Euler's equations of inviscid motion are the system of partial differential equations describing fluid flow in the absence of viscosity, given by

where is the fluid velocity, is the pressure, and is the fluid density.

The regularity problem for the three-dimensional incompressible Euler equations asks whether smooth divergence-free initial data can produce finite-time blow-up. Alpöge and Buckmaster (2026a) claim finite-time blow-up on from smooth axisymmetric initial data with swirl and a force that is smooth in space and time and supported in a fixed solid torus. In their construction, the circulation and meridional velocity stay bounded, while the circulation gradient and full vorticity become unbounded and

Buckmaster (2026) reports substantial assistance from Claude and Codex across the project. An accompanying Lean 4 development formalizes the claim, but independent human verification had not been reported as of Sep. 9, 2026 (Alpöge and Buckmaster 2026b).

OpenAI (2026ab) announced a proposed resolution on Sep. 8, 2026. Its paper claims a smooth, divergence-free initial velocity having compact support on whose maximal smooth solution has finite lifetime. The claimed solution has unbounded velocity gradient and satisfies the divergent vorticity integral condition in the Beale-Kato-Majda criterion.

Coiculescu (2026ab) reported nonuniqueness for the unforced two-dimensional incompressible Euler equations on . The construction gives a radial nonnegative initial vorticity belonging to for every finite , but not to , with two distinct renormalized weak solutions. Weak solutions satisfy the equations after integration against smooth functions with compact support. Renormalization additionally requires the transport equation to hold for smooth functions of the vorticity. One solution is stationary. The other has three-fold rotational symmetry and preserves the distribution of vorticity values. The solutions are distinct in for every on sufficiently long time intervals. The velocity is not square-integrable, so this is not a finite-energy nonuniqueness result.

Coiculescu (2026ab) reports Claude Opus 5.5 assistance with detailed proofs, estimates, certificate computation, and formalization. Part II (Coiculescu 2026b) gives the nonlinear construction and its computer-assisted estimates. Lean formalizes the analytic implication and the soundness of the certificate checker. The recorded evaluation of the certificate also relies on the Lean compiler, runtime, and GMP. Paperlane (2026) independently checked that the Lean project compiled at the earlier revision 1df5e7953d9e.

Choi (2026) obtained finite-energy nonuniqueness in two dimensions. The initial vorticity has compact support, has integral 0, and belongs to for every finite . Two distinct renormalized solutions with these same initial data have velocities continuous in time in and preserve the norms of the vorticity. This does not assert nonuniqueness for bounded initial vorticity. Choi (2026) reports AI assistance with the technical proof of the quadrupole-patch cascade.

Hou et al. (2026b) established a sharp envelope criterion within a specified class of stationary arrays of disjoint radial zero-circulation vortices. If the vorticity norms are bounded by , with and finite and nondecreasing, the threshold is

Divergence gives uniqueness in that class even against bounded energy-admissible weak velocity solutions. Convergence permits a compactly supported datum in the class with two distinct unforced evolutions described by Lagrangian flows. The criterion is not asserted for arbitrary vorticity.

Hou et al. (2026a) also supplied an independent construction of nonuniqueness for each prescribed , with initial vorticity in and two evolutions on . Both are strongly continuous in these spaces and exactly transported by their measure-preserving regular Lagrangian flows. The datum may depend on , so this does not strengthen Choi's all-finite-, global-time result above. Hou et al. (2026b) credit GPT-6 with the dyadic comparison and transfer estimates used in the envelope analysis.

See also

Beale-Kato-Majda Criterion, Boussinesq System, Convective Derivative, Euler Differential Equation, Finite-Time Blow-Up, Incompressible Porous Media Equation, Navier-Stokes Equations, Vorticity

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References

Alpöge, L. and Buckmaster, T. "Blowup for the Euler Equations with Smooth Forcing." Sep. 8, 2026a. https://cims.nyu.edu/~tristanb/euler.pdf.Alpöge, L. and Buckmaster, T. "Euler Blowup." Sep. 8, 2026b. https://github.com/tristanbuckmaster/fluid_lean/tree/main/euler-blowup.Buckmaster, T. "Statement." Sep. 8, 2026. https://cims.nyu.edu/~tristanb/statement.pdf.Choi, H. "Nonuniqueness of Solutions to 2D Incompressible Euler Equations with Unbounded Vorticity." 27 Oct 2026. https://arxiv.org/abs/2610.37653.Coiculescu, M. P. "Non-Uniqueness of Weak Solutions to the Unforced Two-Dimensional Euler Equations with Vorticity in , Part I: A Compactness Criterion and the Linear Analysis." 22 Oct 2026a. https://arxiv.org/abs/2610.30655.Coiculescu, M. P. "Non-Uniqueness of Weak Solutions to the Unforced Two-Dimensional Euler Equations with Vorticity in , Part II: The Nonlinear Construction." 6 Nov 2026b. https://arxiv.org/abs/2611.10211v1. Project materials: https://github.com/mcoiculescu/NonUniqueEuler/tree/b0deb9f061460bd3b83791cbf80fbaed1b6736be/manuscript/part2.Hou, T. Y.; Qin, X.; Song, P.; and Zhang, W. "Nonuniqueness of the D Euler Evolutions with Vorticity." 29 Oct 2026a. https://arxiv.org/abs/2610.40553.Hou, T. Y.; Qin, X.; Song, P.; and Zhang, W. "A Sharp Uniqueness-Nonuniqueness Dichotomy for Neutral Vortex Arrays in D Euler Equations." 29 Oct 2026b. https://arxiv.org/abs/2610.40552.Landau, L. D. and Lifschitz, E. M. Fluid Mechanics, 2nd ed. Oxford, England: Pergamon Press, p. 3, 1982.OpenAI. "Finite Time Blowup for the Euler Equation." Sep. 8, 2026a. https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf.OpenAI. "On the Navier-Stokes Millennium Prize Problem." Sep. 8, 2026b. https://openai.com/index/navier-stokes-solution/.Paperlane. "NonUniqueEuler: Lean Build." Oct. 27, 2026. https://pragyaangaur.dev/Paperlane/lean/mcoiculescu/NonUniqueEuler.html.Quanta Magazine. "Why Navier-Stokes Pushes Math and Physics to the Edge." Sep. 17, 2026. https://www.youtube.com/watch?v=PsWR1rQEWYo.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997.

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Euler's Equations of Inviscid Motion

Cite this as:

Weisstein, Eric W. "Euler's Equations of Inviscid Motion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulersEquationsofInviscidMotion.html

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