Euler's Equations of Inviscid Motion

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

(1)

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

(2)

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.

The author reports Claude assistance with detailed proofs, estimates, certificate computation, and formalization. The analytic implication and certificate-checker soundness are formalized in Lean. The recorded certificate evaluation also trusts the Lean compiler, runtime, and GMP. As of Oct. 25, 2026, Part II remained a working draft and independent specialist verification had not been reported (Coiculescu 2026b).

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

(3)

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. (2026ab) report substantive GPT-6 assistance with mathematical arguments and verification.