Bibliography

Grouped by the role each work plays relative to the construction, not alphabetically. The map draws the same works with their relations.

The construction and its line

• OpenAI (2026). Finite Time Blowup for Navier–Stokes. Manuscript, 166 pp. PDF · Lean.
  The object of this site. Sections 2 and 3 are readable on their own and are what the guide pages paraphrase.
• Córdoba, D. and Martínez-Zoroa, L. (2023). Blow-up for the incompressible 3D-Euler equations with uniform $C^{1,1/2-\epsilon}\cap L^2$ force. Duke Math. J., to appear. arXiv:2309.08495.
  Forced blow-up by amplification across scales, non-axisymmetric, with the force held in a Hölder class. The method every 2026 paper descends from.
• Córdoba, D., Martínez-Zoroa, L. and Zheng, F. (2024). Finite time blow-up for the hypodissipative Navier–Stokes equations with a force in $L^1_tC^{1,\epsilon}_x\cap L^\infty_tL^2_x$. arXiv:2407.06776.
  Small fractional dissipation. Shows what the full Laplacian had been obstructing.
• Alpöge, L., Buckmaster, T. and Coiculescu, M. P. (2026). Extending the Córdoba–Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing. arXiv:2609.16470.
  Smooth forcing for the porous medium equation, with Boussinesq and Euler in companion preprints and Lean proofs. The human-led result of the same week.
• Tao, T. (2016). Finite time blowup for an averaged three-dimensional Navier–Stokes equation. J. Amer. Math. Soc. 29(3):601–674. doi:10.1090/jams/838.
  Modifies the bilinear term by averaging over rotations and Fourier multipliers, keeps the energy identity, and blows up. Often read alongside the forced results; it is a different move, and its conclusion does not transfer to the original equations.

The problem statement and the classical bounds

• Fefferman, C. L. (2000). Existence and smoothness of the Navier–Stokes equation. Clay Mathematics Institute. PDF.
  Alternatives (A) to (D). The force in (C) and (D) must be smooth and decay with all derivatives faster than any power; it need not be small or analytic.
• Leray, J. (1934). Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63:193–248. doi:10.1007/BF02547354.
  Global weak solutions with the energy inequality. Also the first to propose self-similar blow-up, which Nečas, Růžička and Šverák later excluded in $L^3$.
• Caffarelli, L., Kohn, R. and Nirenberg, L. (1982). Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. 35(6):771–831. doi:10.1002/cpa.3160350604.
  The singular set has zero one-dimensional parabolic Hausdorff measure. Isolated singular points are allowed, and the construction produces exactly one.
• Escauriaza, L., Seregin, G. A. and Šverák, V. (2003). $L_{3,\infty}$-solutions of Navier–Stokes equations and backward uniqueness. Russian Math. Surveys 58(2):211–250. doi:10.1070/RM2003v058n02ABEH000609.
  Bounded $L^3$ norm prevents blow-up of the unforced equations. The construction's $\int|u|^3$ over the core grows like $\tau^{-4h}$; the exponent $h$ is what escapes this criterion.
• Beale, J. T., Kato, T. and Majda, A. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94(1):61–66. doi:10.1007/BF01212349.
  Euler blow-up requires the time integral of maximum vorticity to diverge.

Nonuniqueness and instability in similarity variables

• Buckmaster, T. and Vicol, V. (2019). Nonuniqueness of weak solutions to the Navier–Stokes equation. Ann. of Math. 189(1):101–144. doi:10.4007/annals.2019.189.1.3.
  Convex integration for Navier–Stokes. The use of oscillations to realize a prescribed stress, central to the 2026 annulus, comes from this tradition.
• Albritton, D., Brué, E. and Colombo, M. (2022). Non-uniqueness of Leray solutions of the forced Navier–Stokes equations. Ann. of Math. 196(1):415–455. doi:10.4007/annals.2022.196.1.3.
  Two Leray–Hopf solutions from rest with the same force, built on an unstable vortex in similarity variables; the force is singular at the initial time. Instability of a self-similar vortex is also what seeds the 2026 pulses.
• Daneri, S. and Székelyhidi, L. (2017). Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 224(2):471–514. doi:10.1007/s00205-017-1081-8.
  Oscillatory building blocks that realize a prescribed Reynolds stress. The paper cites it as the source of that idea.

Shear-flow waves, the physics of the pulses

• Craik, A. D. D. and Criminale, W. O. (1986). Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier–Stokes equations. Proc. R. Soc. Lond. A 406:13–26. doi:10.1098/rspa.1986.0061.
  Exact finite-amplitude waves on affine background flows, exploiting the cancellation of the wave's quadratic self-interaction. The 2026 pulses are these waves, localized and viscously damped.
• Lifschitz, A. and Hameiri, E. (1991). Local stability conditions in fluid dynamics. Phys. Fluids A 3(11):2644–2651. doi:10.1063/1.858153.
  Evolution of wavevectors and polarizations along a background flow, the WKB ancestor of the amplitude equation on the pulses page.
• Billant, P. and Gallaire, F. (2005). Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542:365–379. doi:10.1017/S0022112005006464.
  When a rotating flow amplifies non-axisymmetric disturbances. The paper's growth rate $\lambda_0^2 = -2F_0N_\theta(2F_0N_\theta+|g_0|)$ is a criterion of this kind.

Responses to the construction

• Constantin, P., Ignatova, M. and Vicol, V. (2026). Regularity of asymptotically axisymmetric solutions to the 3D Navier–Stokes equations with analytic forcing. arXiv:2609.20803.
  Analytic force plus the construction's bounds implies regularity. The force must be non-analytic and active at the singular point.
• Cao, S., Chi, Z. and Nie, P. (2026). Density of Forces Producing Navier–Stokes Blowup. arXiv:2609.10262.
  Blow-up forces are dense near any smooth solution in $L^1_tH^s_x$, $s<1/2$.
• Duraiswami, R. (2026). Self-similar swirl between contracting porous walls. arXiv:2609.17642.
  The core geometry against an exact classical swirl; the mechanism is not reachable in a flow one computes or builds.

Fluctuating fluids

• Agresti, A. (2026). On the absence of blow-up in the 3D Navier–Stokes equations with transport noise. arXiv:2607.15140.
  Global smooth solutions with high probability for large data, and by Remark 4.4 for a smooth force. The theorem behind the transport-noise row of the verdict table.
• Flandoli, F. and Luo, D. (2021). High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations. Probab. Theory Related Fields 180:309–363. doi:10.1007/s00440-021-01037-5.
  Transport noise with high spectrum and large intensity bounds vorticity with high probability; the scaling limit is a deterministic equation with eddy viscosity. Records that additive-noise attempts remained inconclusive.
• Landau, L. D. and Lifshitz, E. M. (1959). Fluid Mechanics, Chapter on hydrodynamic fluctuations. Pergamon; 2nd ed. 1987. doi:10.1016/C2013-0-03799-1.
  The random stress with variance fixed by the fluctuation-dissipation relation. The physically correct correction to the equations above the mean free path, and the source of the thermal estimates on the seeds page.
• Bell, J. B., Nonaka, A., Garcia, A. L. and Eyink, G. (2022). Thermal fluctuations in the dissipation range of homogeneous isotropic turbulence. J. Fluid Mech. 939:A12. doi:10.1017/jfm.2022.188.
  Fluctuating hydrodynamics simulations showing thermal noise dominating the far dissipation range. Evidence that the scales the construction reaches are thermal in a real fluid.
• Braunstein, S. L. (2026). Physical completion of the Navier–Stokes equations. arXiv:2605.21357.
  Thermal noise and a molecular cutoff: finite-dimensional, Lyapunov function, globally well posed.

Averaging

• Dubrulle, B. and Frisch, U. (1991). Eddy viscosity of parity-invariant flow. Phys. Rev. A 43(10):5355–5364. doi:10.1103/PhysRevA.43.5355.
  Multiscale expansion of Navier–Stokes around a small-scale flow gives an eddy viscosity. The deterministic cousin of the transport-noise scaling limit, and the reason averaging the pulses drops a leading term.
• Cotton, P. (2026). Homogenization. homogenization.microprediction.org.
  Averaging a fast exogenous variable and computing what the average leaves out. The switching model of the fluctuations page, and why its corrector does not apply to oscillations the flow generates itself.