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Math-2838 · mathematics · Pde

Stable amplitude blowup for a supercritical defocusing torus equation

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does the defocusing power NLS i $partial_t$ u+Delta $u-|u|^{p-1}u=0$ on $T^12$ admit an odd $p\geq 3$, finite integer $k>8$ and a nonempty open set in $H^k$ such that every initial datum in that set has finite-time classical $H^k$ blowup? Require a point $x_*$ and $c_0>0$ with $(T-t)^(1/(p-1))|u(t,x_*)|\to c_0$.

Short English statement

Does the defocusing power NLS i $partial_t$ u+Delta $u-|u|^{p-1}u=0$ on $T^12$ admit an odd $p\geq 3$, finite integer $k>8$ and a nonempty open set in $H^k$ such that every initial datum in that set has finite-time classical $H^k$ blowup? Require a point $x_*$ and $c_0>0$ with $(T-t)^(1/(p-1))|u(t,x_*)|\to c_0$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2838-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2838

Stable defocusing Schrödinger blowup: The equation is defocusing on $\mathbb T^{12}$, with a selected odd power and a high but finite Sobolev index. The claimed open set has a fixed spatial point whose rescaled amplitude tends to a positive limit, so the conclusion is classical finite-time blowup of the required type.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Stable amplitude blowup for a supercritical defocusing torus equation

OpenAI's claim

Stable defocusing Schrödinger blowup: The paper claims finite-time self-similar blowup on the twelve-dimensional torus for suitable large odd powers of a defocusing nonlinear Schrödinger equation. Blowup occurs on an open high-Sobolev set and has positive probability for sufficiently regular Gaussian Fourier data.

Mathematical statement of the claimed result

Mathematical claim 1.
For the defocusing equation $i\partial_tu+\Delta u=|u|^{p-1}u$ on $\mathbb T^{12}=(\mathbb R/2\pi\mathbb Z)^{12}$, the claim gives some sufficiently large odd $p\ge3$, fixed integer $k>8$ and nonempty open $\mathcal U\subset H^k$ such that every $u_0\in\mathcal U$ has a unique local solution in $C([0,T);H^k)\cap C^1([0,T);H^{k-2})$ with finite $T>0$ and no continuous $H^k$ extension through $T$. More precisely some $x_*,c_0>0$ obey \[\lim_{t\uparrow T}(T-t)^{1/(p-1)}|u(t,x_*)|=c_0.\] The exponent and Sobolev order are chosen by the construction, not quantified over every supercritical power.

Outline of the argument

Stable defocusing Schrödinger blowup: A smooth profile with a pressurized core is matched to a free exterior, and exact finite arithmetic checks the relevant profile and spectral signs. Outgoing spectral control gives linear decay in the rescaled equation. Nonlinear stability on expanding tori produces an open set of blowup data, after which Gaussian support gives the positive-probability consequence.

Formal verification

Family 371 describes Lean coverage for stable blowup for arbitrarily large selected odd powers and the positive-probability Gaussian consequence.

Stable defocusing Schrödinger blowup: The manuscript links DefocusingNLS.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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