The paper was not written for human readers. The companion page asks the questions a reader would ask (why rotation beats viscosity, how a wave can push the mean flow, why two wave families and a cone, why pulses grow and stop) and answers each with a toy model checked by script, a dependency tree of the paper's results, and the questions the paper leaves open.
Theorem 1.1. For every viscosity ν > 0 there are a smooth force f, compactly supported in space and time, a compact set K, and smooth u, p on ℝ³ × [0, 1) solving the Navier–Stokes equations with u(·, 0) = 0, supported in K, with kinetic energy bounded for all t < 1 and with supx|u(x, t)| → ∞ as t ↑ 1.
So the fluid starts at rest. A smooth push, switched on and off inside a fixed region, drives it to infinite speed at time 1, while its total energy stays bounded. No smooth bounded-energy solution can continue past that moment. Clay's problem statement (Fefferman, 2000) asks for a proof of any one of four alternatives. This is alternative (C), breakdown on ℝ³ with a smooth, decaying force allowed. Because everything has compact support, the same flow also gives (D), breakdown on the torus (Corollary 10.6). CITED
With a force allowed, any flow at all "solves" the equations: pick u and p, and call whatever is left over the force, f = ∂tu + (u·∇)u − νΔu + ∇p. The paper calls this the momentum residual R(u, p). The whole difficulty moves into one requirement: f must stay smooth, with every derivative bounded, right through the blowup time, even though u and its derivatives become infinite there. The individual terms of R diverge. Their sum must not.
So the paper is a cancellation problem. It builds a flow whose singular terms cancel to every order at the singular point, and then checks that what is left over is a perfectly ordinary smooth force (§2, §3).
The flow has four parts (§2):
Everything in the core is a power of the remaining time τ = 1 − t. A negative power grows without bound as τ → 0; a positive power vanishes. Move the slider to change the small exponent h and see which quantities blow up. The paper takes 0 < h < 1/100. Energy stays bounded and dissipation stays integrable only while h < 1/6, which is why the paper writes "Since h < 1/6" (§3.5). Each exponent is re-derived by the verify script below.
Marks on the slider scale below: the paper's range ends at 1/100; energy control fails at 1/6.
Similarity coordinates X = r²/(2q), η = z/qD, with τ = q(1 − η²), turn the shrinking core into fixed profiles E, U, Π for swirl, axial velocity and pressure. The profiles are built near the axis (analytic), joined to the heat exterior, and five "radial moments" are matched so that the leftover residual lives only in the annulus and can be written as the divergence of a stress T.
Theorem 4.6; Appendices A (heat exterior, moment matching) and B (profiles near the axis).
A wave family produces momentum flux only in certain directions. With two families, fluxes v₁, v₂, the target must satisfy T = c₁v₁ + c₂v₂ with c₁, c₂ > 0: it must lie inside the cone they span. The profile is tuned (a small oscillation with phase N log X) so that this "admissible stress cone" condition holds everywhere in the annulus.
Lemma 4.5; Proposition C.2, C.3 (Appendix C).
Corrections in powers q2nh absorb the radial momentum equation, axial viscosity and lower-order terms. The result is an axisymmetric background whose residual is exactly the annular stress divergence plus a remainder that is "flat": every derivative is O(qN) for every N.
Proposition 5.5.
Divergence-free oscillations w = ∇ × A, written as waves a cos(ξ·x + φ) with a ⟂ ξ, grow by drawing energy from the shear and then decay as the shear tilts their wavevector and viscosity takes over. Their averaged products ⟨wrwθ⟩, ⟨wrwz⟩ equal T to leading order, so their flux cancels the background's leading residual. An auxiliary torus variable keeps different pulses from interacting.
Lemmas 7.1, 7.4, 7.7; Propositions 7.5, 9.5; Section 6 (auxiliary torus).
Every correction creates new errors (linear terms, wave–wave interactions, cutoff errors). A fixed cycle of four operations removes them and improves the decay exponent by 1/10 each time, σj+1 = σj + 1/10 from σ0 = 1/5. Summing with shrinking cutoffs gives a local flow whose residual is flat at the singular point.
Propositions 9.1, 9.6, 9.9; Section 8 (mean corrections).
Cut off in space and time, define f as the residual, and show every derivative of f has a limit at t = 1, so f extends to a smooth compactly supported force. Bounded energy follows from the scalings. A uniqueness lemma then shows any smooth bounded-energy solution with the same force and data must equal this one before t = 1, so none survives past it.
Proposition 10.1; Lemmas 10.2–10.5; Theorem 1.1, Corollary 10.6.
The local result behind moves 1–5 is stated as Theorem 3.1: a divergence-free field, smooth at the axis, with a flat residual near the singular point, zero residual outside a fixed radius, and swirl growing like τ−1/2−h. CITED
| Part | Pages | What it does | First reading |
|---|---|---|---|
| 1 Introduction | 1–2 | The theorem and the history: Leray, Caffarelli–Kohn–Nirenberg, Escauriaza–Seregin–Šverák, Tao's averaged equation, convex integration, Córdoba–Martínez-Zoroa. | read |
| 2 Physical description | 3–6 | The core, annulus, pulses and exterior, in words and two figures. | read |
| 3 Proof outline | 6–23 | The whole argument in 18 pages, with the flow charts (Figures 5, 6), Theorem 3.1 and a symbol table. | read |
| 4 Leading order flow | 24–44 | Similarity variables, the stress profile, the admissible stress cone, Theorem 4.6. | skim |
| 5 Background to every order | 45–61 | The q2nh expansion and divergence-preserving summation; Proposition 5.5. | skim |
| 6 Auxiliary torus | 62–72 | Dyadic charts and the trick that keeps pulses from interacting. | reference |
| 7 Oscillatory realization | 73–87 | Phases, pulse equations, growth then decay, the covariance that matches T. | skim |
| 8 Mean corrections | 88–99 | Angular-mean residuals, pressure reconstruction, five moment equations. | reference |
| 9 Residual improvement | 100–115 | The correction cycle and its summation into the local field. | skim |
| 10 Compact forcing | 116–125 | Localization, smooth extension of the force, energy, uniqueness, the torus. | read |
| Appendices A–C | 126–164 | Heat exterior and moment matching; analytic profiles near the axis; realizing the stress cone. | reference |
A good first pass: §§1–3 and §10, about 35 pages. Then Theorem 4.6, Lemma 4.5, Proposition 5.5, Lemma 7.4, Proposition 7.5 and Proposition 9.6 by their statements only.
These are the places where an argument of this kind can fail. They are a reading list, not a report of errors; we have not found any.
The paper places itself in a line of work (§1.1). Leray (1934) built global weak solutions. Caffarelli, Kohn and Nirenberg limited the size of the singular set. Escauriaza, Seregin and Šverák proved regularity under a scale-invariant bound. Tao (2016) proved blowup for an averaged version of the equations. Buckmaster and Vicol, and then Albritton, Brué and Colombo, proved non-uniqueness of weak solutions. Córdoba and Martínez-Zoroa, later with Zheng, produced forced singularities for Euler and hypodissipative Navier–Stokes by amplifying ever-smaller vortex layers. The wave tools go back to Craik and Criminale's exact waves on shear flows and to the Lifschitz–Hameiri and Friedlander–Vishik geometric optics for fluids. What is new here, in the paper's words: "oscillatory pulses generate a mean momentum flux that supplies the missing force on a collapsing background vortex." CITED
The verify script re-derives every exponent on this page from the paper's stated scalings, checks the viscosity rescaling symbolically, and, if you give it the PDF, confirms that the quoted sentences are in the paper. It checks the arithmetic of the sketch, not the proof.
python3 index-verify.py --pdf navier-stokes.pdf