero angular mean, but their quadratic products ⟨wrwθ⟩, ⟨wrwz⟩ do not: outward motion carrying a swirl surplus and inward motion carrying a deficit both transport angular momentum outward. Two families with different flux ratios span a cone; the profiles are built so T lies strictly inside it, and positive squared amplitudes reproduce it.§2.2, §3.3, Lemma 4.5, Proposition 7.5 · stress
  4. Grow each pulse from an exponentially small seed on the background shear, then let viscosity kill it. A swirl surplus in a flow whose angular velocity falls fast enough with radius is flung outward, and moving outward increases the surplus: exponential growth when this beats viscous damping. The same shear tilts the wave and shortens its radial wavelength, so damping eventually wins. Growth, then decay — a Gaussian envelope P(v) whose tails are small enough that cutting the pulse off costs nothing.§7.1–7.2, (7.5), (7.12), Lemma 7.4 · pulse
  5. Correct in a cycle, then sum. Each correction cancels a source and creates linear remainders and quadratic interactions; a fixed four-step cycle (wave harmonics, signed covariance increments, mean flow with pressure rebuilt, five radial moment equations) improves the residual's decay exponent σj = ⅕ + j/10 each round. Sum with shrinking cutoffs on vector potentials, curl afterwards, and incompressibility survives exactly.§3.4, Propositions 9.6 and 9.9 · bands
  6. Cut off, extend the force through t = 1, compare. Outside a radius Xext the flow is a pure swirl K(r,τ) = r−1−2hHext(τ/r²) solving the radial heat equation exactly, so at any fixed r > 0 everything has smooth limits at τ = 0. That lets space and time cutoffs be applied without disturbing the core. The residual f has uniform limits of every derivative at t = 1; a Borel-style sum extends it smoothly to t > 1 with support in K×[0,2]. Any smooth bounded-energy solution with the same data must agree with this one, so none exists globally.§3.5, §10, Lemmas 10.2–10.5 · exterior

Viscosity is scaled out at the end: uν(x,t) = √ν·u(x/√ν, t) solves the equation at viscosity ν with force √ν·f(x/√ν, t) and the same singular time; energy scales by ν5/2 (10.22–10.23). The periodic case shrinks the support into a cube and sums integer translates, which never touch (Corollary 10.6).

The pages

/core/

The concentrating core

Every scaling law against τ with h live; the similarity map q(z,τ); true meridional streamlines from the paper's axis profile; the viscosity and torus rescalings.

§2.1 · §3.1 · (4.7) · (10.22) /pulse/

The pulse

Integrates the wave-amplitude equation and overlays the envelope P(v); the exact Orr shearing wave; why vs > 2 is Rayleigh's centrifugal criterion with axial shear.

§7.1–7.2 · Lemma 7.4 · Craik–Criminale 1986 /stress/

Zero-mean waves, nonzero flux

Reynolds stress from oscillations that average to nothing; the admissible stress cone drawn exactly from (ts, vs); drag the target and watch c₁, c₂.

§2.2 · §4.3 · Figure 4 · Proposition 7.5 /exterior/

The heat exterior

H(Z) by quadrature, K(r,τ), the pressure integral, the swirl heat equation's residual to machine precision, and a finite-difference solver checked against it.

(3.5) · (4.29) · (10.7) · Lemma A.6 /bands/

The ledger of scales

Dyadic bands Q = 2−ℓ, ε = Qh, S* = ℓ²; carrier frequency, wavelength and amplitude ratios; the correction cycle; how small q must be before h = 1/100 does anything.

§3.6 · (6.1) · (7.2) · (9.8) /solver/

A 2D Navier–Stokes solver

Pseudo-spectral, hand-written FFT, runs in the tab. A shear-plus-wave experiment measures ⟨uv⟩ and the mean-flow feedback — the mechanism in its simplest setting — and shows why two dimensions cannot blow up.

contrast to §1 · Orr 1907 · Leray 1934 /explorer/

The collapse, in three dimensions

A seeded, exactly divergence-free specimen of the leading-order field drawn live in WebGL: a section plane with pressure as colour and streaks of the meridional flow, spiralling streamlines, particles following the collapse, and the low-pressure core as a glow. Every seed is a permalink.

(3.2) · (4.5)–(4.7) · (4.25) · (4.29) · bismuth’s section trick

Where it sits

1757Euler writes the inviscid equations.
1822–45Navier, then Stokes, add the viscous term.
1934Leray: global finite-energy weak solutions in 3D. Whether smooth data stay smooth is left open — the question ever since.
1982Caffarelli–Kohn–Nirenberg: the singular set of a suitable weak solution has zero one-dimensional parabolic measure. Isolated singular points are not excluded.
1986Craik–Criminale: exact finite-amplitude waves on affine shear flows — the wave's quadratic self-interaction cancels. The pulses here are their descendants.
1991Lifschitz–Hameiri; Friedlander–Vishik: local instability criteria by following wavevectors along a background flow.
2000Clay Mathematics Institute poses the problem; Fefferman's statement lists alternatives (A)–(D).
2003Escauriaza–Seregin–Šverák: regularity under a bounded L∞tL³x norm.
2016Tao: finite-time blowup for an averaged Navier–Stokes equation keeping the energy cancellation.
2017–19Daneri–Székelyhidi (Euler), Buckmaster–Vicol (Navier–Stokes): convex integration — oscillations realise a prescribed stress; non-uniqueness of weak solutions.
2022Albritton–Brué–Colombo: two distinct Leray–Hopf solutions from rest with the same (singular-in-time) force.
2023–26Córdoba–Martínez-Zoroa (–Zheng): forced Euler blowup, then hypodissipative Navier–Stokes, by amplifying ever finer vortex layers while keeping the force smooth.
2026This paper: full viscous Navier–Stokes, every ν > 0, smooth compactly supported force, bounded energy, unbounded velocity at t = 1. Claimed, unverified here.

Questions people ask first

Is the Millennium Prize problem solved?

The official statement asks for a proof of any one of (A)–(D). The paper claims (C) and (D). Whether the argument holds is for referees and the community; a 166-page construction with a four-operation correction cycle iterated to infinite order is exactly the kind of thing that takes a long time to check. This site takes the paper at its word only about what it says, not about whether it is right.

Does this mean water can blow up?

No. The force is engineered: it is whatever residual the designed flow leaves behind, arranged so that residual is smooth. It says nothing about what a fluid does on its own from smooth initial data — alternatives (A) and (B) — and nothing about physical fluids, where the continuum model fails long before speeds become infinite.

How can the velocity be unbounded while the energy is bounded?

The speeds diverge like τ−½−h on a core of volume τ3/2−h. Energy is speed squared times volume: τ1/2−3h, which tends to zero. Even the total dissipation ∫‖∇u‖² is finite, because its integrand ≍ τ−½−3h is integrable when h < 1/6. That is why the exponent is small.

Why can the force be smooth if the flow is singular?

Because individual terms of the momentum equation may diverge while their sum does not. The core balances itself to leading order; the annulus is balanced by the pulses' averaged momentum flux; the exterior is an exact solution; and everything left over is "flat" — every derivative is O(qN) for every N. Flat functions extend smoothly by zero, and Lemma 10.3 extends the force through t = 1 with the matching Taylor data.

Where does the small exponent h come from?

It is the mismatch between radial and axial contraction, ℓz/ℓr ≍ τ−h. It makes axial diffusion weaker than radial diffusion by the factor q2h, which is the expansion parameter for the whole background correction series, and it gives the swirl Reynolds number Reθ ≍ τ−h → ∞ while the radial one stays O(1). The price is that q2h with h < 1/100 is only small when q is astronomically small — see bands.