Result 371, Partial differential equations

Stable blowup for the defocusing Schrödinger equation

For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in Hk(T12)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.

Lean formalization Other

The bigger picture

Why it matters

Defocusing wave equations are associated with spreading, yet this does not automatically rule out singularities. The unreviewed manuscript reports finite-time blowup for one such equation in a specific high-dimensional regime, even when the starting wave is slightly perturbed.

What changes?

The scalar defocusing nonlinear Schrödinger equation describes a single wave field; here it lives on a twelve-dimensional torus, meaning twelve periodic coordinates. For a sufficiently large odd nonlinearity power, the manuscript reports self-similar blowup, with structure that repeats under rescaling. Initial data producing finite-time blowup contain a nonempty open set in a Sobolev space of order k greater than 8, which measures regularity using derivatives. Openness means blowup survives sufficiently small perturbations measured in that same space.

What does that help mathematicians do?

A reported consequence concerns random starting waves: Gaussian Fourier initial data can have positive probability of finite-time blowup even at arbitrarily high Sobolev regularity. Thus strong initial regularity does not, in this supercritical regime, guarantee a solution that remains regular forever. The open-set construction also shows that singularity formation need not depend on choosing one exceptional, finely tuned starting configuration.

Are there practical applications?

The immediate value is foundational: the construction supplies a robust test case for theories of long-time behavior in nonlinear dispersive equations. It identifies a regime where defocusing does not prevent singularity formation. The setting is specific, so the result does not by itself establish analogous behavior in lower-dimensional physical models.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus

September 24, 2026 56 pages

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.

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Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/371.md.

Stable blowup for the defocusing Schrödinger equation

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization constructs stable self-similar finite-time blowup for defocusing nonlinear Schrödinger equations on the twelve-dimensional torus. For every prescribed lower bound on the power, it selects an odd power at least that large and a Sobolev index k>8k>8 for which a nonempty open set of HkH^k initial data has the stated classical blowup behavior. The selected quantifier gives arbitrarily large odd powers, rather than every sufficiently large odd power.

For each such construction, Gaussian Fourier data with decay exponent α>k+6\alpha>k+6 assign positive probability to the blowup set, and almost-sure global existence fails. The exponent α\alpha may be arbitrarily large.

Comparator links

Result Comparator statement
Stable self-similar blowup and positive-probability Gaussian blowup DefocusingNLS.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.