1 · The essentials
The manuscript studies a Schrödinger equation on a twelve-dimensional space with periodic boundaries. It shows that, under particular conditions, solutions can become singular after a finite amount of time rather than remain regular indefinitely. This behavior is called blowup. The result concerns a defocusing equation, with a nonlinear power that must be odd and sufficiently large. Crucially, blowup occurs for a whole open collection of starting states, so sufficiently small changes to those states do not remove it. The manuscript also shows that certain randomly chosen starting states can have a positive chance of blowup, even when they have arbitrarily high regularity. The main stable-blowup result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, think of a patch on a map where every starting point leads to a route that ends abruptly. Moving your starting point a little while staying inside the patch does not avoid that ending; this pictures stability of blowup, not the equation’s actual dynamics.
Why it matters
The result establishes finite-time blowup for an open set of starting states, rather than just for isolated choices. Its stability is specifically measured in a Sobolev space, which controls the size and regularity of changes to the data. The random-data consequence also connects this behavior to Gaussian Fourier initial data, under the conditions stated in the manuscript.
2 · Going further
The precise setting is a scalar, supercritical, defocusing nonlinear Schrödinger equation on the twelve-dimensional torus . For a sufficiently large odd nonlinearity power, the repository summary states that there is a nonempty open set of initial data in , with , whose solutions blow up in finite time. Openness is the precise sense of stability here: each datum in that set has a sufficiently small Sobolev neighborhood consisting entirely of blowup data.
The main manuscript describes the blowup as self-similar. This identifies the form of the behavior being constructed, but the supplied abstract does not describe the construction, estimates, or stability argument. Thus the available account supports the conclusion that a self-similar blowup construction is stable, without supplying a more detailed mechanism for that stability.
As a consequence, the manuscript shows that Gaussian Fourier initial data of arbitrarily high Sobolev regularity can have positive probability of finite-time blowup. Only the main manuscript is supplied, so no relationship to companion manuscripts can be established from this source. The main result has a Lean formalization; the supplied information does not specify whether that formalization also covers the random-data consequence.
Useful notions
- Initial data and evolution equations
- Periodic domains and the torus
- Sobolev spaces and regularity
- Open sets and small perturbations
- Self-similar behavior
What was known before / what is new
Before
The supplied source gives no account of earlier results or historical development. It identifies the setting as a supercritical defocusing Schrödinger equation, but does not state what blowup or stability results were previously available.
New
The main result constructs stable self-similar finite-time blowup on the twelve-dimensional torus for a sufficiently large odd nonlinear power. The blowup data contain a nonempty open set in the high Sobolev space specified by the repository summary. The manuscript further derives a positive-probability blowup consequence for Gaussian Fourier initial data of arbitrarily high Sobolev regularity.
Limits and assumptions
The result is restricted to the stated scalar defocusing equation on , a sufficiently large odd nonlinearity power, and a high Sobolev space with . The source gives neither a numerical threshold for that power nor a more detailed range of admissible Sobolev indices. Stability means openness of the blowup data in that Sobolev space; it does not assert blowup for all initial data. The main result has a Lean formalization, but the supplied material does not delimit its coverage of every additional claim. No specific remaining open problem or companion manuscript is described.
Mini glossary
- Finite-time blowup —
- The development of a singularity after a finite amount of evolution time.
- Twelve-dimensional torus —
- The periodic domain used for the equation in this result.
- Sobolev space —
- A function space, denoted here by , that measures size and regularity and specifies which perturbations count as small.
- Open set of blowup data —
- A collection in which every starting datum has a sufficiently small neighborhood whose data also produce blowup.
- Self-similar blowup —
- Blowup whose evolving form is described through rescaling.
- Gaussian Fourier initial data —
- Random starting data described using Fourier coefficients with Gaussian randomness.
Keywords : Defocusing Schrödinger equation · Stable blowup · Self-similarity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in , 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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus
24/09/2026We 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.