Non-uniqueness of unforced two-dimensional Euler flows with unbounded integrable vorticity

A two-part manuscript by Matei P. Coiculescu, whose detailed proofs, code and text were largely produced with Claude Opus 5.5 under his direction, claims that the incompressible Euler equations in the plane, with no external force, admit two distinct renormalized weak solutions from the same radial initial vorticity, which lies in $L^1 \cap L^q$ for every $q < \infty$ but is unbounded; Part I is on arXiv and the whole proof is formalized in Lean.

What was found

One solution is the stationary radial vorticity itself. The other is 3-fold rotationally symmetric, equimeasurable with the datum at every time, and leaves it in finite time; the two differ in $L^\infty_t (L^1 \cap L^q)_x$ for every $q \in (4/3, \infty)$ and every large enough time horizon. Part I proves two ingredients: a compactness criterion, under which radial data approximated by 3-fold symmetric solutions whose non-radial moment stays bounded below yield a second solution, and a linear analysis showing an unstable eigenvalue for the linearization around the basic building block plus a small point vortex. Part II, the nonlinear construction, is in the repository but not on arXiv. The repository's Lean 4 development, about 1,280 files and 170,000 lines, proves that a three-part Hypothesis (N) implies the main theorem, proves (N3) directly, and proves sound a checker for the computer-assisted parts (N1) and (N2). Those were certified twice: by a Python program in rigorous ball arithmetic (FLINT/Arb), and by a compiled run of the Lean checker on 2026-10-22 that took 8.75 hours and printed true on all 261 cells.

Novelty check

arXiv, 2026-11-05: abs:"Euler" AND abs:"non-uniqueness" AND abs:"vorticity" returned 10 records, among them Vishik's forced non-uniqueness (1805.09426), an exposition of it (2112.04943) and forced vanishing-viscosity non-uniqueness (2507.19257); none is unforced. The closest result is concurrent: Palomar record PALOMAR-2026-10-28-000008, registered 2026-10-28 by Tao Tao and Haiqi Zhang, six days after this paper's arXiv version 1, formalizes non-uniqueness for the unforced Euler equations on the two-dimensional torus with vorticity in $L^p$ for each finite $p$, which its abstract identifies as the two-dimensional finite-$L^p$ question in the first part of Buckmaster and Vicol's Problem 6. The setting differs (the torus against the plane, a different construction) and the two appear independent; that development declares human authors working with ChatGPT and an Aristotle formalization, and is not in this registry. Its sources also cite Bruè, Colombo and Kumar (arXiv:2408.07934) and Mengual (2024) on non-uniqueness with integrable or $L^p$ vorticity, whose exact relation to this result was not assessed here. Within this registry 2026-09-08-euler-unforced-blowup is a finite-time blowup claim, a different question, and 2026-09-15-ipm-forced-blowup is earlier work by the same author.

Also recorded at

Graded claimed for verification and ai-assisted for autonomy. What these mean.

Cite this entry

whataifound.org. (2026). Non-uniqueness of unforced two-dimensional Euler flows with unbounded integrable vorticity. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-22-euler-2d-unforced-nonuniqueness

@misc{whataifound-independent-2026-nonuniqueness,
  title        = {Non-uniqueness of unforced two-dimensional Euler flows with unbounded integrable vorticity},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Claimed. Autonomy: AI-assisted.},
  url          = {https://whataifound.org/finding/2026-10-22-euler-2d-unforced-nonuniqueness}
}

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