Browse › Partial differential equations › 371 Stable blowup for the defocusing Schrödinger equation
Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove stable self-similar finite-time blowup for a supercritical defocusing nonlinear Schrödinger equation on the twelve-dimensional torus. For a sufficiently large odd power, the blowup initial data contain a nonempty open set in a high Sobolev space. As a consequence, Gaussian Fourier initial data of arbitrarily high Sobolev regularity can have positive probability of finite-time blowup.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 371: Stable blowup for the defocusing Schrödinger equation (lean/docs/371.md, October 2026). The scope note describes a nonempty open blowup set in $H^k(\mathbb T^{12})$ for arbitrarily large selected odd powers (rather than every sufficiently large odd power) and corresponding $k>8$, together with positive-probability Gaussian blowup for every decay exponent $\alpha>k+6$, with comparator file
DefocusingNLS.lean. [family-level] - punk01-shaka (GitHub), OpenAI's Comparator configs leave 62 fully-written definitions unchecked (definition holes) — fake proofs accepted for 3 results (GitHub issue, 12 Oct 2026). A third-party audit of the release's comparator configurations; its scope table lists the
DefocusingNLSconfig with 1 of 3 definition holes having a real body, but this config is not among the three fake-proof demonstrations. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Roland Donninger, Lorenz Lichtnecker, Self-similar blowup for mass supercritical Schrödinger equations (arXiv:2509.16600, Sep 2025). A focusing-NLS self-similar blowup result (three dimensions, powers near three) that extends the cubic case; relevant to the paper's discussion of rigorous focusing profile constructions, where only Dahne–Figueras and Donninger–Schörkhuber are cited.
- Feng Shao, Dongyi Wei, Zhifei Zhang, On blow-up for the supercritical defocusing nonlinear scalar wave equation (arXiv:2610.19757, 14 Oct 2026). For the defocusing wave equation, the authors verify self-similar imploding-profile conditions for all sufficiently large odd p; this parallels the paper's large-power limit a = 1/(p−1) → 0 for the Schrödinger profile.
- M. Cadiot, A. D. Ionescu, S. Palasek, Finite-time blow-up for the real-valued defocusing energy-supercritical quintic NLW (arXiv:2610.06678, 5 Oct 2026). A computer-assisted (Newton–Kantorovich) construction of discretely self-similar defocusing supercritical blowup for the wave equation, giving wave-equation context for the paper's certificate-based profile analysis.
Digests and expositions
- Latent Seminar, Stable blowup for a supercritical defocusing Schrödinger equation (6 October 2026). This exposition explains the claimed profile construction, spectral decay, expanding-torus localization, and passage from physical-data openness to positive Gaussian probability, explicitly distinguishing its discussion from independent verification of the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 371
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026,
author = {{OpenAI}},
title = {{Stable self-similar blowup for a supercritical defocusing Schr{\"o}dinger equation on the torus}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/paper.pdf}{OAI:Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026}},
year = {2026}
}From OpenAI's README for this paper:
Verification
The exact-arithmetic verification requires Python 3.8 or newer with only its
standard library, and GNU sha256sum. Follow the
verification guide for the verifier run,
the strict comparison with historical results, and the current file-integrity
check. Success requires all three steps to pass. These checks recompute the
finite integer and rational comparisons; they do not verify the analytic
arguments connecting those comparisons to the blowup theorem.
Lean formalization
OpenAI provides Lean files for this family: lean/docs/371.md. The Lean statements have not been checked by people against the paper's claims.
Family 371: Stable blowup for the defocusing Schrödinger equation
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in $H^k(\mathbb T^{12})$, with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.
This is the only paper in this family.